UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Wall confinement effects for spheres in the Reynolds number range of 30-2000 Akutsu, Toshinosuke 1977

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1977_A7 A48.pdf [ 11.45MB ]
Metadata
JSON: 831-1.0094042.json
JSON-LD: 831-1.0094042-ld.json
RDF/XML (Pretty): 831-1.0094042-rdf.xml
RDF/JSON: 831-1.0094042-rdf.json
Turtle: 831-1.0094042-turtle.txt
N-Triples: 831-1.0094042-rdf-ntriples.txt
Original Record: 831-1.0094042-source.json
Full Text
831-1.0094042-fulltext.txt
Citation
831-1.0094042.ris

Full Text

WALL CONFINEMENT EFFECTS FOR SPHERES IN THE REYNOLDS NUMBER RANGE OF 30-2000 by TOSHINOSUKE AKUTSU B.A. Sc., Kanto Gakuin U n i v e r s i t y , 1970 M.A. Sc., Kanto Gakuin U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF:GRADUATE STUDIES Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1977 (cP) Toshinosuke Akutsu, 1977 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s under-stood that publication, i n part or in whole, or the copying of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. TOSHINOSUKE AKUTSU Department of Mechanical Engineering The University of B r i t i s h Columbia Vancouver, B r i t i s h Columbia Canada, V6T 1W5 ABSTRACT T h i s t h e s i s s t u d i e s i n d e t a i l : formation, developmen and i n s t a b i l i t y o f the v o r t e x r i n g ; 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 ; and drag f o r a f a m i l y of spheres i n the Reynolds number range of 30-2000 and the blockage r a t i o of 3-30%. In the beginning, 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 experimental program 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 model support system, p r e s s u r e measuring i n s t r u m e n t a t i o n , drag balance and t e s t procedures. An approach to the data r e d u c t i o n , so c r i t i c a l a t 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 of the pressure 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 v olved. F i n a l l y , the t e s t data are analyzed as f u n c t i o n s of the confinement 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 the r a t i o of the model to v e r t i c a l stem support should be a t l e a s t 10 to make stem e f f e c t s n e g l i g i b l e . I n f l u e n c e o f 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 < 1000. However, f o r the model wi t h the • n h i g h e s t blockage r a t i o of 30.6%, the p r e s s u r e continues to show Reynolds number dependency f o r R n as high as 2300 ( l i m i t o f the tunnel c a p a b i l i t y f o r a g l y c e r o l - w a t e r i i i concentration used). In general, the e f f e c t of Reynolds number i s to increase the minimum as well as the wake pressure. On the other hand, the e f f e c t of an increase in the blockage r a t i o i s just the opposite. The wall confine-ment tends to increase the drag c o e f f i c i e n t , however, the c l a s s i c a l dependence of skin f r i c t i o n on the Reynolds 1/2 number, C-, c a R ' , i s maintained. The results v i v i d l y a, r n showed inadequacy of Maskell's correction procedure p a r t i c u l a r l y at higher blockage (S/C > 5%). An extensive flow v i s u a l i z a t i o n study using dye i n j e c t i o n i n conjunction with high speed photography complements the test program. i v TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . 1 1.1 Preliminary Remarks 1 1.2 Survey of Literature 2 1.3 Sphere Flow F i e l d 12 1.4 The Plan of Study 14 2. EXPERIMENTAL APPARATUS AND TEST PROCEDURES 21 2.1 Glycerol Tunnel 22 2.2 Hot Film Anemometry and Velocity P r o f i l e s 27 2.3 Models and Support System 39 2.4 Pressure Measurements 44 2.5 Drag Measurements 59 2.6 Flow V i s u a l i z a t i o n 65 3. RESULTS AND DISCUSSION 70 3.1 Choice of Reference Velocity and Pressure . 71 3.2 E f f e c t of Reynolds Number 89 •3.3 Wall Confinement Ef f e c t s 97 3.4 Drag C o e f f i c i e n t 114 3.5 Blockage Correction Using Maskell's Theory 122 3.6 Flow V i s u a l i z a t i o n and Near-Wake Analysis 126 Chapter Page 3.7 C l o s i n g Comments . . 138 3.7.1 Concluding remarks 13 8 3.7.2 Recommendation f o r f u t u r e work 141 BIBLIOGRAPHY 144 APPENDIX I - A PROCEDURE FOR DRIFT CORRECTION USING DIRECT MEASUREMENT OF THE DIFFERENTIAL PRESSURE 152 APPENDIX I I - CONVENTIONAL PRESSURE COEFFICIENT C IN TERMS OF MEASURED INFORMATION . . 155 P v i LIST OF FIGURES Fi g u r e Page 1-1 A summary of l i t e r a t u r e i n d i c a t i n g the scope of r e c e n t important c o n t r i b u t i o n s i n the f i e l d of flow p a s t a sphere . . . . . . 15 1-2 Starr-Edwards p r o s t h e s i s and i t s exploded view: (a) cage; (b) b a l l or poppet; (c) seat; (d) suture r i n g . . . . . . . . . 16 1-3 A schematic diagram showing the S t a r r -Edwards p r o s t h e s i s occupying a o r t i c p o s i t i o n and p r e s e n t i n g a l a r g e blockage 18 1- 4 The p l a n of study 20 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 25 2-2 C a l i b r a t i o n p l o t f o r the sharp edge o r i f i c e meter 26 2-3 A photograph showing the r o t a t i n g d i s h arrangement used i n c a l i b r a t i o n o f the h o t - f i l m probe: C, c o n s t a n t temperature anemometer; D, d r i v e wheel; M, d r i v e motor; P, probe; R, r o t a t i n g d i s h ; V, d.c. d i g i t a l voltmeter . . . . 30 2-4 C a l i b r a t i o n data f o r h o t - f i l m probe i n the g l y c e r o l - w a t e r s o l u t i o n of C n = 55.7% . 32 2-5 C a l i b r a t i o n p l o t s showing the e f f e c t o f temperature on probe's c o l d r e s i s t a n c e when immersed i n g l y c e r o l - w a t e r s o l u t i o n o f d i f f e r e n t c o n c e n t r a t i o n 33 2-6 Instrumentation l a y o u t used dur i n g v e l o c i t y p r o f i l e measurements . 35 2-7 A s e t of t y p i c a l v e l o c i t y p r o f i l e s a t the s t a t i o n y = 83 cm i n absence o f the s p h e r i c a l ^ m o d e l s 36 v i i Figure Page 2-8 E f f e c t of wall confinement on the upstream v e l o c i t y p r o f i l e : (a) S/C = 7.6%; . . 37 (b) S/C = 30.6% 38 2-9 Variation of the centerline v e l o c i t y as measured by the hot-film probe with the average v e l o c i t y as given by the flowmeter data 40 2-10 A photograph showing spherical models used in the experimental program covering the blockage r a t i o range of 2.7 - 30.6% 42 2-11 E f f e c t of the stem diameter on measured sur-face pressure d i s t r i b u t i o n over a sphere: (a) Sphere diameter = 5.08 cm; • . . 45 (b) Sphere diameter = 6.35 cm 46 2-12 Plots showing stem-effects to be r e l a t i v e l y independent of the Reynolds number in the range 300-1000: (a) D = 5.08 cm, d. = 1.32 mm, d =4. 76 mm; • • 47 (b) D = 5.08 cm, d^ = 4.31mm, d^= 12.70 mm . . 48 2-13 Variation of the minimum and base pressures showing the c r i t i c a l value of d /D 49 3 o 2-14 A schematic diagram of the Barocel pressure transducer 52 2-15 A procedure for compensation of the e l e c t r o n i c d r i f t of the pressure measuring system . . . 55 2-16 A l i n e drawing of the instrumentation set-up used for pressure measurements . . . . . . . . . 58 2-17 A schematic diagram showing the spherical model and i t s support system during pressure measurements 60 2-18 An exploded view of the drag measuring balance: (a) supporting stem; (b) intermediate connection; (c) central suspension block; (d) cantilever with s t r a i n gages; (e) needle bearings support-ing the central block 62 v i i i F i g u r e P a g e 2-19 (a) Drag balance assembly w i t h b r i d g e . . . . 63 a m p l i f i e r meter; (b) Close-up view o f the drag balance . . . . 64 2-20 Dye i n j e c t i n g probe used d u r i n g flow v i s u a l i z a t i o n .• 66 2- 21 A sketch showing the equipment l a y o u t d u r i n g flow v i s u a l i z a t i o n 67 3- 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 75 3-2 T y p i c a l p r e s s u r e p r o f i l e s f o r a sphere u s i n g the c o n v e n t i o n a l d e f i n i t i o n of 2 p r e s s u r e c o e f f i c i e n t , Cp = (Pg - P a ) / ( 1 / 2 ) p u . Note the pressure c o e f f i c i e n t i s zero i n the v i c i n i t y of 6 = 60° 78 3-3 P l o t s showing s e n s i t i v i t y of d i f f e r e n t d e f i n i t i o n s f o r p r e s s u r e c o e f f i c i e n t to changes i n v e l o c i t y p r o f i l e : (a) C p = (P 0 - P 0 0 ) / ( l / 2 ) pU 2; 80 (b) C p = (P 9 - P 6 Q O ) / ( l / 2 ) pU 2; 81 ( c ) c p = ( p e - p 6 0 - , / ( p o - p 6 o « > ) • - ' 8 3 3-4 E f f e c t o f Reynolds number on 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 terms of:„ (a) C p = (P e - P j / ( l / 2 ) pU 2; 84 (b) C p = (P Q - P j / d / 2 ) pU2,; 85 ( C ) % = ( P e - P 6 0 ° ) / ( P 0 - P 6 0 o } 86 3-5 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 showing 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 blockage e f f e c t s : (a) 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 (Cp) based on average v e l o c i t y , C = (P f l -P )/ (1/2) p U ; P .' . . . °°. . 87 (b) suggested 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 a s c p = ^ e - ^ o ^ / ^ o - ^ o ^ • • 8 8 i x F i g u r e Page 3-6 Surface p r e s s u r e d i s t r i b u t i o n on spheres u s i n g P^QO a s r e f e r e n c e . Note the p l o t s show very l i t t l e dependence on (a) v e l o c i t y p r o f i l e ; (b) Reynolds number; (c) blockage . . 3-7 Surface pressure d i s t r i b u t i o n as a f f e c t e d by Reynolds number a t a s m a l l blockage r a t i o o f 4.9%: ( a ) CP = ( p e - p 6 0 ° ) / ( p o - > ° ) ; (b) C p = (P e - P j / ( l / 2 ) plT; . . (c) comparison with r e c e n t data by ot h e r i n v e s t i g a t o r s . Note the r e s u l t s by Maxworthy and Achenbach are near the ^ c r i t i c a l Reynolds number ( R n , c r = 3.7 x10 , Reference 9) 96 3-8 Pres s u r e p l o t s showing t h e i r r e l a t i v e i n s e n -s i t i v i t y to Reynolds number _> 1000 and f o r in t e r m e d i a t e v a l u e s of blockage: (a) C p, S/C = 11.0%; 98 '(b)- C , S/C = 15.0%; • • 99 (c) C , S/C = 19.6%; . . 100 (d) C^ ", S/C = 19.6% 101 3-9 Reynolds number e f f e c t on the pressure d i s -t r i b u t i o n a t higher blockage r a t i o s : (a) Cp, S/C = 24. 6%; 102 (b) C p, S/C = 30. 6% ; • . 103 (c) Cp, S/C = 30.6% . . . . . 104 3-10 Re p r e s e n t a t i v e p l o t s showing n e g l i g i b l e e f f e c t o f w a l l confinement f o r blockage r a t i o s up to 11% . . . . 106 3-11 Pressure p l o t s as a f f e c t e d by higher blockage: (a) R n i 950; . 107 (b) R n = 1250; 108 (c) R n = 1600 109 X F i g u r e Page 3-12 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 , 1000 < R n < 1600. Note both Cp and C p are e s s e n t i a l l y c o n s t a n t up to "b the blockage r a t i o o f around 13% 110 3-13 P l o t s showing dependence of Cp (and hence Cp, - Cp ) on w a l l confinement, meven when S/C i s l e s s " 1 than 131, at R n < 1000 112 3-14 Condensation of the base p r e s s u r e data showing the i n f l u e n c e o f Reynolds number and blockage 113 3-15 V a r i a t i o n o f the measured drag c o e f f i c i e n t w i t h Reynolds number and blockage: (a) p r e s s u r e drag c o e f f i c i e n t ; 116 (b) t o t a l drag c o e f f i c i e n t . The drag c o e f f i c i e n t i s based on average v e l o c i t y i n the t e s t - s e c t i o n 117 3-16 Comparison of the pr e s s u r e and t o t a l drag c o e f f i c i e n t s with the standard drag curve and r e c e n t data r e p o r t e d i n l i t e r a t u r e . Note the r e s u l t s are based on the c e n t e r -l i n e v e l o c i t y : (a) p r e s s u r e drag c o e f f i c i e n t ; 118 (b) t o t a l drag c o e f f i c i e n t 119 3-17 F r i c t i o n f o r c e as a percentage of the t o t a l drag 121 3-18 C o r r e c t e d drag c o e f f i c i e n t s showing inadequacy o f M a s k e l l ' s procedure, p a r t i c u l a r l y a t hig h e r blockage 124 3-19 E m p i r i c a l c o r r e c t i o n formulae f o r sphere drag 125 3-20 A t y p i c a l photograph i l l u s t r a t i n g formation o f a v o r t e x r i n g behind a sphere 127 x i F i gure Page 3-21 A flow v i s u a l i z a t i o n study showing develop-ment and i n s t a b i l i t y of vortex r i n g w i t h Reynolds number: (a) R = 3 0 ; 128 (b) = 65; 128 (c) R =. 115; 128 (d) R n = 165; . , . . 128 (e) R n = 221; 129 (f) R n = 265; 129 (g) R n - 280; 129 (h) R n = 280 . . 129 n 3-22 T y p i c a l c y c l e of i n i t i a t i o n , development and shedding of the r i n g vortex at Reynolds number R = 360 133 n. 3-23 P o s i t i o n of separation as a f f e c t e d by Reynolds number and w a l l confinement . . . 134 3-24 T y p i c a l photographs showing downstream movement of the separation p o s i t i o n due to blockage 136 x i i LIST OF APPENDIX FIGURES Figure Page I I - l A d r i f t c o r r e c t i o n measurement of the procedure using d i r e c t d i f f e r e n t i a l pressure . 156 ACKNOWLEDGEMENT I would l i k e to take t h i s opportunity to express my gratitude and sincere thanks to Professor V.J. Modi for the enthusiastic guidance given throughout the research program and help f u l suggestions during the preparation of the thesis. His help and encouragement have been invaluable. The cheerful assistance of the technical s t a f f i s gr a t e f u l l y acknowledged. Their s k i l l f u l assistance greatly accelerated the research program. The investigation was supported by the National Research Council of Canada, Grant No. A-2181. F i n a l l y , s p e c i a l appreciation i s extended to my parents, Mr. and Mrs. Toshi Akutsu, for t h e i r encourage-ment, love and understanding during d i f f i c u l t times. x i L I S T OF SYMBOLS c r o s s - s e c t i o n a l area of the wake i n Maskell's theory v e l o c i t y of sound tunnel c r o s s - s e c t i o n a l area F t / ( l / 2 ) p U 2 S pressure drag c o e f f i c i e n t based on average v e l o c i t y pressure drag c o e f f i c i e n t based on c e n t e r l i n e v e l o c i t y t o t a l drag c o e f f i c i e n t , + ^ based on average v e l o c i t y »P / t o t a l drag c o e f f i c i e n t , C d + t based on c e n t e r l i n e v e l o c i t y — ' — ' s k i n f r i c t i o n component of t o t a l drag, based on average v e l o c i t y s k i n f r i c t i o n component of t o t a l drag, based on c e n t e r l i n e v e l o c i t y percentage concentration of g l y c e r o l - w a t e r s o l u t i o n by weight ^ e - V ^ o - V (Pg " P j / ( l / 2 ) (P e - P J / ( 1 / 2 ) P U 2 (P e - P r ) / ( l / 2 ) p U 2 base pressure c o e f f i c i e n t , (P^ - P r) / (Pg - P ) ' minimum pressure c o e f f i c i e n t , ( P m - P r ) / ( P Q - P r) 2 base pressure c o e f f i c i e n t , (P^ - P ^ ) / ( 1 / 2 ) p U XV D sphere diameter d. inside diameter of stem 1 d outside diameter of stem o f frequency of shedding vortex sheet F sectional pressure drag F^_ t o t a l sectional drag k size of roughness element M Mach number, U/c n . P s t a t i c pressure Pfl s t a t i c pressure on the surface of the sphere at an angle 6 from the front stagnation point P s t a t i c pressure of undisturbed stream P^ base pressure P s t a t i c pressure at reference tap, i n the present case r = 60° 2 q dynamic pressure, (l/2)pU R ,R cold and operating resistance of the hot f i l m probe, respectively R n Reynolds number, UD/v R c r i t i c a l Reynolds number n, cr J S diametral cross-sectional area S fD/U n T temperature of solution U average v e l o c i t y i n the test-section based on a flow rate as given by the o r i f i c e meter U centerline v e l o c i t y u 1 rms value of v e l o c i t y fluctuations i n the down-stream d i r e c t i o n l o c a l v e l o c i t y as measured by a hot f i l m probe d.c. voltage output of a constant temperature anemometer probe distance from the tunnel i n l e t l o c a tion of model from the tunnel i n l e t v e r t i c a l coordinate with o r i g i n at the bottom of the test section zero d r i f t of the el e c t r o n i c pressure measuring system angular location of a pressure tap with reference to the front stagnation point angular l o c a t i o n of the separating shear layer with respect to the rear stagnation point roughness parameter, k/D dynamic v i s c o s i t y kinematic v i s c o s i t y , u/p density component of turbulent i n t e n s i t y i n the downstream d i r e c t i o n , u 1/U skin f r i c t i o n , shear stress at the wall d i f f e r e n t i a l pressure at o r i f i c e meter 1 1. INTRODUCTION 1.1 P r e l i m i n a r y Remarks G e o m e t r i c a l symmetry has h e l d s p e c i a l f a s c i n a t i o n f o r t h e human mind s i n c e m i l l e n n i a . P y r a m i d s / t h e c o l o s s a l monu-ments o f p h a r a o h s , d e p i c t symmetry a t i t s b e s t t h r o u g h s t r a i g h t l i n e s . However, c u r v i l i n e a r symmetry i n t h e f o r m o f 'sun-d i s c ' has a l w a y s been g i v e n s p e c i a l p o s i t i o n i n t h e a n c i e n t p a n t h e o n . S p h e r e i s m e r e l y a s y m m e t r i c a l e x t e n s i o n o f c i r c l e i n s p a c e . T h e r e a r e numerous s i t u a t i o n s o f p r a c t i c a l i m p o r t a n c e where b o d i e s o f r e v o l u t i o n i n g e n e r a l and s p h e r i c a l o b j e c t s i n p a r t i c u l a r o p e r a t e i n f l u i d f i e l d s w i t h r e l a t i v e l y low R e y n o l d s number. Towed s o n a r s o r s t a t i o n a r y h y d r o p h o n e s u s e d i n s u b m a r i n e d e t e c t i o n s y s t e m s , o c e a n o g r a p h i c p l a t f o r m s e mployed i n h y d r o g r a p h i c s u r v e y s , p r o p o s e d c o n f i g u r a t i o n s o f u n d e r w a t e r h a b i t a t s , o i l - s t o r a g e t a n k s , m e t e o r o l o g i c a l s t u d i e s o f r a i n d r o p s and b a l l o o n s , s p r a y d r y i n g i n t h e c h e m i c a l i n d u s t r y , e t c . b e l o n g t o t h i s c l a s s o f p r o b l e m s . F o r l a b o r a t o r y s i m u l a t i o n , model o f a g i v e n s y s t e m i s u s u a l l y t e s t e d i n w i n d o r w a t e r t u n n e l s where c o n f i n e d c o n d i t i o n i s c r e a t e d e i t h e r u n i n t e n t i o n a l l y o r t h r o u g h c h o i c e f o r g e o m e t r i c s i m i l a r i t y . 2 In the present case, however, attention on the spherical geometry was focussed due to the i n t e r e s t i n under-standing and improving hydrodynamic performance of the prosthetic a o r t i c heart valve. The Starr-Edwards prosthesis, when implanted i n an aorta, has been observed to work under a highly confined condition with the blockage r a t i o i n the range of 15-50% depending on the size of the aorta and model of the prosthesis used. The a o r t i c flow, even i n absence of blockage, i s extremely complicated, being non-Newtonian and p u l s a t i l e with complex swirling component of v e l o c i t y superposed. The a o r t i c wall confinement would further complicate the problem. Unfortunately, surprising as i t may seem, blockage e f f e c t s for even a uniform flow past a sphere at low Reynolds number remains v i r t u a l l y unexplored to date. This thesis represents a modest step towards better understanding of t h i s d i f f i c u l t problem. 1.2 Survey of Literature Interest i n the behaviour of a sphere moving through a f l u i d goes back to the days of Newton who i s credited with the f i r s t recorded measurements on sphere drag. Following th i s but p r i o r to 1930, numerous experiments on the drag of f a l l i n g spheres were conducted and a body of information — 1 6 generated for Reynolds numbers in. the range 10 - 10 . These 3 data, i n general, show a s i g n i f i c a n t degree of scatter and hence are approximated by the f a m i l i a r "standard" drag curve. I t apparently applies only to smooth spheres i n steady motion in a.non-turbulent, isothermal, incompressible continuum f l u i d of e f f e c t i v e l y i n f i n i t e extent. Ever since, 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 in the subject has resulted i n a large volume of l i t e r a t u r e , and the contributions up to 1960 have been c i t e d by Torobin and Gauvin 1 i n t h e i r comprehensive review of the f i e l d . The drag c o e f f i c i e n t appears to depend primarily on the magnitude of the r e l a t i v e turbulence i n t e n s i t y and Reynolds number. Increasing turbulence i n t e n s i t i e s cause a systematic regression of the t r a n s i t i o n region of the drag c o e f f i c i e n t curve towards lower Reynolds number together with a moderate increase of the drag c o e f f i c i e n t i n both s u b c r i t i c a l and s u p e r c r i t i c a l regions. In 1963, Heinrich 2 et a l . c a r r i e d out sphere drag measurements i n a wind tunnel for 2 x l O 3 <R < 2 x 10 4 and 0. 078 <M <0.39. Their n n data, however, are s i g n i f i c a n t l y higher than the standard values. The discrepancy was attributed to the free stream 3 turbulence. S i v i e r has measured the drag of magnetically supported spheres i n a wind tunnel with a free stream turbulence i n t e n s i t y up to 8%, and reported a d e f i n i t e increase i n C_ for R >200, the increase qrowing with i n -D n ' ^ -a creasing R . However, for R <200, he observed l i t t l e or 4 no change i n C D compared to results at lower turbulence l e v e l (- 1%) . His results are also considerably higher than 4 • the standard drag values. Zarin refined the magnetic balance system used by S i v i e r and varied the free stream turbulence i n t e n s i t y l e v e l . Even at a turbulence l e v e l less 3 than 1%, he found, for R ' >10 , drag to be markedly greater 3 than the standard values. However, for Rn <10 , the results were i n good agreement with the standard values. From th i s study Zarin concluded that i n the higher Reynolds number 3 range (R >10 ), a small degree of free stream turbulence results i n higher drag values. 5 Ross and Willmarth conducted drag measurements for spheres moving r e c t i l i n e a r l y through the glycerine-water 5 mixture for 5 <R <10 . Their results agree f a i r l y well n 3 with the standard data for R <2 x10 but are somewhat greater n 3 for the Reynolds number exceeding t h i s value. The study revealed that the drag on a sphere i s not s i g n i f i c a n t l y affected by the vortex shedding (5% v a r i a t i o n ) . On the 6 other hand, Baily and Hiatt carried out sphere drag measure-5 ments i n the b a l l i s t i c range for 0.1 < M < 6 and 20 < R <10 . ^ n n There i s a reasonable agreement between t h e i r low speed data 7 and the c l a s s i c a l standard drag curve. Goin and Lawrence studied subsonic drag on spheres i n the Reynolds number range of 200-10,000 using a test range with controlled 5 environmental c o n d i t i o n . The r e s u l t showed c o m p r e s s i b i l i t y e f f e c t on drag to be evident f o r the Mach number as small as 0.2. Of some i n t e r e s t are the r e s u l t s of V l a j i n a c and 8 4 5 Covert i n the laminar range of 2 x 1 0 - 2 . 6 x 1 0 . They found t h a t the c l a s s i c a l wind tunnel c o r r e c t i o n s as discussed by Pankhurst and Holder do not completely account f o r model s i z e and w a l l i n t e r f e r e n c e . Although the authors do not s p e c i f y a c t u a l t e s t blockage values, t h e i r data show con-s i d e r a b l e v a r i a t i o n from r e s u l t s by other i n v e s t i g a t o r s . In the higher Reynolds number range of 4 6 9 5 x 10 <_ R n _< 6 x 10 , Achenbach's c o n t r i b u t i o n i s s i g n i f i -cant. Based on the measured t o t a l drag, l o c a l s t a t i c pressure and s k i n f r i c t i o n d i s t r i b u t i o n he estimated p o s i t i o n s of boundary l a y e r t r a n s i t i o n and separation. Furthermore, the r e s u l t s s u b s t a n t i a t e d a dependence of the f r i c t i o n force on the Reynolds number. In another study"*"*"*, Achenbach has i n v e s t i g a t e d the e f f e c t of surface roughness and tunnel blockage f o r the flow past spheres i n the above range of Reynolds number. I t was observed that an increase i n roughness parameter leads to a decrease i n the c r i t i c a l Reynolds number, however, the t r a n s c r i t i c a l drag c o e f f i c i e n t showed a d e f i n i t e r i s e . The blockage e f f e c t , i n the range of 25-80%, was to cause an increase i n both the drag c o e f f i c i e n t and the c r i t i c a l 6 Reynolds number. P r e l i m i n a r y experiments showed the turbu-lence to i n i t i a t e a premature t r a n s i t i o n from l a m i n a r ' t o t u r b u l e n t flow. These r e s u l t s , i n g e n e r a l , s u b s t a n t i a t e the c o n c l u s i o n s o f an e a r l i e r i n v e s t i g a t i o n by Maxworthy i n a s l i g h t l y d i f f e r e n t range of the Reynolds number 4 5 ( 6 x 1 0 - 2 x10 ), and the blockage v a r i a t i o n over 5-25%. I t would be o f i n t e r e s t to review here r a t h e r l i m i t e d and c o n f l i c t i n g i n f o r m a t i o n a v a i l a b l e on the frequency 12 o f s e p a r a t i n g shear l a y e r , although M o l l e r i n i t i a t e d such a study as e a r l y as i n 1938. A l i t t l e l a t e r , i n 1957, 13 Commetta extended M o l l e r ' s study oT: the S t r o u h a l number 5 v a r i a t i o n w i t h the Reynolds number to R f l= 5 x 1 0 , but c o u l d d e t e c t p e r i o d i c s e p a r a t i o n o f v o r t i c e s o n l y up to 4 14 R n < 4 x10 . More r e c e n t l y , Majumdar and Douglas as w e l l 15 as C a l v e r t have r e p o r t e d vortex shedding from spheres i n 3 4 the Reynolds number ranges of 5.6x10 <R < 1.16x10 and 4 4 2 x 1 0 <R n <6 x10 , r e s p e c t i v e l y . C a l v e r t ' s r e s u l t s showed the base pr e s s u r e c o e f f i c i e n t to be s u b s t a n t i a l l y dependent on R with the v a r i a t i o n of -0.270 to -0.356 over n 4 4 R =1.5x10 - 6 x 1 0 . The e f f e c t o f t r i p wire was to s h i f t the o r i g i n o f the wake, l e a v i n g the s c a l e unchanged. One must p o i n t out o c c a s i o n a l d i s c r e p a n c i e s i n r e s u l t s as r e p o r t e d by the d i f f e r e n t a uthors. For example, M o l l e r 4 measured, a t R =10 , a S t r o u h a l number of 2 while Majumdar n J and Douglas r e p o r t e d the value an order o f magnitude lower (S = fD/U=0.2), which i s the value t y p i c a l of c i r c u l a r c y l i n d e r s i n c r o s s - f l o w . I t was a l s o suggested t h a t i n a t u r b u l e n t flow there i s no r e g u l a r v o r t e x shedding. On the other hand, Commetta d e t e c t e d c o e x i s t e n c e o f v o r t e x shedding i n two modes: the lower mode at S=0.2 and the h i g h e r mode, a s s o c i a t e d w i t h t r a n s i t i o n of the v o r t e x sheet from laminar to t u r b u l e n t , a t S = 0.8 -1.4. Recent s t u d i e s X 6 17 by Achenbach ' , i n the Reynolds number range of 4 0 0 - 5 x 1 0 ^ confirms M o l l e r ' s r e s u l t s a t lower Reynolds number, however, the lower c r i t i c a l Reynolds number was found to be 5 x l O 3 . In the range 6 x 1 0 3 < R < 3 x 1 0 5 ^ n s t r o n g p e r i o d i c f l u c t u a t i o n s i n the wake flow were observed Beyond the upper c r i t i c a l Reynolds number of 3.7 x10^ no p e r i o d i c v o r t e x shedding was d e t e c t e d . Experimental i n v e s t i g a t i o n i n v o l v i n g flow v i s u a l i z a t i o n and photographing of the wake behind a sphere i n the low Reynolds number range of 5 < R n - 300 was c a r r i e d out by 18 Taneda u s i n g a water tank. The r e s u l t s showed t h a t the c r i t i c a l R n a t which the permanent " v o r t e x - r i n g " begins to form i n the r e a r o f a sphere i s about 24, s i z e of the r i n g i s n e a r l y p r o p o r t i o n a l to the l o g a r i t h m of the R , and the wake behind the r i n g begins to o s c i l l a t e f o r R n ~ 1 3 0 . 19 Magarvey and Bishop s t u d i e d the t r a n s i t i o n ranges f o r three dimensional wakes produced by the motion of a drop of an immiscible l i q u i d i n the Reynolds number range 0 < R n < 2500 They d i s t i n g u i s h e d the observed wakes as steady o r p e r i o d i c w i t h s e v e r a l s u b c l a s s i f i c a t i o n s i n each o f the c a t e g o r i e s , an concluded (as can be a n t i c i p a t e d ) t h a t the wake p a t t e r n depends e n t i r e l y on the Reynolds number r e g a r d l e s s of the l i q u i d - l i q u i d system employed. Furthermore, i t was observed t h a t the g e n e r a l values o f the t r a n s i t i o n Reynolds numbers cannot be o b t a i n e d as they depend on the drop deformation. However, f o r a l l the cases c o n s i d e r e d the t r a n s i t i o n i n the wake p a t t e r n s were l i m i t e d to Reynolds number spread of l e s s than 20. A q u a l i t a t i v e i n t e r p r e t a t i o n o f heat and mass t r a n s f e r mechanisms i n the wake r e g i o n o f a sphere i n low 20 speed flows (R < 410) i s presented by Lee and Barrow who employ measurements of the v e l o c i t y f i e l d i n the wake through flow v i s u a l i z a t i o n by dye i n j e c t i o n . The observed flow p a t t e r n s g e n e r a l l y confirmed Taneda's r e s u l t s . An important c h a r a c t e r i s t i c o f the near-wake i s the r e v e r s e d flow, a t a v e l o c i t y much s m a l l e r than the f r e e stream v e l -o c i t y , along the a x i s o f the sphere towards the r e a r s t a g n a t i o n p o i n t . The wake t r a n s i t i o n and S t r o u h a l number f o r the i n c o m p r e s s i b l e wake of v a r i o u s bodies was s t u d i e d 21 by Goldburg and F l o r s h e i m . Based on the experimental r e s u l t s , i t was suggested t h a t the t r a n s i t i o n c o u l d be approximately c o r r e l a t e d f o r a range of spheres and cones 9 by the Reynolds number based on t o t a l wake momentum t h i c k -ness. Furthermore, i t was found t h a t f o r r e g u l a r vortex shedding the data f o r spheres and cones could be c o r r e l a t e d w i t h Rayleigh-Strouhal formula based on the same c r i t e r i o n . Before moving to the review of a n a l y t i c a l approaches, i t would be appropriate to mention here a recent and r a t h e r 22 s i g n i f i c a n t c o n t r i b u t i o n by Modi and Ammzadeh . I t i s p a r t i c u l a r l y r e l e v a n t as the present p r o j e c t represents an extension of t h e i r i n v e s t i g a t i o n . Using a g l y c e r o l - w a t e r s o l u t i o n tunnel and by a p p r o p r i a t e l y c o n t r o l l i n g concentra-t i o n of the working f l u i d , the authors were able to c o r r e l a t e the progress of formation, e l o n g a t i o n , asymmetry and i n s t a b i l i t y of the vortex r i n g w i t h the surface pressure d i s t r i b u t i o n i n the Reynolds number range of 70-6000. Of p a r t i c u l a r i n t e r e s t i s a s p e c t a c u l a r r i s e i n the minimum pressure i n the range R n = 240 - 275, which was found to be a s s o c i a t e d w i t h the onset of i n s t a b i l i t y of the r i n g vortex l e a d i n g to i t s p e r i o d i c shedding. In general, Reynolds number e f f e c t s were confined to the region near and down-stream of the minimum pressure p o i n t . An extensive flow v i s u a l i z a t i o n program complemented t h e i r t e s t data. T h e o r e t i c a l i n v e s t i g a t i o n of even a steady viscous incompressible flow past a sphere i s very complex. I t was 2 3 f i r s t considered by Stokes (1851) , and has been discussed by many authors s i n c e then. A l a r g e p o r t i o n of these s t u d i e s 10 have been concerned with the solutions for vanishingly small R . Stokes solved the problem by neglecting the i n e r t i a 24 of the f l u i d . Later, Whitehead t r i e d to improve upon thi s solution by introducing higher approximations to the flow when the Reynolds number i s not n e g l i g i b l e . But as i s now well known, his solution i s not v a l i d i n problems of 25 2 6 uniform streaming . Oseen solved Whitehead's paradox by assuming that the sphere caused a small perturbation i n the uniform p a r a l l e l flow and neglected second order perturbation v e l o c i t i e s , thus taking the i n e r t i a terms into account to a l i m i t e d extent. Oseen's solution for l i n e a r i z e d 2 7 28 equationshas been improved by Goldstein , Tomotika et a l . , 29 and Pearcey et a l . However, as can be anticipated, l i n e a r i z a t i o n renders these analyses inadequate for R n >2. Of considerable i n t e r e s t are two independent 30 solutions: one by Kawaguti who s a t i s f i e d an integrated form of the Navier-Stokes equation for f i r s t and second-order terms when expanded by Legendre Polynomials and the other by 25 Proudman and Pearson who l i n e a r i z e d the Navier-Stokes equation by two approximations, one v a l i d at a distance from the sphere, and the other v a l i d near the surface of the sphere. 31 Kawaguti has also developed an alternative procedure to solve the Navier-Stokes equation using the f i n i t e difference method. Unfortunately, the technique, v a l i d for flow around 11 spheres up to R n = 20, proves to be extremely l a b o u r i o u s . 32 33 34 Fox e t a l . ' and A l l e n e t a l . have p a r t i a l l y a l l e v i a t e d t h i s d i f f i c u l t y by t r a n s f e r r i n g the technique i n t o a r e l a x a -35 t i o n procedure. On the other hand, Jenson a p p l i e d the r e l a x a t i o n method d i r e c t l y to the governing equations f o r v o r t i c i t y and stream f u n c t i o n i n m o d i f i e d s p h e r i c a l c o o r d i n -ates to o b t a i n s o l u t i o n s f o r flow around spheres a t R n = 5, 36 — 38 10> 20, 40. Hamielec e t a l . have a l s o used a s i m i l a r method, but w i t h f i n e r g r i d s i z e to o b t a i n numerical s o l u t i o n s o f the Navier-Stokes equations f o r slow v i s c o u s flow around spheres. F o u r i e r expansions f o r the flow v a r i a b l e s were used to s o l v e the problem over a wide range o f the Reynolds 39 40 number by Dennis and Walker . Rimon and Cheng d e r i v e d steady s t a t e s o l u t i o n s f o r 1 < R n < 1000 by i m p u l s i v e l y s t a r t -i n g a sphere from r e s t with uniform v e l o c i t y and used a time dependent i n t e g r a t i o n to c a r r y the s o l u t i o n to the steady 41 s t a t e . More r e c e n t l y , Dennis and Walker have presented • 42 a s e r i e s t r u n c a t i o n method, f i r s t proposed by Van Dyke , employing a f a m i l y of Legendre f u n c t i o n s to s o l v e the N a v i e r -Stokes equations f o r flow around spheres i n the Reynolds number range of 1-40. 12 1.3 Sphere Flow F i e l d From the b r i e f l i t e r a t u r e review given above, impor-t a n t c h a r a c t e r i s t i c s of the flow f i e l d a s s o c i a t e d with a sphere become c l e a r . At a very low Reynolds number, R n < 0.1, the flow near the sphere i s dominated by v i s c o u s f o r c e s r e s u l t i n g i n the f o r e - a f t symmetry. In the range 0.1<R njc24, i n e r t i a l e f f e c t s i n c r e a s e w i t h i n c r e a s i n g Reynolds number and the s t r e a m l i n e p a t t e r n no longer conforms to the above mentioned symmetry. The f i r s t evidence of flow s e p a r a t i o n near the r e a r s t a g n a t i o n p o i n t appears a t a Reynolds number s l i g h t l y g r e a t e r than 20(R n - 24), although there i s some disagreement as to the p r e c i s e v alue of R n corresponding to i t s onset. The s e p a r a t i o n r e g i o n grows wi t h Reynolds number, as suggested by the growth o f the vortex r i n g , accompanied by a r e d u c t i o n i n i t s s t a b i l i t y . In the range R n = 130-210, o s c i l l a t i o n o f the bubble ensues and becomes g r a d u a l l y s t r o n g e r . For 210 <R n <270, an asymmetrical s e p a r a t i o n bubble i s observed, f o l l o w e d by d i s c r e t e v o r t e x loops shedding p e r i o d i c a l l y from o p p o s i t e s i d e of the s e p a r a t i o n bubble i n the range extending up to 700. The value of R n a t which the vo r t e x shedding begins, o f t e n r e f e r r e d to as lower c r i t i c a l Reynolds number, s i g n i f i e s the appearance of a wake i n which the flow around the sphere i s no longer c l o s e d . The z i g -zag or h e l i c a l path of the f r e e f a l l i n g spheres a t R^ > 210 13 i s d i r e c t l y a t t r i b u t e d to t h i s c h a r a c t e r of the wake. I t i s of i n t e r e s t to note here t h a t the angle between the r e a r s t a g n a t i o n p o i n t and the s e p a r a t i o n c i r c l e s t e a d i l y i n c r e a s e s from a value of zero at R =24 to 72° a t R = 450. The S t r o u h a l n n number f o r sphere has been found to be around 0.2 f o r 5.6 x 1 0 3 < R <11.6 x 1 0 3 . n Beyond the value of the lower c r i t i c a l Reynolds number, nature o f the flow remains e s s e n t i a l l y the same u n t i l the Reynolds number (often r e f e r r e d to as the upper c r i t i c a l Reynolds number) o f 3.7 x l O 5 i s reached. Now the boundary l a y e r upstream o f the s e p a r a t i o n p o i n t changes from laminar to t u r b u l e n t . The r e s u l t i s a rearward s h i f t of the s e p a r a t i o n p o i n t , c a u s i n g a decrease i n the s i z e of the separated r e g i o n and c h a r a c t e r i s t i c sharp drop i n the drag. No w e l l d e f i n e d v o r t e x shedding frequency has been recorded beyond the upper c r i t i c a l Reynolds number. The flow d e s c r i p t i o n so f a r r e l a t e s to the "standard c o n d i t i o n " devoid of t u r b u l e n c e , s u r f a c e roughness, compres-s i b i l i t y , r a r e f a c t i o n , and heat t r a n s f e r e f f e c t s . The l a s t t h ree parameters are not s i g n i f i c a n t i n the p r e s e n t study. In g e n e r a l , i n c r e a s e i n t u r b u l e n c e i n t e n s i t y r e s u l t s i n a s y s t e m a t i c r e g r e s s i o n of the t r a n s i t i o n r e g i o n towards lower Reynolds numbers, together with a moderate i n c r e a s e o f the drag c o e f f i c i e n t f o r both the s u b c r i t i c a l and s u p e r c r i t i c a l Reynolds numbers. At h i g h e r Reynolds numbers, s u r f a c e 14 roughness a f f e c t s the flow i n the s i m i l a r manner: i t causes e a r l y t r a n s i t i o n to a t u r b u l e n t boundary l a y e r r e s u l t i n g i n a range of Reynolds number over which the drag c o e f f i c i e n t d i m i n i s h e s (compared to the standard drag curve v a l u e s ) . However, i t may i n c r e a s e the drag a t low Reynolds numbers. F i g u r e 1-1 b r i e f l y summarizes the scope of important c o n t r i -b u t i o n s s i n c e a comprehensive review of the f i e l d by Torobin and Gauvin"^. 1.4 The P l a n o f Study As p o i n t e d out b e f o r e , an i n v e s t i g a t i o n aimed at s t u d y i n g the f l u i d mechanics of p r o s t h e t i c h e a r t v a l v e s has been i n progress i n t h i s department s i n c e 1969. P r i m a r i l y , the a t t e n t i o n i s focussed on the Starr-Edwards c o n f i g u r a t i o n , which e s s e n t i a l l y c o n s i s t s o f : (i) a metal cage of h i g h l y p o l i s h e d uncoated c a s t i n g o f S t e l l i t e 21, a c o b a l t a l l o y noted f o r h i g h s t r e n g t h and c o r r o s i o n r e s i s t a n c e ; ( i i ) a s p h e r i c a l b a l l of s i l i c o n e rubber w i t h diameter ranging from 1.2 - 2.2 cm; ( i i i ) a metal seat normally c a l l e d o r i f i c e ; (iv) a sewing margin of k n i t t e d T e f l o n c l o t h . The main o b j e c t i v e has been to i d e n t i f y f a c t o r s c a u s i n g : 15 10 10' 1 0 ' Rn 10' 10 v 10' 3x10 Wake Geometry 10. 2.5x10 C d ,0.2<Mn<3.17 4.1 xio Wake" Observat ion 25. 4x10 a <8%, M n 2x10' C d ,0.2 <Mn< 0.98 o 6x10 2x10 4x10 5x10 Cp,S/C C d , 0.1 <Mn<0.57 , a<13% S n 5 J x 10 10: a 5x10 6x10 C d , C p , T.v , a<0.45°/o 2xl03 2.6x10 20 105 C d , 0.02<Mn<0.31 cd 0.1 <Mn<6 1 _ 7 0 4 6xlOa Cpb . S n 4x10 'n 3 xlO4 6x10 C d ,S/C Fiaure 1-1 A summary of l i t e r a t u r e indicating the scope of recent important contributions i n the f i e l d of flow past a sphere. 18 Taneda [1956] 2 Heinrich et al. [1965] 20 Lee & Barrow [1965] 3 Sivier [19 6 7] 7 Goin & Lawrence [1968] M a x w o r t h y 1 1 [ l 9 6 9 ] Zar in [1969] 14 Mujumdar & Douglas [1 970] 5 Roos & Willmarth [1971] 9 Achenbach [1972] 8 Vlajinac & Covert [1972] Bailey & Hiatt [1972] 15 Calver t [1972] A c h e n b a c h 1 7 [ 1 9 7 4 ] A c h e n b a c h 1 0 [1974] Figure 1-2 Starr-Edwards prosthesis and i t s exploded view: (a) cage; (b) b a l l or poppet; (c) seat; (d) suture r i n g 17 (a) d e s t r u c t i o n o f red blood c e l l s ; (b) d i s s o c i a t i o n of the b l o o d c o n s t i t u e n t s and t h e i r d e p o s i t i o n on the cage r e s u l t i n g i n the v a l v e f a i l u r e . S u i t a b l e m o d i f i c a t i o n s i n the v a l v e geometry t h a t would a l l e v i a t e or a t l e a s t minimize these problems and l e a d to an improvement i n the v a l v e performance were a l s o o f i n t e r e s t . The f l u i d dynamics o f b l u f f bodies i n p u l s a t i l e flows a t low Reynolds number r e p r e s e n t s a c h a l l e n g i n g task. Hence one i s f o r c e d to approach the problem i n stages of i n c r e a s i n g d i f f i c u l t y . Although s t u d i e s by Aminzadeh and •22 43-47 Modi ' have p r o v i d e d c o n s i d e r a b l e u s e f u l i n f o r m a t i o n , there are a number o f aspects to the problem which remain unexplored. One of them p e r t a i n s to the e f f e c t of blockage as imposed by the poppet of the Starr-Edwards v a l v e occupying a o r t i c p o s i t i o n (Figure. 1 _3) . Depending upon the s i z e of the a o r t a and p r o s t h e s i s , 47 the blockage o f f e r e d by the poppet may be s u b s t a n t i a l l e a d i n g to a l a r g e change i n the flow c h a r a c t e r . With t h i s as background, i t was decided to e x p l o r e w a l l e f f e c t s on such a h e a r t v a l v e p r o s t h e s i s i n the p u l s a t i l e flow r e p r e -s e n t i n g a c a r d i a c c y c l e . However, a d e t a i l e d l i t e r a t u r e review r e v e a l e d t h a t the c o r r e s p o n d i n g i n f o r m a t i o n f o r a sphere by i t s e l f even i n a uniform flow i n the Reynolds number range of i n t e r e s t remains unrecorded. 18 F i g u r e 1-3 A schematic diagram showing the Starr-Edwards p r o s t h e s i s occupying a o r t i c p o s i t i o n and p r e s e n t i n g a l a r g e blockage 19 This t h e s i s , t h e r e f o r e , s t u d i e s : (i) formation, development and i n s t a b i l i t y of vortex r i n g ; ( i i ) a s s o c i a t e d pressure d i s t r i b u t i o n ; ( i i i ) drag; and (iv) near wake geometry f o r a f a m i l y of spheres re p r e s e n t i n g blockage r a t i o of 3 - 30% i n the Reynolds number range of 30 - 2000. In the beginning, the e f f e c t of stem used i n support-i n g the spheres i s s y s t e m a t i c a l l y i n v e s t i g a t e d , which provides a c r i t e r i o n f o r t h e i r s e l e c t i o n . This i s followed by a d e t a i l e d study of the surface pressure d i s t r i b u t i o n and drag. An approach to 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 discussed and i t s merit assessed compared, to the conventional procedure. F i n a l l y , the t e s t r e s u l t s are analyzed as f u n c t i o n s of the confinement c o n d i t i o n and Reynolds number. An extensive flow v i s u a l i z a t i o n study i n conjunction w i t h s t i l l and high speed movie photography complements the t e s t program. Figure 1-4 summarizes the plan of study. 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 S P H E R E S IN T H E R E Y N O L D S N U M B E R R A N G E O F 3 0 — 2 0 0 0 S t e m E f f e c t M e a n P r e s s u r e D i s t r i b u t i o n N e a r W a k e C o n f i g u r a t i o n F l o w V i s u a l i z a t i o n Figure 1-4 The plan of study o 21 2. EXPERIMENTAL APPARATUS AND TEST PROCEDURES This chapter introduces the test f a c i l i t i e s used in the experimental program. Some of the instrumentation em-ployed constitutes the standard equipment i n any well equipped f l u i d mechanics laboratory and hence needs no elaboration. On the other hand, design and constructional d e t a i l s involved i n the development of s p e c i f i c equipments are often numerous and hence, though important and relevant, cannot be covered in t h e i r e n t i r e t y . The attention i s , therefore, focussed on more s a l i e n t features. The t e s t procedures employed are conceptually well known but t h e i r implementation often a t t a i n complexity of a higher order, mainly because of the character of the working f l u i d (glycerol-water solution). Often p e c u l i a r i t i e s of s p e c i f i c experiments make certa i n measurements quite d i f f i -c u l t . Throughout, the emphasis i s on p r a c t i c a l considerations involved i n executing the experimental programme. At times the factors involved are,seemingly, so t r i v i a l that one would seldom give them a second look. However, a common experience of most experimenters i s that resolution of apparently simple problems occasionally takes days, i f not weeks or months. This i s p a r t i c u l a r l y true i n the case where l i q u i d i s the working f l u i d . The glycerol-water solution tunnel representing a fundamental f a c i l i t y for the entire test program and i t s 22 c a l i b r a t i o n using the hot-film anemometry are described f i r s t . This i s followed by an introduction of the models and t h e i r support system. Next, the highly sensitive pressure transducing system capable of determining surface pressure d i s t r i b u t i o n i s discussed leading to the arrange-ment used i n drag measurements. F i n a l l y , d e t a i l s of the flow v i s u a l i z a t i o n procedure, which proved extremely useful in obtaining physical appreciation as to the character of the flow, are presented. Wherever appropriate, c a l i b r a t i o n procedures employed are explained and corresponding charts included. 2.1 Glycerol Tunnel The tests were conducted i n a glycerol-water solution tunnel designed to produce Reynolds number in the range 30 - 6000 (based on sphere diameter and representative average v e l o c i t y i n the t e s t - s e c t i o n ) . The choice of concentration of the working f l u i d provided a degree of f l e x i b i l i t y , but only to a c e r t a i n extent, as governed by the c h a r a c t e r i s t i c s of the power unit. Primarily the tunnel consists of three subassemblies: the test section; the f l u i d return system; and the power unit consisting of a pump and a drive motor. The test-section i s b u i l t of four plexiglas walls 2.44 m (8 ft) long, 1.9 cm (0.75 in) thick and wide enough to produce an inside cross-section of 20 . 32 cm x 20. 32 cm ( 8 i n x 8 in) . 23 D e f l e c t i o n annular vanes together w i t h s e v e r a l s e c t i o n s of honeycombs, brass screens and nylon wool gave e x c e p t i o n a l l y f l a t v e l o c i t y p r o f i l e s . There are three accesses to the i n s i d e of the t e s t - s e c t i o n , through each end and v i a a p o r t - h o l e a t the top. The p o r t h o l e , 12.7 cm (5 in) i n diam-e t e r , i s j u d i c i o u s l y l o c a t e d 0.84m (33 in) from the entrance to admit arm to reach, p o s i t i o n and a d j u s t models. 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 1.6cm (5/8 i n N-C) plugs were d r i l l e d and tapped i n the top w a l l of the t e s t - s e c t i o n . These openings were used to mount models, take out pressure conducting Tines and to support a h o t - f i l m probe i n the t e s t - s e c t i o n . Two g l a s s p l a t e s , 63.5 x14 x1.27 cm ( 2 5 x 5 1/2x1/2 i n ) , recess-mounted i n the s i d e s of the t e s t - s e c t i o n p r o v i d e d o p t i c a l l y f l a t , homogen-eous and t h e r m a l l y s t a b l e w a l l s f o r i n s p e c t i o n and photography. Located between the end of the p l e x i g l a s t e s t -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 comprising of heat exchanger, P o l y v i n y l c l o r i d e (PVC) p i p e s and elbows wi t h c o n n e c t i n g f l a n g e s and r a d i a t o r hose. A copper p i p e , 3 mx 7.62 cm (10 f t x 3 i n ) , i n c o n j u n c t i o n w i t h a 2.4 mx 15.24 cm (8 f t x 6 in) PVC p l a s t i c pipe formed an annular 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, i t was p o s s i b l e to maintain temperature of the working f l u i d w i t h i n ± 0.2°C. PVC elbows and s e c t i o n s of 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, 24 a n t i - c o r r o s i o n and v i b r a t i o n free connection between the t e s t - s e c t i o n and heat exchanger. 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: Aurora type GAPB, 12.7 £/s (200 gal/min), 7.6 mhead, 1750 rpm. I t i s d r i v e n by a three horsepower v a r i a b l e speed d.c. motor. The pump i m p e l l e r and housing are of c a s t brass to guard agai n s t p o s s i b l e c o r r o s i o n . The motor i s energized by a three phase g r i d , the voltage being adjusted through an autotrans-former and r e c t i f i e d by selenium diodes. No f u r t h e r smoothing of the d.c. output was re q u i r e d . I t was important to minimize d i r t contamination of the tunnel f l u i d . This was achieved by i n c o r p o r a t i n g a 10 u f i l t e r i n a bypass c i r c u i t across 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 twenty-four hours of o p e r a t i o n . The tunnel i s s c h e m a t i c a l l y shown i n Figure 2-1. Flow r a t e i n the tunnel was monitored using a sharp edge o r i f i c e p l a t e mounted 0.61m (2 f t ) upstream of the pump i n l e t . The p l a t e l o c a t i o n was so s e l e c t e d as to make i t s reading r e l a t i v e l y independent of the upstream and downstream disturbances i n the form of elbows, 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 f i n a l assembly the o r i f i c e p l a t e and as s o c i a t e d plumbing were c a l i b r a t e d , Under simulated t e s t c o n d i t i o n s , by pumping water from a larg e sump i n t o a weighing tank. The c a l i b r a t i o n p l o t thus obtained i s presented i n Figure 2-2. radiator hose honeycombs portholes test section brass s c r e e n s heat e x c h a n g e r fice plate c o o l i n g water in vent pipe F i g u r e 2-1 A schematic diagram showing the g l y c e r o l - w a t e r s o l u t i o n tunnel 26 Figure 2-2 Cal i b r a t i o n p l o t for the sharp edge o r i f i c e meter 27 2.2 Hot Film Anemometry and Velocity P r o f i l e s As mentioned before, an average v e l o c i t y i n the t e s t -section was determined from the volume flow rate given by the o r i f i c e meter. However, during the course of the experi-mental work i t was also required to measure v e l o c i t y pro-f i l e s arid, i n p a r t i c u l a r , the centerline v e l o c i t i e s in the range of 2.5-15 cm/s. Measurements of f l u i d v e l o c i t i e s at low values of Reynolds number has long been known to be exceptionally d i f f i c u l t . Apart from laser-doppler anemometer, which was s t i l l i n the early stage of acceptance when c a l i b r a t i o n of the tunnel was undertaken (197 3), a hot f i l m probe appeared to meet the requirements of high resolution in time and space of flow v e l o c i t i e s quite adequately. Hence a quartz coated wedge shaped platinum f i l m probe (Thermo-Systerns Inc., model 1239W) was used i n conjunction with the standard constant temperature anemometric equipment (DISA model 55A01). Despite the existence of a comprehensive l i t e r a t u r e on measurements i n gases, r e l a t i v e l y few papers deal with the use of hot f i l m anemometry for investigation of slow l i q u i d flow. I t i s mainly because of several d i f f i c u l t i e s involved i n adapting the anemometer to use i n water or other l i q u i d s : (i) E l e c t r o l y s i s i s by far the worst source of trouble causing corrosion of the probe, generation of gases and i n s t a b i l i t y i n the 28 el e c t r o n i c control c i r c u i t r y . This p a r t i c u l a r problem does not arise i n non-conducting l i q u i d s , such as d i s t i l l e d water or kerosene. Another way of avoiding serious corrosion could be the use of high frequency alternating current to heat the probe, and/or coating the probe to provide e l e c t r i c a l i n s u l a t i o n from the l i q u i d . ( i i ) Often the formation of bubbles on the sensor causes incorrect and unstable operation of the 48 probe . Bubble formation can be reduced by cleaning the probe i n a solvent, e.g., methyl alcohol, with the anemometer in "stand-by" condition, and/or by adding some surface reactants to reduce f l u i d ' s surface tension, thus prevent-ing the formation of bubbles and th e i r attachment to the sensor. In the case of water a "wetting agent" (Kodak Photo-Flo 200) can be used. ( i i i ) Contamination of the probe by dust p a r t i c l e s or . 4 9 other deposits reduces and modifies i t s s e n s i t i v i t y To eliminate d i r t contamination the surface of the l i q u i d should be shielded. Continuous f i l t r a t i o n of a part of the c i r c u l a t i n g f l u i d should also help in minimizing the problem. Both these methods and frequent cleaning of the probe were found necessary i n the present set of experiments. 29 There are many ways to ca l i b r a t e hot-film probes ' The choice of method depends on the a v a i l a b i l i t y of a suitable standard of comparison, the ease of measurement, and the desired degree of accuracy. In most cases, the v e l o c i t y measured by mechanical means at a s p e c i f i c point i n the fl u i d ' f i e l d i s compared with the e l e c t r i c a l signal of the anemometer. The degree of accuracy then depends mainly on the accuracy with which the reference v e l o c i t y i s known. In the present case the probe was held stationary in a rotating dish, 30.5 cm diameter and 25.4 cm high, mounted hor i z o n t a l l y on a turntable with i n f i - n i t e l y variable speed drive (Figure 2-3). This arrangement was found s a t i s f a c t o r y over the v e l o c i t y range of i n t e r e s t . S u f f i c i e n t time had to 52 be allowed for a quasi-steady state of motion to be set-up The motion obtained was very cl o s e l y s o l i d body rotation when the probe was not too far from the c y l i n d r i c a l or bottom walls of the dish. The c i r c u l a r dish had to be s u f f i c i e n t l y large to allow for the d i s s i p a t i o n of v o r t i c i t y generated by the probe between successive passes; obviously the time 2 constant of t h i s e f f e c t i s of the order v/r (r = distance 5 3 from the probe to the axis of rotation) . Absolute cleanliness was found to be es s e n t i a l in these tests. The complete r i g was kept i n a glass enclosure which greatly reduced the frequency of probe cleaning required to produce repeatable r e s u l t s . Figure 2-3 A photograph showing the rotating dish arrangement used i n c a l i b r a t i o n of the hot-film probe: C, constant temperature anemometer; D, drive wheel; M, drive motor, P, probe; R, rotating dish, V, d.c. d i g i t a l voltmeter co o 31 C a l i b r a t i o n t e s t s were c a r r i e d out i n the g l y c e r o l -water s o l u t i o n o f c o n c e n t r a t i o n 55.7% by weight (Figure 2-4). As a n t i c i p a t e d , the experimental p o i n t s c l u s t e r e d around a s t r a i g h t l i n e down to q u i t e low v a l u e s o f the t r u e v e l o c i t y . The i n d i c a t e d s t r a i g h t l i n e i s a l e a s t mean square f i t through the measured data. The maximum d e v i a t i o n from the f i t i s 2.3% f o r the overheat r a t i o of 0.09 7. The s c a t t e r i n the experimental r e s u l t s i s of the order t h a t can be expected from the a n c i l l a r y equipment alone. Since the h o t - f i l m temperature T i s kept constant *• m by v i r t u e o f the overheat r a t i o , a change of R c (probe's c o l d r e s i s t a n c e ) d u r i n g the measurements would imply a change of .T . I t i s , t h e r e f o r e , u s e f u l to i n v e s t i g a t e d r i f t i n the overheat r a t i o induced by v a r i a t i o n s i n R . c T h i s would g i v e some a p p r e c i a t i o n as to the changes i n the ambient temperature t h a t can be t o l e r a t e d d u r i n g a given t e s t . To t h i s end, dependence of probe c o l d r e s i s t a n c e on f l u i d temperature was measured u s i n g a constant temperature bath. F i g u r e 2-5 shows these r e s u l t s f o r v a r i o u s concen-t r a t i o n s o f g l y c e r o l s o l u t i o n . A l l the curves have almost the same slope s u g g e s t i n g the constant c o e f f i c i e n t o f r e s i s t i v i t y . In the worst case, the maximum d e v i a t i o n was observed to be about 1.2%. The f i r s t step i n the t e s t programme was to c a l i b r a t e the t u n n e l , i . e . , to o b t a i n i n f o r m a t i o n about the 2 5 0 2 0 0 (Volt)' 1 5 0 1 0 0 0 1 7 , , 0 . 5 - .0 .5 U ; ( c m / s e c ) Figure 2-4 Calibr a t i o n data for hot-film probe i n the glycerol-water solution of C. =55.7% n Figure 2-5 Cal i b r a t i o n plots showing the e f f e c t of temperature on probe's cold resistance when immersed i n g l y c e r o l -water solution of d i f f e r e n t concentration 34 boundary layer growth as r e f l e c t e d i n the v e l o c i t y pro-f i l e s along the t e s t section. To t h i s end, the tunnel was f i l l e d with the working l i q u i d of a fixed concentration. A l l a i r pockets and bubbles were removed from the tunnel by c i r c u l a t i n g the test f l u i d , with the wetting agent, for at least eight hours at around 30°C. Velocity p r o f i l e at a given st a t i o n was then obtained using the calib r a t e d hot f i l m probe i n conjunction with a traversing gear, which can p o s i t i o n the probe with an accuracy of around ±0.25 mm. I t should be pointed out that the probe movement i s confined to the v e r t i c a l d i r e c t i o n i n the central plane of the tunnel. Step size for the probe movement was regulated according to the v e l o c i t y gradient so as to provide an accurate p r o f i l e near the wall. Figure 2-6 shows instrumen-tati o n layout used during the v e l o c i t y p r o f i l e measurements. Velocity p r o f i l e s were measured at a station 8 3 cm down stream of the entrance to the test section (y = 83 cm) i n the Reynolds number range of 960 - 3900 based on the hydraulic diameter of the test section and the average v e l o c i t y as deduced from the flowmeter data. Two d i s t i n c t cases were considered: tunnel without a model and with models of d i f f e r e n t blockage r a t i o s located at y =100 cm. ^ 1 m Typical plots are presented in Figures 2-7 and 2-8. I t i s apparent that 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 f l a t at l e a s t over the central 15 cm of the tunnel height. The constant temperature annemometer d.c. digi t a l v o l t m e t e r . o s c i l l o s c o p e h o n e y c o m b s •TS h o t . f i l m probe , 1 23 9 w , s u p p o r t e d by a tr a v e r s i n g t e s t s e c t i o n b r a s s s c r e e n s F i g u r e 2-6 Instrumentation l a y o u t used d u r i n g v e l o c i t y p r o f i l e measurements CO ui 36 2 0 . 0 1 7 . 5 1 5 . 0 o o o o o o o o A A A A A A A A 1 2 . 5 1 0 . 0 Z , c m 7 . 5 5 . 0 2 5 Y p = 8 3 c m U , c m / s e c • 3 - 3 3 o 6 . 6 0 A 1 0 . 0 2 o o o o o o A A A A A A A O A 0 . 0 0 5 1 0 U z , c m / s e c F i g u r e 2-7 A s e t o f t y p i c a l v e l o c i t y p r o f i l e s a t the s t a t i o n y p = 8 3 cm i n absence of the s p h e r i c a l models 20-0 1 7.5 15.0 12.5 10.0 Z,cm 7.5 5.0 2.5 < o • 37 < O • U .cm / sec • 12.70 o 11.30 < 9 .80 v p = 100 cm o • < o a < o n < o a < o n i o • o . • o • o • o • o a o • o • o o • o • o o . • o a o a o • • —4 9-B-0.0 4.0 < o • I  8.0 12.0 16.0 20.0 U z ,cm /sec Figure 2-8 E f f e c t of wall confinement on the upstream velocity-p r o f i l e : (a) S/C = 7.6% 2 0 . 0 38 1 7 . 5 1 5 . 0 o o o o o o o o A A A A A 1 2 . 5 o o 1 0 . 0 c m 7 . 5 5 . 0 2 . 5 Y p = 8 3 c m U , c m / s e c • 3 . 1 2 o 6 - 9 5 A 9 . 7 2 o o o o o o o o o o A A A A A o . o ' ' ' I I I I I 0 10 U z , c m / s e c 15 Figure 2-8 E f f e c t of wall confinement on the upstream v e l o c i t y p r o f i l e : (b) S/C = 30.6% 39 presence of the model does a f f e c t to some extent i t s uniform c h a r a c t e r , however, the maximum d e v i a t i o n from the average value was found to be l e s s than 8%. As we w i l l see l a t e r , any v a r i a t i o n s i n the v e l o c i t y p r o f i l e can be accounted f o r by a modified d e f i n i t i o n of the pressure c o e f f i c i e n t . For the purpose of comparison of the experimental r e s u l t s w i t h a v a i l a b l e i n f o r m a t i o n i n the l i t e r a t u r e , i t was a l s o necessary to have d e t a i l s of c e n t e r l i n e v e l o c i t y over the op e r a t i n g range of the mean flow. This i s shown i n Figure 2-9. I t i s i n t e r e s t i n g to note t h a t at a very low value of the average v e l o c i t y (based on flowmeter data, U <3cm/s), the c e n t e r l i n e v e l o c i t y (U c) i s e s s e n t i a l l y the same as U. However, with an increase i n the flow r a t e , the r a t i o U/Uc g r a d u a l l y drops and tends to a t t a i n a uniform value of around 0.74. 2.3 Models and Support System A f a m i l y of seven spheres ranging i n diameter from 3.8-12.7 cm were c a r e f u l l y machined from p l e x i g l a s w i t h a tol e r a n c e of 0.0025mm. Any d e v i a t i o n from s p h e r i c i t y was checked using two procedures: (i ) micrometer; ( i i ) p r o j e c t i n g a photograph of the model on a screen. Figure 2-9 Variation of the centerline v e l o c i t y as measured by the hot-film probe with the average ve l o c i t y as given by the flowmeter data 41 Maximum deviation from the mean diameter was found to be less than 0.18%. The accuracy was considered quite adequate for the test program. Figure 2-10 shows the models, t h e i r dimensions and the r e s u l t i n g blockage r a t i o s . As against construction of the models, which turned out to be r e l a t i v e l y simple, the design of t h e i r support system presented several i n t e r e s t i n g problems from f l u i d dynamics considerations. Ideally one would l i k e to hold a model i n p o s i t i o n without introducing any supporting structure i n the f l u i d f i e l d . Although such 'non-contact' magnetic support systems are available commercially, they tend to be p r o h i b i t i v e l y c o s t l y . One i s , therefore, forced to turn to conventional stem type of support. This poses two questions: (i) What i s the desirable orientation of the stem r e l a t i v e to the f l u i d f i e l d ? ( i i ) What i s the e f f e c t of a stem on the f l u i d f i e l d ? To put i t d i f f e r e n t l y , what i s the c r i t e r i o n for selection of the stem size so that i t s e f f e c t on the f l u i d f i e l d becomes negligible? A horizontal stem l y i n g i n the wake of the model appears a t t r a c t i v e , however, i t suffers from two disadvan-tages. Here, to cover the spherical surface, pressure taps would be required on the horizontal meridional section, r 1 2 3 4 5 6 7 D (cm) 3.81 5.08 7.62 8.89 10.16 11.43 12.70 j S/C (%) 2.7 4.9 11.0 15.0 19.6 24.9 30.6 43 the number of p r e s s u r e taps being governed by the d e s i r e d accuracy o f the p r e s s u r e p l o t . The stem may be used to convey the p r e s s u r e t u b i n g s , hence i t s s i z e would be a d i r e c t f u n c t i o n o f the number of pressure taps used. P r e l i m i n a r y experiments w i t h tubes of d i f f e r e n t diameters suggested t h a t the i n s i d e diameter should be a t l e a s t 1.3 mm to have reasonable time constant (< 5 minutes). C o n s i d e r i n g a minimum number o f p r e s s u r e taps to be f i f t e e n ( t h i s i s g r o s s l y inadequate to p r e c i s e l y d e s c r i b e l o c a l v a r i a t i o n s ; as w i l l be seen l a t e r , a c t u a l experimental program recorded pressure a t t h i r t y "locations to p r o v i d e a w e l l - d e f i n e d p r o f i l e ) , leads to i n s i d e diameter of the stem to be a t l e a s t 2 cm, i . e . , the o u t s i d e diameter of around 2.4 cm! O b v i o u s l y t h i s i s unacceptable when the s m a l l e s t sphere has a diameter of 3.8cm. Furthermore, the stem would i n t e r f e r e w i t h the near-wake, one of the items of i n t e r e s t i n the p r e s e n t study. On the other hand, a v e r t i c a l stem support presents an a t t r a c t i v e a l t e r n a t i v e . A s i n g l e p r e s s u r e tap, through a s y s t e m a t i c r o t a t i o n about the v e r t i c a l a x i s , can cover the e n t i r e h o r i z o n t a l plane and hence, through symmetry, the e n t i r e s u r f a c e o f the sphere i f l o c a t e d on the h o r i z o n -t a l m e r i d i o n a l s e c t i o n . Furthermore, the stem, i f connected to the tap, can serve as a conduit f o r t r a n s f e r r i n g the p r e s s u r e s i g n a l t o 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 . How-ever, we s t i l l need to answer the q u e s t i o n concerning an 44 appropriate size of the stem that would not disturb the pressure f i e l d . To a r r i v e at a c r i t e r i o n for stem siz e , an extensive test program with four stems of equal inside diameter (d^ = 1.32mm) but varying outside diameter (d =1.58, 4. 76 , 6.35, and 12.70mm) was undertaken using two spherical models of diameters 5.08 and 6.35 cm. The results are presented in Figures 2-11 to 2-13. The e f f e c t of outside stem diameter on pressure p r o f i l e s at a given Reynolds number as plotted i n Figure 2-11 c l e a r l y shows that a stem diameter < 4.76 mm does not a f f e c t the data sub s t a n t i a l l y . Furthermore, the stem e f f e c t seems to be e s s e n t i a l l y independent of the Reynolds number i n the range investigated (Figure 2-12). The r e s u l t s on minimum pressure and the base pressure when plotted against the outside stem diameter to sphere diameter r a t i o (dQ/D) c l e a r l y e s t a b l i s h i t s c r i t i c a l value as shown i n Figure 2-13. Note that for dQ/D <_ 0.1 the stem e f f e c t i s v i r t u a l l y n e g l i g i b l e . In the present t e s t -program, depending on the size of the sphere, dQ/D varied in the range 0.0125-0.083. 2.4 Pressure Measurements The mean pressure component, being extremely small 2 (of the order of 0.6898 N/m ) demanded a highly sensitive instrumentation for i t s measurement. This was accomplished 45 o • A 2 o • Rn= 9 6 0 D = 5 . 0 8 c m d j ' •= 1 . 3 2 m m d 0 o 1 . 5 8 m m • 4 . 7 6 a 6 . 3 5 A 1 2 . 7 0 o • o A ° a a a A A A A. • A • A • A O a A • A 9 • • A • 0 3 0 6 0 9 0 0 ° 120 150 180 Figure 2-11 E f f e c t of the stem diameter on measured surface pressure d i s t r i b u t i o n over a sphere: (a) Sphere diameter = 5.08 cm 46 • w o 9 o Rn= 9 6 0 D = 6 . 3 5 c m d i - 1 . 3 2 m m d o o 1 . 5 8 m m • 4 . 7 6 ° 6 . 3 5 A 1 2 . 7 0 o ft o 8 Q sr • • • * A * ® A A A fi • A ft • A fi D A a A o a A o 0 3 0 6 0 9 0 0 ° 120 1 5 0 180 Figure 2-11 E f f e c t of the stem diameter on measured surface pressure d i s t r i b u t i o n over a sphere: (b) Sphere diameter = 6.35 era 47 D = 5 . 0 8 c m d j = 1 - 3 2 m m d 0 = 4 . 7 6 m m Rn = )• 6 0 7 9 5 5 X A # A A A * A A « A • 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 2-12 Plots showing stem-effects to be r e l a t i v e l y independent of the Reynolds number i n the range 300-1000: (a) D = 5.08 cm, d. = 1.32 mm, d =4.76 mm o 48 1 . .8 .4 Cp .2 .0 .2 - . 4 8 0 D = 5 . 0 8 c m * d j = 4 - 3 1 m m d o = 1 2 . 7 0 m m o 3 1 4 Rn= {• 6 6 4 * 9 6 7 o 9 A 8 3 o 8,° e.S88««.»« O o 0 A A A A ° A ^ % A £ ° . A • O o 0 ft© o 0 3 0 6 0 9 0 1 2 0 150 180 e° Figure 2-12 Plots showing stem-effects to be r e l a t i v e l y independent of the Reynolds number i n the range 300-1000: (b) D = 5.08 cm; d ± = 4.31 mm, d = 12.70 mm o 49 - . 3 r o - . 2 Cn f - . 0 5 0 . 0 5 o« R n = 9 6 0 • D = 5 . 0 8 c m o D = 6 - 3 5 c m • D = 5 0 8 c m • D = 6 - 3 5 c m C p , min C p b 1. 1 . 5 2 . -i 2 . 5 x 1 0 D Figure 2-13 Variation of the minimum and base pressures showing the c r i t i c a l value of d /D 3 • o 50 us i n g a " B a r o c e l Modular Pressure Transducing System" developed by D a t a m e t r i c s l n c . o f Watertown, Massachusetts. The type 550-5 B a r o c e l sensor i s designed to operate with 2 f l u i d s over the pr 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 high 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 of 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 of 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 pressure medium from the sensor diaphragm-capacitance system, 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 bellows, i s o l a t o r and sensor diaphragm i s f i l l e d w i t h degassed s i l i c o n e o i l which serves both as pr 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 . The pressure 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 bellows 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 a t 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 d e t e r -mines an output v o l t a g e dependent on the r a t i o o f the cap 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 --5 i n g to the i n p u t 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 pr 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 imp 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 51 pockets from the pre s s u r e d u c t i n g f o r s 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 steady p r e s s u r e s . F i g u r e 2-14 prese n t s a schematic diagram o f the pre 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 of ambient temperature e x c u r s i o n s on the B a r o c e l ' s performance. T h i s was achieved by mounting the tr 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 of temperature t r a n s i e n t s . A f t e r a complete removal of 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 46 cm downstream of the l a s t screen. Next, the pr 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 s e t of p o l y e t h y l e n e and Mylar tubings a f t e r removal of 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 balanced t o read zero output i n the no-flow c o n d i t i o n . With the pump o p e r a t i n g a t a p r e s e l e c t e d speed to gi v e 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 a t a cons t a n t temperature, the mean p r e s s u r e d i s t r i b u t i o n around the h o r i z o n t 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. For each run the v e l o c i t y p r o f i l e upstream of. the sphere was a l s o recorded. The h o t - f i l m probe, mounted on a t r a v e r s i n g gear, was p o s i t i o n e d 25 cm upstream of the sphere. The procedure was repeated over a range of mean flow r a t e s . .52 Power input P 2 Modulated output Stationary Diaphragm capcitor plates Figure 2-14 A schematic transducer diagram of the Barocel pressure 53 A p o i n t concerning an a p p r o p r i a t e c h o i c e of the s i z e of the p r e s s u r e tubings must be emphasized here. A s y s t e m a t i c study w i t h tubes of 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 steady 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 diameter l e s s than 1.6mm to have an e x c e s s i v e l y l a r g e time constant (>20min) . Of course, as suggested by s e v e r a l t h e o r e t i c a l and experimental s t u d i e s on the dynamic response o f f l u i d l i n e s " * 4 the time constant would depend on a number of parameters i n c l u d i n g the diameter 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 of 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 of p r e s s u r e s i g n a l s , e t c . For the mean pre s s u r e measure-ments under c o n s i d e r a t i o n , i t was convenient to use f l u i d l i n e s of 4.6 - 6.4 mm r e s u l t i n g i n the time c o n s t a n t (T) of around 3 - 5 minutes. To i n s u r e accuracy as w e l l as r e p e a t a b i l i t y of the measured data, i t was o f utmost importance to minimize 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 of -4 the p r e s s u r e s i g n a l s (10 p s i ) together with the r e l a t i v e l y l o n g time i n v o l v e d i n r e a c h i n g the steady s t a t e made t h i s a l l the more necessary. 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 2 4-48 hours showed them to be q u i t e s i g n i f i c a n t , at times as l a r g e as 50% o f the a c t u a l s i g n a l , but of 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 three s u c c e s s i v e measurements at equal i n t e r v a l s of 54 time c o r r e s p o n d i n g to the time c o n s t a n t of 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. pressure P^-P^, where P^ r e p r e s e n t s pressure on the s u r f a c e of the sphere a t p o i n t 'a' and P corresponds to the p r e s -sure a t a r e f e r e n c e p o i n t . L e t the a r b i t r a r y zero d r i f t of 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 2-15. The diagram a l s o shows the co r r e s p o n d i n g d r i f t of the d i f f e r e n t i a l p r e s s u r e s AP and AP , where AP =P -P and c a r a a w AP = P - P . Here P r e p r e s e n t s p r e s s u r e a t a s u i t a b l e r r w w c c l o c a t i o n , taken to be on the tunnel w a l l i n the p r e s e n t case. Thus, the d e s i r e d P -P =AP - AP . a r a r L e t 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 Now, from F i g u r e 2-15, AP + 6 , , r 1 AP + 6 , + 6 a ( A P r ) 1 + ( A P r ) 3 (AP r + 6 1 ) + (AP r + c%1 + 6 2 + 6 3) 2 2 AP + 5, r 1 + Hence, 55 Figure 2-15 A procedure for compensation of the electronic d r i f t of the pressure measuring system 56 (AP r ' l ' — r ' 3 - - 62 + 63 ( A P J , + ( A P ^ ) a'2 = ( A P a ) 1 - A P r - 5 1 -P - P + a r 6 2 - 6 3 Assuming that the elec t r o n i c zero setting d r i f t s l i n e a r l y during the i n t e r v a l marked by the pressure measure-ments (AP ) , , (AP ) ~ then r 1 r 2 i . e . , 6 2 - 63 = 0 , (AP ) + (AP ) Thus determination of the d i f f e r e n t i a l pressure i n -volved the measurement of (AP ) , , (AP )0 and (AP ) i n that r x a £ r o order. The procedure gave data that can be reproduced within an accuracy of ±2%. Note, the evaluation of the d i f f e r e n -t i a l pressure P^ - at some d i f f e r e n t location on the sphere would follow the same procedure. Thus (AP ) + (AP ) . P b " P r - ( A Pb>4 " ' 57 A question may a r i s e : why not measure the d i f f e r e n -t i a l pressure P a - d i r e c t l y . As shown in Appendix I, although t h i s can be done, i t involves necessarily more measurements. Furthermore, the r e s u l t i n g formula i s more involved and lacks recursion character leading to a sub-s t a n t i a l increase i n time and e f f o r t . Furthermore, any f l u c t u a t i o n i n the l i n e voltage would be r e f l e c t e d on the pump speed and hence on the pressure signals from the model. The speed fluctuations were moni-tored through variations i n the o r i f i c e meter data. The output voltage from the pressure transducer was damped using a DISA type 550 d i g i t a l d.c. voltmeter equipped with a r-c damping c i r c u i t to provide an adjustable time constant of up to 100 seconds. A schematic diagram of the instrumentation layout i s shown in Figure 2-16. The tests were conducted on a family of spheres, ranging i n diameter from 3.8 -12.7cm, i n the Reynolds number range of 280-2200. In a l l the cases, the model was supported by a v e r t i c a l stem, a s t a i n l e s s s t e e l tubing, which also served as a pressure conducting l i n e . Its outside diameter was dictated by the r e l a t i v e size of the sphere and the stem influence on the pressure f i e l d . On the other hand, the inside diameter was governed by the time constant to reach the steady state pressure as discussed before. The stem adopted for the experiments had an inside diameter of 1.32 mm u a i r s u p p l y t o f l u s h i q u i d i n l i n e d .c . d i g i t a l v o l t m e t e r o s c i l l o s c o p e u - v . r e c o r d e r f i l t e r _ 1 Figure 2-16 A l i n e drawing of the instrumentation set-up used for pressure measurements 59 and an outside diameter of 1.83mm re s u l t i n g i n D/dQ i n the range of 22.8-66.6. The pressure measurements were confined to the horizon-t a l meridional section of the model. A 1/16 in. pressure tap connected the stem through a groove (1/16 in.dia.) d r i l l e d i n the body of the sphere (Figure 2-17). The entire horizon-t a l plane was covered by a controlled rotation of the stem in a step size that varied between 4° to 10° depending upon the gradient of the pressure p r o f i l e . The measurements i n general were confined to only one side of the sphere, except for occasional checks to confirm flow symmetry. 2.5 Drag Measurements The balance used for drag measurements e s s e n t i a l l y consists of three components: (i) removable stem supporting the spherical model; ( i i ) central suspension block supported by a pair of needle bearings; ( i i i ) interchangeable cantilever type sensing unit with s t r a i n gages a f f i x e d near i t s root. The stem supporting the model i s attached to the central block by a thread and nut arrangement, which proved to be quite convenient i n changing the models without a f f e c t -ing the rest of the balance. Interchangeable character of 60 t o p r e s s u r e t r a n s d u c e r 1 /1 6 i n . d i a . h o r i z o n t a I m e r i d i o n a l s e c t i o n Figure 2-17 A schematic diagram showing the spherical model and i t s support system during pressure measurements 61 the s e n s i n g element was p u r p o s e l y i n t r o d u c e d to achieve a d e s i r e d degree of accuracy i n a given range of drag. S e v e r a l beams of v a r y i n g f l e x u r a l r i g i d i t y and l e n g t h were c o n s t r u c t e d f o r measurement of drag i n the range of 0.1-15 -3 grams w i t h the s e n s i t i v i t y o f 10 gram. F u r t h e r improve-ment i n s e n s i t i v i t y can a l s o be a t t a i n e d by a d j u s t i n g the gain of a b r i d g e a m p l i f i e r meter. TWO s t r a i n gages, one on each s i d e of the beam, were used f o r temperature compen-s a t i o n . The t i p of the beam r e s t e d a g a i n s t a f i x e d wedge shaped support. Alignment of the stem w i t h the l o c a l v e r t i c a l being c r i t i c a l f o r e l i m i n a t i o n of any c o n t r i b u t i o n of the model weight to the drag, the wedge support was mounted on a micrometer w i t h p o s i t i o n a l accuracy of 0.025 mm (0.001 in) . S e n s i t i v e c h a r a c t e r o f the beam demanded t h a t c a l i -b r a t i o n o f the balance be c a r r i e d out under a c t u a l t e s t arrangement w i t h a s p h e r i c a l model l o c a t e d i n the t u n n e l . T h i s i s p a r t i c u l a r l y important to e f f e c t i v e l y compensate f o r any c o n t r i b u t i o n from the weight of the sphere due to d e v i a t i o n of the s u p p o r t i n g stem from v e r t i c a l . T h i s can a r i s e i n s p i t e of the i n i t i a l alignment assured by the wedge because o f the f l e x i b i l i t y of the stem, no matter how small i t may be. The arrangement i s s c h e m a t i c a l l y shown i n F i g u r e 2-18 w h i l e F i g u r e 2-19 shows the a c t u a l assembled u n i t . Figure 2-18 An exploded balance: mediate block; (d) (e) needle view of the drag measuring \ (a) supporting stem; (b) i n t e r -connection; (c) central suspension cantilever with s t r a i n gages; bearings supporting the central block Figure 2-19 (a) Drag balance assembly with bridge amplifier meter 65 2.6 Flow V i s u a l i z a t i o n To better appreciate the physical character of the f l u i d f i e l d associated with spherical models under confined condition, flow v i s u a l i z a t i o n was undertaken. The dyed glycerol-water solution of the same concentration as that of the test f l u i d was injected approximately 10 cm upstream of the model. The dye employed was an imitation cochineal food colour. Appropriate volumes of the dye and pure gl y c e r i n were mixed to produce a glycerol-water solution of the same density as that of the test f l u i d . A dye i n j e c t -ing probe consisting of seven #23 syringe needles (0.38mm) placed 0.5 -1.0 cm apart on a streamlined support, was con-structed (Figure 2-20). "Intramedic" tubings (0.6 mm inside diameter) were used to connect the needles to a manifold. The rate of i n j e c t i o n was controlled with brass needle valves. To ensure adequate flow through each needle, i . e . , to provide s u f f i c i e n t head, the supply bottle 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 schematic diagram of the complete set-up i s shown in Figure 2-21. At times, p a r t i c u l a r l y while studying the near wake geometry, i t was convenient to i n j e c t dye d i r e c t l y through the pressure tap located on the surface of a sphere vi a the supporting stem. I t was now possible to introduce dye at any desired location on the sphere. The procedure proved quite e f f e c t i v e i n i d e n t i f y i n g separation position of the ri n g vortex. 66 Figure 2-20 Dye i n j e c t i n g probe used during flow v i s u a l i z a t i o n Figure 2-21 A sketch showing the equipment layout during flow v i s u a l i z a t i o n 68 I t would be appropriate to point out here the type of l i g h t i n g system used as i t played a c r i t i c a l role i n the photographing process. A combination of three variable i n t e n s i t y photo floods (maximum 5 0 0 watts, 3 4 0 0 ° K ) back-illuminated the subject through the tunnel glass window. To eliminate hot spots, the l i g h t beam was evenly diffused by masking the te s t section wall with a tracing paper. A set of t r i a l runs helped arrive at the appropriate aperture s e t t i n g and exposure time for the type of f i l m used (Kodak high speed Ektachrome type E H B - 1 3 5 ( s t i l l ) or EF-7242(movie), tungsten, 3 2 0 0 ° K , A S A 1 2 5 , f i l t e r 8 1 A ) . During the course of v i s u a l i z a t i o n study, i t was discovered that i n spite of the large volume of the test f l u i d ( 4 0 U.S. gallons), a r e l a t i v e l y small amount of dye ( 8 f l u i d oz) was s u f f i c i e n t to pollute the working f l u i d to the point that no clear photographs could be taken. This presented a rather serious problem i n terms of time, e f f o r t and cost involved i n replenishing the working f l u i d . C l early, i t was necessary to find an agent that would neutralize the dye without attacking the tunnel material 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 physical properties of the te s t f l u i d . Unfortunately, no such agent has been reported i n the l i t e r a t u r e . A considerable amount of patient t e s t i n g with numerous oxidizing agents led to sodium hypochlorite which has a l l the desirable a t t r i b u t e s . Only 3 0 0 cc of the agent was 69 s u f f i c i e n t to completely neutralize the dye. To keep the concentration of the test f l u i d constant, s u f f i c i e n t amount of gl 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. 70 3. RESULTS AND DISCUSSION With some appreciation of background to the problem, instrumentation used and the experimental procedures adopted, we are ready to look into the test results and t h e i r i n t e r p r e t a t i o n . The amount of experimental data obtained i s rather enormous, thus d i c t a t i n g a compromise i n presentation between conciseness and comprehensibility. The guiding p r i n c i p l e has been to include only those results which have immediate relevance to the study i n hand, and help i n e s t a b l i s h i n g d e f i n i t e trends. In general, the sequence i n which the r e s u l t s are presented also denote the chronological order of the t e s t s . To begin with, an approach to data reduction, so c r i t i c a l at low Reynolds number, i s discussed. This i s followed by presentation of the surface pressure d i s t r i b u t i o n results as affected by Reynolds number and blockage. Next, measured sphere drag values are analyzed as functions of system parameters and compared with integrated pressure drag to e s t a b l i s h skin f r i c t i o n contribution. F i n a l l y , near-wake structure i s studied using flow v i s u a l i z a t i o n i n conjunction with s t i l l and 16 mm movie photography. Available results from l i t e r a t u r e are included when appropriate for comparison and to a s s i s t i n emphasizing the influence of blockage. 71 3.1 Choice of Reference V e l o c i t y and Pressure Before proceeding 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 of the t e s t r e s u l t s , one must address to s e v e r a l fundamen-t a l 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 flow 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 uniform stream there i s no ambiguity concerning 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 : I t i s the constant v e l o c i t y and p r e s s u r e of the stream f a r away from the model. For low Reynolds number flow 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 vary s i g n i f i c a n t l y along the a x i s of the t e s t s e c t i o n , even i n absence of the model due to boundary l a y e r growth along the w a l l s . Presence of the model and a s s o c i a t e d wake would o n l y accentuate the problem. Ob 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 of these parameters. 57 5 8 Grove e t 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 e n c e 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 a t the same s e t t i n g of 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 wit h a small blockage t h i s choice of r e f e r e n c e p r e s s u r e may prove to be adequate but with a l a r g e r blockage, due to a c c e l e r a t i o n of the flow a t the model l o c a t i o n , the reference pressure i s indeed affected and becomes a function of wall confinement (besides other parameters). To put i t d i f f e r e n t l y , the choice of reference pressure as suggested above has a degree of optimism i m p l i c i t i n i t . I t assumes that e f f e c t s of the upstream adverse pressure gradient created by presence of the model exactly cancels the influence of acceleration i n gaps at the model location thus giving the desired P T O . One possible improvement in the choice of would be to take i t as the pressure at the model location (but without the model) with operating condition of the tunnel kept the same as that used with the model in p o s i t i o n . However, t h i s s t i l l cannot account for the changes in v e l o c i t y p r o f i l e from section to section i n a given tunnel, and between tunnels used by d i f f e r e n t investigators. Usefulness of the centerline 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 also poses several questions. In general, 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 affected by location, boundary layer growth, screen's mesh size, blockage, pump speed and the t o t a l c i r c u i t resistance. 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 hardly be considered a suitable reference. Another possible compromise would be to take uniform portion of the v e l o c i t y p r o f i l e far upstream 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 used, model and i t s l o c a t i o n . 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 t h e r 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 permit comparison. 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 com-promise 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 flow 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 -ates most of the problems mentioned above. Obvio u s l y , t e s t s conducted w i t h and without model (but a t the same meter s e t t i n g as with the model) would leav e 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. Thus, 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 of model l o c a t i o n , type o f t u n n e l , flow 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 blockage. 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 . Furthermore, i t s measure-ment i s q u i t e simple 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 essure e f f e c t s due to l o c a t i o n of the model. This brings us to that elusive task of s e l e c t -ing P . As discussed e a r l i e r , the P advocated by Grove et a l . has l i t t l e meaning here i n view of the large blockage presented by the model. From the point of view of r e p e a t a b i l i t y and comparison of data, the use of pressure at a s p e c i f i e d tap on the surface of the model as reference appears quite a t t r a c t i v e . Although th i s cannot account for l o c a l variations due to blockage e f f e c t s (from point to point at the surface of the model), i t could e f f e c t i v e l y compensate for i t i n an average fashion. Thus one way to present pressure data i n co-2 e f f i c i e n t form would be as C = (P. -P ) / ( p U /2) where p 9 r H ' P r corresponds to the pressure at a s p e c i f i e d tap on the surface of the poppet and U as calculated from the average flow rate (average flow rate/test-section area of 20. 32 cmx 20. 32 cm) . However, t h i s d e f i n i t i o n i s s t i l l susceptible to errors introduced by non-uniformity of the v e l o c i t y p r o f i l e (at a pressure tap and the reference location) p a r t i c u l a r l y because the denominator remains unaffected by t h i s change. One way to v i r t u a l l y eliminate t h i s shortcoming i s to express pressure c o e f f i c -ient as explained below (Figure 3-1). 75 Figure 3-1 An i l l u s t r a t i o n showing possible errors introduced by non-uniformity of the v e l o c i t y p r o f i l e Let errors i n pressure due to non-uniformity of the v e l o c i t y p r o f i l e be at P Q , e Q at P Q and at P . Expressing pressure c o e f f i c i e n t as the r a t i o of the d i f f e r e n t i a l pressures, between that at a tap i n question and the stagnation point with respect to the reference pressure, gives P ( P 9 + £ 9 ) - ( P r + £ r } (P 0 +-&Q) - (P r + e r) where P Q, P , P„ correspond to pressures with uniform H r u ve l o c i t y p r o f i l e . Thus P. - P = ( 0 r 1 + ( e e - e r ) / ( P e - P r ) 1 + ( e 0 - e r ) / ( P 0 - P r ) Note that en - e and e_ - e are l i k e l y to be very small. 0 r 0 r On the other hand, P D - P and P A - P represent r e l a t i v e l y o r u r large quantities compared to the respective error d i f f e r e n t i a l s . Therefore, £ 0 - £ r . £ 0 " £ r E . - — — - and e_ = = =r— 6r P Q - P r Or P Q - P r are l i k e l y to be vanishingly small. Consequently, the term 1 + e Q P Q - P 0 r -. , „ 0 r " 1 a n d C p P -P 1 + £ 0 r P 0 r Both numerator and denominator being se n s i t i v e , the proposed d e f i n i t i o n of the pressure c o e f f i c i e n t promises to provide adequate compensation for errors introduced by non-uniformity of the v e l o c i t y p r o f i l e . The reference location was taken to be at 0 = 60°, The choice was prompted by the test data which showed C 77 to reach zero i n the g e n e r a l v i c i n i t y o f 6 = 60°, i . e . , Pggo ~ (Figure 3-2). Of course, 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 data presented i n t h i s chapter use the 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 as P - P C = 9 60° P P - P 0 60° I t i s easy to r e c o g n i z e the term P Q - P^QO as an approximation 2 o f (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 n t r o d u c e d by n o n - u n i f o r m i t y of 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 , blockage e f f e c t s , 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 s o f the p r e s s u r e sensing 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 2). Furthermore, i n c o n j u n c t i o n w i t h the Reynolds number (based on average flow v e l o c i t y and sphere diameter), i t promises to a s s i s t i n comparison with s i m i l a r data by other 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 . 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 78 1.0 0 . 5 0 . 0 - 0 . 5 - 1 . 0 - 1 . 5 - 2 . 0 2 . 5 2 6 . 6 © a D 5 . 0 8 c m S / C = 4 . 9 % R n • 3 0 2 o 6 4 0 A 9 5 3 2 o 6 AD 6 o o . • . . • 0 3 0 6 0 9 0 6 ° 1 2 0 1 5 0 1 8 0 Figure 3-2 Typical pressure p r o f i l e s for a sphere using the conventional d e f i n i t i o n of pressure c o e f f i c i e n t , 2 Cp = ( P Q - Pa,) / (1/2) pu c . Note the pressure co-e f f i c i e n t i s zero i n the v i c i n i t y of 9 = 60° 79 data w i t h other published 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 present any problem. As shown i n Appendix I , conventional pressure c o e f f i c i e n t C can be w r i t t e n i n P terms of measured informa t i o n as c r = P e " P " = ( p e - p 6 0 " ) - ( p o - p 6 o ° ) + 1 P " ^ P " 2 l / 2 p u 2 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 here. ~ Representative surface pressure data as presented i n Figures 3-3 to 3-5 d r a m a t i c a l l y emphasize e f f e c t i v e n e s s of t h i s new d e f i n i t i o n of the pressure c o e f f i c i e n t . The e f f e c t of d i f f e r e n c e s i n v e l o c i t y p r o f i l e s , as encountered by a model l o c a t e d at 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 i s shown i n Figure 3-3. The conventional pressure c o e f f i c i e n t C~ shows l a r q e v a r i a t i o n s almost P over the e n t i r e surface except f o r a small region i n the v i c i n i t y of the stagna t i o n (Figure 3-3a). Although f a i r s b e t t e r i n the region 8 > 60°, the p l o t s are considerably d i s t o r t e d i n the upstream d i r e c t i o n (Figure 3-3b). S u r p r i s i n g l y , the new pressure c o e f f i c i e n t remains q u i t e i n s e n s i t i v e to the v e l o c i t y changes over 80 1.Q 0 . 5 0 . 0 h o - 0 . 5 - 1 .0 - 1 .5 - 2 . 0 - 2 . 5 o D = 8 . 8 9 c m S / C = 1 5 . 0 % R n = 1 0 0 3 M o d e l L o c a t i o n f r o m T u n n e l I n l e t • 1 0 0 c m o 1 6 3 . c m • • • • * • • • * o o o o o ° o o ° ° o o o o o o o o 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0° Figure 3-3 Plots showing s e n s i t i v i t y of d i f f e r e n t d e f i n i t i o n s for pressure c o e f f i c i e n t to changes i n v e l o c i t y p r o f i l e : „ (a) C p = ( P e - Poo)/(1/2) plT 81 3 . 0 • o 2 . 5 [ D = 8 . 8 9 c m S / C = 1 5 . 0 % 2 -0 1 . 5 A C p | O 1 .0 O 0 . 5 o R n = 1 0 ° 3 M o d e l L o c a t i o n f r o m T u n n e l I n l e t • 1 0 0 c m o 1 6 3 c m 0.0 \ • o < * » © « . 2 » ® 9 •o 0 . 5 h 9*oh 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 Figure 3-3 Plots showing s e n s i t i v i t y of d i f f e r e n t d e f i n i t i o n s for pressure c o e f f i c i e n t to changes i n vel o c i t y p r o f i l e : (b) C p = ( P Q - P 6 0 o ) / ( l / 2 ) p U 2 the e n t i r e surface even w i t h the enlarged s c a l e used i n p l o t t i n g the data (Figure 3-3c). Figure 3-4 summarizes e f f e c t of the Reynolds number on the surface pressure f o r three d i f f e r e n t d e f i n i t i o n s of the pressure c o e f f i c i e n t . Note th a t both the conventional d e f i n i t i o n C as w e l l as i t s P m o d i f i c a t i o n as given by the use of c e n t e r l i n e v e l o c i t y are q u i t e s u s c e p t i b l e to the i n f l u e n c e of Reynolds number (Figure 3-4a,b). On the other hand, the proposed d e f i n i t i o n (Figure 3-4c) shows only s l i g h t s e n s i t i v i t y i n and near the wake region. Note t h a t the s c a l e used i n Figure 3-4c magnifies d e v i a t i o n s by a f a c t o r of 2.5. Thus the new d e f i n i t i o n performs e x c e p t i o n a l l y w e l l and makes the pressure d i s t r i b u t i o n v i r t u a l l y independent of the Reynolds number i n the range i n v e s t i g a t e d . The r e l a t i v e independence of the w a l l confinement e f f e c t s f o r the blockage r a t i o as l a r g e as 30.6% as shown i n Figure 3-5 makes the proposed d e f i n i t i o n extremely a t t r a c t i v e . As pointed out before, the choice of reference pressure i s q u i t e a r b i t r a r y . However, e f f e c t i v e com-pensation of e r r o r s introduced through various sources mentioned e a r l i e r w i l l indeed depend on a given P r f o r a s p e c i f i c pressure gradient, v e l o c i t y p r o f i l e , blockage, 83 1.0 . 8 . 6 . 4 C p . 2 . 0 - . 2 .4 o D = 8 . 8 9 c m S / C = 1 5 . 0 % R n = 1 0 0 3 M o d e l L o c a t i o n f r o m T u n n e l I n l e t • 1 0 0 c m o 1 6 3 c m o O o a 6 9 • 9 « ® ® o • ° _ o« 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 6 ° Figure 3-3 Plots showing s e n s i t i v i t y of d i f f e r e n t d e f i n i t i o n s for pressure c o e f f i c i e n t to changes i n v e l o c i t y p r o f i l e : ( C ) CP = <P e-P 6 0.>/<P 0-P 6 0o> 84 1.0 0 . 5 0 . 0 - 0 . 5 C p - 1 . 0 - 1 . 5 - 2 . 0 - 2 . 5 o A D = 5 . 0 8 c m S / C = 4 . 9 % 9 R n • 3 0 2 6 o 6 4 0 A 9 5 3 o A o A fi A A 0 6 o 2 2 2 2 . o 2 6 ° ri A° a ^ S AAO • • • • • • • 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 3-4 E f f e c t of Reynolds number on surface pressure d i s t r i b u t i o n i n terms of: (a) C p = (P e - P j / ( l / 2 ) pU2 85 1 .0 0 . 5 0 . 0 2 6 6 © fi D - 5 . 0 8 c m S / C = 4 . 9 % R n • 3 0 2 o 6 4 0 A 9 5 3 2 © 6 e, A 6 6 AO 6 Q A A Q 6 O © O © 0 . 5 1 . 0 - 1 . 5 • • • - 2 . 0 2 " 5 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 6 ° Figure 3-4 E f f e c t of Reynolds number on surface pressure d i s t r i b u t i o n i n terms of: 86 1.0 . 8 . 6 . 4 C p . 2 o a i D = 5 . 0 8 c m S / C = 4 . 9 % R n • 3 0 2 o 5 1 6 A 6 4 0 • 9 5 3 • . 0 - . 2 2 ° AO 6 O o O D A N • D A O « • AO • Q A A o • »° - o • 8 § D A - . 4 0 Figure 3-4 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 e o E f f e c t of Reynolds number on surface pressure d i s t r i b u t i o n i n terms of: (c) C ^ P9 ~ P 6 0 ° ^ ^ ^ P0 P 6 0 o ) 87 • Rn D.cm S / C % • 825 3.81 2.7 o 948 7.62 11.0 A 994 10.16 19.6 • 999 12.70 30.6 A a 6 o o o • o 0 ° ° o ° ° o o o o • o • o o A O • , , o o o Q o • o o A A A A A A A • A A A • • • • D 0 30 60 90 120 150 180 9 ° 3-5 Representative pressure plots showing r e l a t i v e i n s e n s i t i v i t y of the proposed pressure c o e f f i c -ient to blockage e f f e c t s : (a) conventional pressure c o e f f i c i e n t (C ) based on average vel o c i t y , C =  ( P e - P j / d / 2 ) p U 2 P 88 i .Or . 8 . 6 . 4 C p .2 . 0 - .2 - . 4 R n D . c m S / C % • 8 2 5 3 . 8 1 2 . 7 o 948 7 . 6 2 1 1 . 0 6 ^ 994 1 0 . 1 6 1 9 . 6 3 • 9 9 9 1 1 . 4 3 2 4 - 8 • 9 9 9 1 2 . 7 0 3 0 . 6 A O 1 O A ° • o » 6 » ft *. 6 ^XD • O • 0 A A A cj 9*:^. s . : i 4 • n~ o A • • ° r* fl A A • A A _ O • • • O D • 0 3 0 6 0 9 0 1 2 0 1 5 0 0° Figure 3-5 Representative pressure plots showing r e l a t i v e i n s e n s i t i v i t y of the proposed pressure c o e f f i c -ient to blockage e f f e c t s : (b) suggested pressure c o e f f i c i e n t defined as C p = < p e - p 6 o - ) / { p o - R 6 0 o ) geometry of test model, etc. Hence, although the proposed d e f i n i t i o n i s l i k e l y to be less dependent on f l u i d dynamical parameters compared to the conventional C , the degree of v a r i a t i o n may indeed depend upon the chosen reference pressure. Data reduced using 0=30°, 60°, and 90° as references substantiate t h i s observation. Thus a question concerning the optimum choice of the reference pressure a r i s e s . In general, i t would be impossible to i d e n t i f y an optimum reference for a l l cases. However, recognizing the fact that for most b l u f f bodies the difference between the minimum and base pressures remains r e l a t i v e l y constant, reference 6 i n the range of around 50° -120° i s l i k e l y to lead to good r e s u l t s . Figure 3-6 shows the e f f e c t of v e l o c i t y p r o f i l e , Reynolds number and blockage on the pressure c o e f f i c i e n t defined using PgQ0 as reference. Note that plots remain e s s e n t i a l l y unaffected except for a small region i n the v i c i n i t y of the stagnation. 3.2 E f f e c t of Reynolds Number Figures 3-7 through 3-9 summarize a rather com-prehensive set of data on the surface pressure d i s t r i -bution for a sphere as affected by the Reynolds number for a given blockage i n the range 4.9 - 30.6%. Results 90 1 . 0 ' D = 8 . 8 9 . c m Q \ 9 S / C = 1 5 . 0 % R n = 1 0 0 3 o M o d e l L o c a t i o n f r o m T u n n e l I n l e t 2 \ - . 2 o • 1 0 0 c m ° 1 6 3 c m a # c v * a » » * 6 • * 9 ft 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 3-6 Surface pressure d i s t r i b u t i o n on spheres using PgQo as reference. Note the plots show very-l i t t l e dependence on;(a) v e l o c i t y p r o f i l e 91 1 .0 .8 •6 A o A D = 5 . 0 8 c m S / C = 4 . 9 % R n • 3 0 2 o 6 4 0 A 953 o A • O A 2 \ . 0 2 . o 6« a # 2 2 o A o A o o o A 2 * . t « 6 2S - .2 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 Figure 3-6 Surface pressure d i s t r i b u t i o n on spheres using PgO° as reference. Note the plots show very l i t t l e dependence on :(b) Reynolds number 92 R n D • 8 2 5 O • • o 994 • 9 9 9 O • a o o D . c m S / C % 3 . 8 1 2 . 7 • o <3 W 8 o " cfl 3 0 6 0 9 0 0° 1 2 0 1 5 0 1 8 0 3-6 Surface pressure d i s t r i b u t i o n on spheres using PgQo as reference. Note the plots show very l i t t l e dependence on:(c) blockage 93 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 f o r comparison when a v a i l a b l e . In most cases, the i n f o r m a t i o n i s p r e -sented u s i n g the new d e f i n i t i o n of 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 , however, t y p i c a l r e s u l t s i n terms o f C are a l s o i n c l u d e d to i l l u s t r a t e i d e n t i c a l trends P p r e d i c t e d by both approaches (Figures 3-7b and 3-8d). At the o u t s e t one can say t h a t the e f f e c t of 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 and even here i t i s l i m i t e d t o R n<1000, except f o r the very h i g h blockage r a t i o of 30.6%. In g e n e r a l , the e f f e c t of 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 . Furthermore, l o c a t i o n of the minimum pressure p o i n t together 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 of the uniform pressure r e g i o n o f the wake) tend to s h i f t a l i t t l e upstream. I t i s of some i n t e r e s t to note t h a t i n the r e g i o n bounded by the f r o n t s t a g n a t i o n and the zero p r e s s u r e p o i n t , the e f f e c t of Reynolds number appears to be j u s t the o p p o s i t e , i . e . , the p r e s s u r e decreases w i t h an i n c r e a s e i n the Reynolds number. F i g u r e 3-7(c) compares the present data with the h i g h e r Reynolds number r e s u l t s as o b t a i n e d by 47 11 Aminzadeh and Maxworthy . Aminzadeh's p r e s s u r e data a t R = 5848 tend to s u b s t a n t i a t e e a r l i e r o b s e r v a t i o n n 94 1.0 .8 . 4 C p . 2 . 0 - . 2 - . 4 o S e • 2 D = 5 . 0 8 c m S / C = 4 . 9 % R n • 3 0 2 o 5 1 6 6 4 0 9 5 3 A • • A 0 ® O • ° • • o • A Q • • • AO * • AO • D • A O ^ • • A ^ b • •°o Ao° • o. . • • • • 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 . e o Figure 3-7 Surface pressure d i s t r i b u t i o n as affected by Reynolds number at a small blockage r a t i o of 4.9%: ( a ) CP - ( p e - p 6 0 ° ) / ( p o - p 6 0 ° } 95 1.0 0 . 5 0 . 0 \ - 0 . 5 - 1 . 0 - 1 . 5 - 2 . 0 - 2 . 5 ° • D = 5 . 0 8 c m S / C = 4 . 9 % A o R n • 3 0 2 o 6 4 0 6 A 9 5 3 o A o A 2 A A 0 6 o 2 2 2 A 2 J>6 6° AO S © 0 ° " , • • • • 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 o 9 F i g u r e 3-7 Surface pressure d i s t r i b u t i o n as a f f e c t e d by Reynolds number a t a s m a l l blockage r a t i o o f 4.9%: (b) C p = (P 0 - P j / ( l / 2 ) pU 2 96 D = 5 . 0 8 c m „ S / C = 4 . 9 % o R n • 3 0 2 • 5 8 4 8 4 ? | A m i n z a d e h g • ° 5 1 6 o 1 3 7 X 1 0 3 S A 6 4 0 M a x w o r t h y • 9 5 3 1 6 2 x 1 0 3 g A c h e n b a c h • • 1 9 8 X 1 0 3 ^ M a x w o r t h y ^ • • • • • • • w i t h T r i p W i r e © • o oo cr> O • A O ^ s A o • • t° ° 3 0 6 0 9 0 1 2 0 1 5 0 180 , 6° Surface pressure d i s t r i b u t i o n as a f f e c t e d by Reynolds number a t a small blockage r a t i o of 4.9%: (c) comparison with r e c e n t data by ot h e r i n v e s t i g a t o r s . Note the r e s u l t s by Maxworthy and Achenbach are near c r i t i c a l Reynolds number (R = 3.7 x 10^, Reference 9) n,cr ' 97 concerning independence of the p r e s s u r e f o r R n> 1000. However, note the change i n the pressure p r o f i l e as one approaches the c r i t i c a l Reynolds number (Maxworthy"'""'", 3 R =137 x10 ). There i s a sudden i n c r e a s e i n the n minimum pre s s u r e value and an upstream s h i f t i n the s e p a r a t i o n p o i n t . T h i s i s a s s o c i a t e d w i t h the f a m i l i a r 3 5 s l i g h t i n c r e a s e i n the drag i n t h i s r e g i o n (R = 6 x 1 0 - 2 x 1 0 ) . 5 However, a t the c r i t i c a l Reynolds number (R v 2 x 1 0 ) n , c r i t i c a l t h ere i s a sudden r i s e i n the base pr e s s u r e and the down-stream movement o f the s e p a r a t i o n p o i n t r e s u l t i n g i n the 11 3 c l a s s i c a l r e d u c t i o n i n drag (Maxworthy , R n =198 x10 ). At h i g h e r blockage r a t i o s of 11-19.6% (Figure 3-8) e s s e n t i a l l y the same t r e n d i s maintained. However, f o r any f u r t h e r i n c r e a s e i n the w a l l confinement the base p r e s s u r e begins to be a l i t t l e s e n s i t i v e to the Reynolds number (Figure 3-9). I t i s i n t e r e s t i n g to note here t h a t 59 Modi and Sherbmy a l s o observed the same tr e n d i n t h e i r study 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. 3.3 Wall Confinement E f f e c t s F i g u r e s 3-10 and 3-11 summarize r e s u l t s on the i n f l u e n c e o f blockage o f f e r e d by the s p h e r i c a l models. I t must be r e c o g n i z e d t h a 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 f i x e d by 98 1.0 .8 •4 C p . 2 - . 2 - . 4 6 D = 7.62 c m 6 S / C = 1 1 . 0 % R n 5 9 4 o 9 4 8 © A 1 2 9 4 o A o A A o .o\ © A < o a £ a 6 S * 6 * 0 * 2 A o AO 6 A ° A O • '6 0 3 0 6 0 9 0 1 2 0 1 5 0 180 0° Figure 3-8 Pressure plots showing t h e i r r e l a t i v e i n s e n s i t i v -i t y to Reynolds number _> 1000 and for i n t e r -mediate values of blockage: (a) C , S/C=11.0% P 99 1.0 .8 . 6 . 4 . 2 .0 - . 2 1 . 4 2 6 D = 8 . 8 9 c m S / C = 1 5 . 0 % R n • 1 0 0 2 o 1 2 8 3 A 1 5 8 5 ^fi ! S • g & * o $ A A <5 AO © o* 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0° Figure 3-8 Pressure plots showing t h e i r r e l a t i v e i n s e n s i t i v -i t y to Reynolds number _> 1000 and for intermediate values of blockage: (b) C , S/C = 15.0% 100 1.0 . 8 . 4 C p . 0 . 2 D = 1 0 . 1 6 c m £ S / C = 1 9 . 6 % R n • 9 9 4 e o 1 2 8 5 A A 1 5 8 9 2 A $ $ ° « « * 4 8 g A • 8 ®2 6 # 8 i A « *$9 . 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0° Figure 3-8 Pressure plots showing t h e i r r e l a t i v e i n s e n s i t i v -i t y to Reynolds number j> 1000 and for intermediate values of blockage: (c)~C , S/C = 19.6% 101 1-0 0.5 0.0 -0.5 C p -1.0 -1.5 -2.0 -2.5 ! D = 10.16 c m ft S / C = 19.6 % R n 994 • Q1285 A 1589 A o 2 5 8 ©* A . 8 8 ° * 8 § 2 ' 5 5 A 8 4 A A £ ® 2 0 30 60 90 120 150 180 9° Figure 3-8 Pressure plots showing t h e i r r e l a t i v e i n s e n s i t i v -i t y to Reynolds number 2^1000 and for intermediate values of blockage: (d) C , S/C =19.6% 102 1.0 . 8 . 6 .4 2[ . 0 - . 2 . 4 2 6 O e D = 1 1 . 4 3 c m S / C = 2 4 . 6 % R n • 9 9 9 6 O 1 2 8 2 A 2 0 1 4 8 6 0 A w 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 3-9 Reynolds number e f f e c t on the pressure d i s t r i b u -t i o n at higher blockage r a t i o s : (a) C , S/C = 24.6% P 103 1.0 .8 . 6 .4 . 2 .0 - . 4 D = 1 2 . 7 0 c m I S / C = 3 0 . 6 % o R n • 9 9 9 o 1 6 0 2 A 2 0 0 6 • • 2 2 9 1 e o o n o 9 O 9 a a 6 A A ^ • • • 8 2 ~ ° • 0 30 60 90 120 150 180 9° Figure 3-9 Reynolds number e f f e c t on the pressure d i s t r i -bution at higher blockage r a t i o s : (b) C , S/C = 30.6% P 1 .0 0 . 5 h o D = 1 2 . 7 0 c m 0.01- ° S / C = 3 0 . 6 % - 0 - 5 - 1 . 0 - 1 . 5 - 2 . 0 2 . 5 - 3 . 0 A R n • 9 9 9 o 1 6 0 2 $ • A 2 2 9 1 A O A o e A A O O Og A 2 o © 104 ° o o ° ° ° Q g 2 2 Q A A A A A A • • 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0 ° Figure 3-9 Reynolds number e f f e c t on the pressure d i s t r i -bution at higher blockage r a t i o s : (c) C , S/C = 30.6% 105 d e s i g n c o n s i d e r a t i o n s . For the presen 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 . Hence, f o r a given blockage, i t wasn't always p o s s i b l e to cover the d e s i r e d range of Reynolds number (300-2000). T h i s has l e d to unavoidable gaps i n the r e s u l t s presented here, however, the trends are reasonably w e l l e s t a b l i s h e d f o r R n > 600. From F i g u r e 3-10 i t i s apparent t h a t f o r up to around 11%, the confinement e f f e c t s are e s s e n t i a l l y n e g l i g i b l e but beyond t h a t the blockage has d e f i n i t e tendency to reduce the minimum and base p r e s s u r e s (Figure 3-11). The minimum pressure p o i n t shows a d i s t i n c t rearward s h i f t . S i m i l a r downstream movement of 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 although i t i s not q u i t e d i s t i n c t . A flow v i s u a l i z a t i o n study d e s c r i b e d l a t e r ( S e c t i o n 3.'6) confirmed 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 blockage e f f e c t remains e s s e n t i a l l y the same f o r R >1000. J n Corresponding r e s u l t s i n terms o f 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 were presented e a r l i e r (Figure 3-4d). F i g u r e 3-12 shows v a r i a t i o n o f the average base p r e s s u r e and the minimum p r e s s u r e w i t h blockage. Up to S/C of around 12 - 15% the base p r e s s u r e as w e l l as the minimum pr e s s u r e remains e s s e n t i a l l y constant, however, beyond t h a t there i s a d i s t i n c t r e d u c t i o n i n 106 1 .0 .8 • 6 .4 Cp •0 - .2 - .4 o W A • 2 o 4 e A Rn D,cm S / C , * • 607 3.81 2.7 o 616 5.08 4.9 A 659 6.35 7.6 D 594 7.62 1 1.0 • A - # 0«5 O * A * & S A 6 a cfi *5 5Q* AP' 0 30 60 90 120 150 180 0° Figure 3-10 Representative plots showing n e g l i g i b l e e f f e c t of wall confinement for blockage r a t i o s up to 11% t o r R n D . c m S / C % 9 • 8 2 5 3 . 8 1 2 . 7 m ° 9 4 8 7 . 6 2 1 1 . 0 6 • A 9 9 4 1 0 . 1 6 1 9 . 6 D • 9 9 9 1 1 . 4 3 2 4 - 8 • • 9 9 9 1 2 . 7 0 3 0 . 6 o I o A o • ° A A ^ ^ f t C ^ A . A B A A • A • • A A • • 0 3 0 6 0 9 0 1 2 0 1 5 0 1* 9° Figure 3-11 Pressure plots as affected by higher blockage: (a) R = 950 n 108 1 .0 . 8 . 6 . 2 . 0 - -2 - -4 s R n D , c m S / C , % • 1 2 2 7 6 . 3 5 7 . 6 o 1 2 8 3 8 . 8 9 1 5 . 0 A 1 2 8 5 1 0 .1 6 1 9 . 6 • • 1 2 8 2 1 1 . 4 3 2 4 . 8 • • 1 2 7 8 1 2 . 7 0 3 0 . 6 • 11 o g • • • • • 0 * 0 0 0 0 0 0 0 o o • A A 2 A 0* - «o ° i O'' <tfO A • • o " • n n A • _ • • D D I 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 3-11 Pressure plots as affected by higher blockage: (b) R = 1250 n 109 1.0 I 8 • • 8 - . R n D , c m S/C,% • 1 5 8 5 8 . 8 9 1 5 . 0 o 1 5 8 9 1 0 . 1 6 1 9 . 6 A 1 5 8 2 1 1 . 4 3 2 4 . 8 n • 1 6 0 2 1 2 . 7 0 3 0 6 • • A • 4 l • • 2 \ 6 Q .o i • • • • . • . • Q o o • o o ° o ° # . Q O O A A A A A A A A ° A A • ° D o D ° ° D ° a A A A • • ' 4 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 9° Figure 3-11 Pressure plots as affected by higher blockage: '(c) R = 1600 n 110 1 0 0 0 < R n < 1 6 0 0 ° -Cpb • ~ C p m A C p b - C p m A I I 8 S j . 1 0 2 0 3 0 S / C % gure 3-12 E f f e c t of wall confinement on the minimum and base pressures, 1000 <R<1600. Note both C D and C D are e s s e n t i a l l y constant up to the blockaV- m r a t i o of around 13% ige the p r e s s u r e s with blockage. Thus there appears to be a c r i t i c a l value of the blockage r a t i o above which the e f f e c t of w a l l confinement tends to become s i g n i f i c a n t . T h i s r e p r e s e n t s i n f o r m a t i o n of c o n s i d e r a b l e p r a c t i c a l s i g n i f i c a n c e . I n t e r e s t i n g l y the d i f f e r e n c e C p - C , which i s b pm a measure o f the pressure r i s e s u s t a i n e d by the boundary l a y e r p r i o r to s e p a r a t i o n , remains v i r t u a l l y independent of the blockage throughout. The near independence of t h i s q u a n t i t y from the confinement e f f e c t s suggests r e l a t i v e i n s e n s i v i t y o f the boundary-layer to the l o c a l changes i n the f r e e stream v e l o c i t y r e s u l t i n g from blockage. However, at lower Reynolds number, due to dominance o f v i s c o u s f o r c e s , one would expect t h i s t r e n d to change. Measurements conducted a t R n =600 confirmed t h i s (Figure 3-13). Although the base p r e s s u r e continues to remain r e l a t i v e l y i n s e n s i t i v e to blockage, there i s a d e f i n i t e drop i n the minimum pr e s s u r e r e s u l t i n g i n a c l e a r i n c r e a s e of C - C . p b pm U s e f u l condensation o f Reynolds number and blockage e f f e c t s on base pressure i s presented i n F i g u r e 3-14. The r e s u l t s suggest t h a t Reynolds number e f f e c t s are c o n f i n e d to the range R <1000 f o r a l l 3 n blockage r a t i o s and R n >1600 f o r the h i g h e r blockage (S/C o f 30.6% i n the p r e s e n t c a s e ) . 112 . 2 .1 • 0 0 R n = 6 0 0 o C p b • - C p m A C p b - C p m o o 5 S / C % 1 0 Figure 3-13 Plots showing dependence of C n (and hence pm C n, ~ C n ) on wall confinement, even when S/C i s less than 13%, at R < 1000 n o D, c m S / C % o 6 . 3 5 7 . 6 • 8 . 8 9 1 5 . 0 A 1 0 . 1 6 1 9 . 6 • 1 1 . 4 3 2 4 . 8 • 1 2 . 7 0 3 0 . 6 3 4 5 6 7 R n X 1 0 2 Figure 3-14 Condensation of the base pressure data showing the influence of Reynolds number and blockage 114 3.4 Drag C o e f f i c i e n t P ressure d i s t r i b u t i o n on the s u r f a c e of a sphere having been e s t a b l i s h e d , the next logical step was to o b t a i n the pressure i n t e g r a t e d value o f drag and assess i t s dependence on blockage. T h i s , o f course, 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 other hand, the d i r e c t measurement of s k i n f r i c t i o n i s c o n f i n e d to the 4 6 h i g h Reynolds number range o f 5 x10 - 6 x10 , by 9 Achenbach . The c o r r e s p o n d i n g i n f o r m a t i o n a t lower 4 Reynolds number (R <5 x10 ) i s t o t a l l y m i s s i n g . With t h i s as background, i t was decided to undertake measure-ment o f the t o t a l drag u s i n g a s t r a i n gage balance d e s c r i b e d i n S e c t i o n 2.5. The i n f o r m a t i o n proved use-f u l i n checking 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 ( i n absence of b l o c k a g e ) . I t a l s o helped e s t a b l i s h the i n f l u e n c e of w a l l confinement on the t o t a l as w e l l as the s k i n f r i c t i o n drag components and t h e i r comparison w i t h s e v e r a l e m p i r i c a l r e l a t i o n s found i n l i t e r a t u r e . One would expect the 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 of the minimum pr e s s u r e p o i n t , the pressure d i s t r i b u t i o n downstream o f i t , as w e l l as the s k i n f r i c t i o n con-t r i b u t i o n . S ince the p r e s s u r e p r o f i l e s do not change beyond R =1000, 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 given blockage i s expected to remain e s s e n t i a l l y c onstant (Figure 3-15a). However, the t o t a l drag would show a drop w i t h an i n c r e a s e i n Reynolds number. T h i s i s p r e c i s e l y the t r e n d shown i n F i g u r e 3-15(b). As can be expected, the e f f e c t o f blockage i s to i n c r e a s e the drag c o e f f i c i e n t because o f l o c a l i n c r e a s e i n the f r e e stream v e l o c i t y . Note, a change i n blockage by approximately 30% changes the drag c o e f f i c i e n t by more than 100%. I t would be u s e f u l t o p o i n t out t h a t both the drag c o e f f i c i e n t and Reynolds number are based here on the average v e l o c i t y i n the t e s t section-. For comparison w i t h a v a i l a b l e r e s u l t s , p l o t s i n F i g u r e 3-16 are i d e a l being based on measured values o f the t o t a l drag and reduced u s i n g the c e n t e r l i n e v e l o c i t y . T h i s i s because most o f the r e s u l t s recorded i n l i t e r a t u r e are obtained u s i n g spheres e i t h e r towed or i n f r e e f a l l c o n d i t i o n . The f i g u r e shows prese n t experimental data together w i t h the standard drag curve 3 4 5 r e s u l t s by S i v i e r , Z a r i n , Ross and W i l l m a r t h , and 6 0 the e m p i r i c a l r e l a t i o n suggested by White . I t was h e a r t e n i n g to note a r a t h e r e x c e l l e n t c o r r e l a t i o n o f the presen t r e s u l t s at s m a l l e r blockage thus s u b s t a n t i a t i n g t h e i r accuracy. 2 . 5 2 . 0 C d p 1 . 5 1 . 0 0 . 5 X X O x x o o • 'O X D, c m S / C % o 3 . 8 1 2 . 7 • 6 . 3 5 7.6 o 7 . 6 2 1 1 . 0 • 8 . 8 9 1 5 . 0 A 1 0 . 1 6 1 9 . 6 • 1 1 - 4 3 2 4 . 8 • 1 2 . 7 0 3 0 . 6 X A m i n z a d e h , S / C < 4 . 9 % • 5 6 7 8 9 10 2 0 3 0 R n x 1 0 ' Figure 3-15 Variation of the measured drag c o e f f i c i e n t with Reynolds number and blockage: (a) pressure drag c o e f f i c i e n t 2 . 5 2 . 0 D , c m S / C % o 3 . 8 1 2 . 7 • 5 . 0 8 4 . 9 o 7 . 6 2 1 1.0 A 1 0 . 1 6 1 9 . 6 • 1 2 . 7 0 3 0 . 6 C d , t 1 . 5 • o • Q 1.0 o mo A A A A A A o °o<5>. : 2 A A A A A O 0 . 5 i i i i i i 11 5 6 7 8 9 1 0 2 0 R n X 1 0 2 3 0 F i g u r e 3-15 V a r i a t i o n o f the measured drag c o e f f i c i e n t w i t h Reynolds number and blockage: (b) t o t a l drag c o e f f i c i e n t . The drag c o e f f i c i e n t i s based on average v e l o c i t y i n the t e s t - s e c t i o n i—• D, c m S/C % o 3.81 2.7 • 6 , 3 5 7.6 o 7.62 11.0 A 1 0.1 6 19.6 • 1 2 . 7 0 30.6 —Standard Drag Curve; Qd t = S iv ier , Zar in ; C^ t 5 x Ross &Wil lmarth j C ^ t 4 5 6 7 8910 20 30 40 50 6070 *10 2 Rn Figure 3-16 Comparison of the pressure and t o t a l drag c o e f f i c i e n t s with the standard drag curve and recent data reported i n l i t e r a t u r e . Note the results are based on the centerline v e l o c i t y : (a) pressure drag c o e f f i c i e n t oo 1.0 D, c m S / C % o 3 . 8 1 2 . 7 • 6 . 3 5 7 . 6 o 7 . 6 2 1 1 . 0 A 1 0 . 1 6 1 9 . 6 • 1 2 . 7 0 3 0 . 6 2 5 10 2 0 3 0 4 0 5 0 6 0 7 0 x i o R n Figure 3-16 Comparison of the pressure and t o t a l drag c o e f f i c i e n t s with the standard drag curve and recent data reported i n l i t e r a t u r e . Note the results are based on the centerline v e l o c i t y : (b) t o t a l drag c o e f f i c i e n t With p r e s s u r e and t o t a l drag i n f o r m a t i o n a t hand i t was convenient to p l o t v a r i a t i o n o f s k i n f r i c t i o n w i t h Reynolds number and blockage. Corresponding r e s u l t s 9 by Achenbach (unconfined flow) near c r i t i c a l end of the Reynolds number range and e m p i r i c a l r e l a t i o n s as suggested by Rosenhead^ and White^^ are a l s o i n c l u d e d f o r comparison (Figure 3-17). The r e s u l t s tend to c o n f i r m the c l a s s i c a l dependence o f s k i n f r i c t i o n on the Reynolds -1/2 number, C-, ^ a R , however, the i n f o r m a t i o n i s not e x t e n s i v e enough to e s t a b l i s h any w e l l d e f i n e d t r e n d f o r the blockage e f f e c t . Achenbach's r e s u l t s near the c r i t i c a l Reynolds number and the p r e s e n t data i n a r e l a t i v e l y lower Reynolds number range can be f i t t e d q u i t e w e l l along the l i n e which corresponds t o C, _ = 2.4 32/(R -6.08) and C J =0. d,f n d,p On the other hand, White and Rosenhead's p r e d i c t i o n s show c o n s i d e r a b l e d i s c r e p a n c y which tends to i n c r e a s e with an i n c r e a s e i n the Reynolds number. n 6.08 R n -0.5 Figure 3-17 F r i c t i o n force as a percentage of the t o t a l drag ISO i—* 122 3.5 Blockage Correction Using Maskell's Theory Maskell developed a theory for blockage correction based on momentum balance between the undisturbed flow upstream of the body and that downstream where e f f e c t i v e wake reaches i t s maximum width, B. For a square plate normal to the flow, and assuming pressure i n the wake to be uniform and equal to the base pressure, P^, the drag c o e f f i c i e n t i s given by C , = m [K 2 - (1 - m S/C) _ 1 ] a where m = B/S. Here K represents r a t i o of the v e l o c i t y on separating streamline to the free stream v e l o c i t y . By hypothesis, he derived an expression for the e f f e c t of blockage on the wake width as JL = 1 _ C d " C d n . / S \ ( K 2 - l ) (K 2 -lAc/ where the subscript c stands for corrected values. This gives the correction formula as K _ i i d .. r /S,2, = 1 + + Q [ ( ) ] K c K 2 - 1 C c 2 where the terms of order (S/C) were considered n e g l i g i b l y small. With th i s the correction for drag and pressure c o e f f i c i e n t s are d i r e c t l y given by 1 - C P_ 1 - C p c I t should be mentioned here t h a t the theory considers i n v a r i a n c e of the separation p o i n t under c o n s t r a i n t , hence Maskell doubted i t s a p p l i c a b i l i t y to well-rounded bodies. However, the c o r r e c t i o n procedure has been q u i t e popular w i t h i n d u s t r i a l aerodynamicists, who have a p p l i e d i t t o s i t u a t i o n s ' t o t a l l y u n r e l a t e d w i t h t h a t considered i n the theory. I t was decided to assess v a l i d i t y of Maskell's c o r r e c t i o n procedure i n the present case. Figure 3-18 shows v a r i a t i o n of pressure i n t e -grated as w e l l as t o t a l measured drag as functions of blockage at R^ = 1000. The c o r r e c t e d values using Maskell's approach are a l s o presented. I t i s apparent t h a t t h i s c o r r e c t i o n procedure i s q u i t e inadequate p a r t i c u l a r l y at higher blockage r a t i o s . The e r r o r was found to vary from 8% to 84% over the blockage r a t i o range of 7.6-30.6%. C o r r e c t i o n r e l a t i o n s f o r t o t a l and pressure drag c o e f f i c i e n t s are summarized i n Figure 3-19. For the convenience of a p p l i c a t i o n , the r e l a t i o n s are s p e c i f i e d i n terms of average as w e l l as c e n t e r l i n e v e l o c i t i e s . 124 1.8 1 .6 1.4 C d . t 1-2 1 .0 0 . 8 1 .4 1.2 1.0 C d , p 0 . 8 0 . 6 • M e a s u r e d T o t a l D r a g o C o r r e c t e d D r a g o 0 o • P r e s s u r e I n t e g r a t e d D r a g o C o r r e c t e d D r a g o o 0 Figure 3-18 1 0 2 0 S / C , % 3 0 Corrected drag c o e f f i c i e n t s showing inadequacy of Maskell's procedure, p a r t i c u l a r l y at higher blockage 3.6 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 provide b e t t e r a p p r e c i a t i o n as w e l l as sub-s t a n t i a t i o n of the c e r t a i n behaviour e x h i b i t e d by the measured data, i t was decided to undertake e x t e n s i v e flow v i s u a l i z a t i o n program. A s e t o f spheres ranging i n diameter from 0.95 - 12.7 cm were used i n the g l y c e r o l -water s o l u t i o n o f 54% c o n c e n t r a t i o n by weight. The main o b j e c t i v e was to observe the formation, development and i n s t a b i l i t y o f the v o r t e x r i n g and the a s s o c i a t e d i n f l u e n c e on the measured pr e s s u r e data. I t was a l s o hoped t h a 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 o f 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 procedure, 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 formation o f the v o r t e x r i n g 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 presented i n F i g u r e 3-20. Numerous photographs were taken a t system-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 formation, symmetric e l o n g a t i o n , onset of 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 presented i n F i g u r e 3-21. The e x i s t e n c e of an axisymmetric, s t a b l e v o r t e x r i n g f o r low Reynolds number i n a stream, c d Figure 3-21 A flow v i s u a l i z a t i o n study showing development and i n s t a b i l i t y of vo r t e x ring with Reynolds number: (a) R =30; (b) R =65; (c) R =115; (d) R =165 M 3 n n n n to 00 130 e s s e n t i a l l y f r e e of macroscopic t u r b u l e n c e i s shown i n F i g u r e s 3-21(a) to 3-21(e). For the Reynolds number above a c r i t i c a l value (corresponding to the f i r s t f ormation of a s t a b l e r i n g , R - 24) , the s t r e a m l i n e s ^' n ' separate from the s u r f a c e and form a c l o s e d r e g i o n immediately behind the sphere. 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 extending to a long d i s t a n c e behind the sphere. The s i z e o f the r i n g i s such as to maintain an e q u i l i b r i u m between the r a t e at 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 stream. As the Reynolds number i s i n c r e a s e d the v o r t e x r i n g becomes elongated i n the flow d i r e c t i o n to m aintain 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 (Figure 3-21). 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 presented e a r l i e r (Figure 3-7). For Reynolds number between 170-230 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 motion i n the r i n g 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 . The s t a t e of unsymmetrical but steady 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 remains 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 e x r i n g c r e a t e s unstable c o n d i t i o n w i t h i n the v o r t e x sheet. B a s i c a l l y , the process 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 r i n g escapes through an opening i n the end o f 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 of the asymmetrical wake about the a x i s o f symmetry. When the v o r t e x s t r e n g t h o f the r i n g reaches a c r i t i c a l v alue, a sudden motion o f the r i n g d i s t u r b s the sheet, which i n t u r n i s r e s p o n s i b l e f o r a r e l e a s e of v o r t i c i t y and a consequent r e t u r n o f the r i n g to i t s o r i g i n a l p o s i t i o n and shape. T h i s phenomenon appears to occur i n the Reynolds number range of about 250 - 300 (Figures 3-21f-h) With f u r t h e r i n c r e a s e i n the Reynolds number, the o s c i l l a t o r y motion of the v o r t e x r i n g assumes higher frequency and the c i r c u l a t i o n w i t h i n the sheet ceases to be symmetrical. In the c y c l e of 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 con 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 flow a x i s w i t h i n the v o r t e x sheet. 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 132 vo r t e x element d i s c h a r g e d i n t o the stream i n t e r a c t s with 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 i g u r e s 3-22 shows a t y p i c a l c y c l e of i n i t i a t i o n , develop-ment and shedding of the r i n g v o r t e x . As mentioned before, 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 concerning 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 analyzed 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 of R n as shown i n F i g u r e 3-23. Note t h a t the s e p a r a t i o n p o i n t moves forward by as much as 20° f o r blockage r a t i o o f 2.7%, over the Reynolds number range of 100-600. For comparison, a v a i l a b l e r e s u l t s by other 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 38 a t t r i b u t e d t o Pruppacher e t a l . r e p r e s e n t s an average 35 value based on h i s own data as w e l l as those by Jensen , Hamielec e t a l . and, Rimon and Cheng 4^. The f i g u r e a l s o shows e f f e c t o f blockage on p o s i t i o n of the s e p a r a t i n g v o r t e x sheet. T y p i c a l photographs of the v o r t e x r i n g a s s o c i a t e d w i t h spheres o f f e r i n g d i f f e r e n t blockage are presented i n F i g u r e 3-24. i n g e n e r a l , f o r a given Reynolds number, the w a l l confinement tends t o move the s e p a r a t i o n p o s i t i o n downstream. I t must be emphasized t h a t the v i s u a l d e t e r m i n a t i o n of s e p a r a t i o n p o i n t i s , 133 Figure 3-22 Ty p i c a l cycle of i n i t i a t i o n , development and shedding of the r i n g vortex at Reynolds number R n = 360 . o • D , c m S/C,% 3 . 8 1 2 . 7 7 - 6 2 1 1 . 0 8 . 8 9 1 5 . 0 1 1 . 4 3 2 4 . 8 A m i n z a d e h T a n e d a 1 ^ 4 7 P r u p p a c h e r e t a l . 3 8 5 6 7 8 9 1 0 R n 1 5 2 0 x 1 0 2 Figure 3-23 Position of separation as affected by Reynolds number and wall confinement 135 b Figure 3-24 Ty p i c a l photographs showing downstream movement of the separation p o s i t i o n due to blockage: (a) R = 170, S/C = 2.7%; (b) R n = 170, S/C = 30.6% -24 T y p i c a l photographs showing downstream movement of the separation p o s i t i o n due to blockage: (c) R = 290, S/C = 2.7%; (d) R = 290, S/C = 30.6% at best, approximate. Considering t h i s and the unstable character of the process, scatter i n the experimental results i s s u r p r i s i n g l y small. 138 3.7 C l o s i n g Comments I t can be s a i d with a measure o f confidence t h a t the experimental programme achieved c o n s i d e r a b l y more than i t s i n i t i a l o b j e c t i v e s . To date the r e s u l t s have been analyzed o n l y w i t h r e f e r e n c e to the immediate goal o f a s s e s s i n g the blockage e f f e c t s . However, there i s a c o n s i d e r a b l e scope f o r f u r t h e r a n a l y s i s and i n t e r p r e t a t i o n of the data which would y i e l d b e t t e r a p p r e c i a t i o n o f a v a r i e t y of aspects a s s o c i a t e d w i t h the fundamental f l u i d mechanics a t low Reynolds number. The t o t a l experience was indeed e x c i t i n g and s a t i s f y i n g because the p r o j e c t p r o v i d e d an exposure to the s o p h i s t i c a t e d experimental i n s t r u m e n t a t i o n and procedures and, more i m p o r t a n t l y , throughout there was a f e e l i n g o f p a r t i c i p a t i o n i n a search f o r knowledge. The awareness o f broader perspec-t i v e s has l e f t me humble f o r I r e a l i z e t h a t a s c i e n t i f i c i n q u i r y i s unending. This i s merely a beginning. 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 o f 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 . 3.7.1 Concluding remarks Important c o n c l u s i o n s based on the experimental r e s u l t s may be summarized as f o l l o w s : 139 (i) The use of average v e l o c i t y i n the test-section (based on the mean flow rate) as a reference v e l o c i t y together with the pressure c o e f f i c i e n t defined as promises to promote r e p e a t a b i l i t y and comparison of data by other investigators regardless of the test f a c i l i t i e s used. This approach tends to compensate for blockage 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 possible errors i n pressure measurements caused by e l e c t r i c a l d r i f t of the pressure measuring system. ( i i ) A v e r t i c a l stem supporting the spherical model has ne g l i g i b l e influence on the pressure d i s t r i b u t i o n i f the sphere to stem diameter r a t i o i s greater than 10. ( i i i ) For pressure d i s t r i b u t i o n on the surface of a sphere, the e f f e c t of Reynolds number i s e s s e n t i a l l y confined to the region downstream of zero pressure point and even here i t i s li m i t e d to Rn < 1000, except for the very high blockage r a t i o of 30.6%. In general, the e f f e c t of Reynolds number i s to increase the minimum as well as the wake pressure. Furthermore, locations of the minimum pressure and separation tend to s h i f t l i t t l e upstream. 140 (iv) 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 of a sphere, the confinement e f f e c t s are e s s e n t i a l l y n e g l i g i b l e up to around 11% blockage. But beyond t h a t i t 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 pressure 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 blockage. (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 pressure 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 Aminzadeh's r e s u l t s ^ and thus tends to s u b s t a n t i a t e 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 . Furthermore, the t o t a l drag compares f a v o u r a b l y w i t h the r e s u l t s , a t small blockage 3-5 . . . . by o t h e r i n v e s t i g a t o r s thus r e i n f o r c i n g c onfidence i n the s t r a i n gage balance. 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 blockage because o f the l o c a l r i s e i n the f r e e stream v e l o c i t y . R e s u l t s a l s o show the c l a s s i c a l dependence of s k i n f r i c t i o n on the Reynolds number, •C„ f « K " 1 / 2 • d, f n (vi) M a s k e l l ' s c o r r e c t i o n procedure f o r drag data i s inadequate to compensate f o r h i g h e r confinement e f f e c t s . ( v i i ) 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 of the flow i n terms o f formation, 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 vortex r i n g . 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 wi t h an i n c r e a s e i n blockage. 141 3.7.2 Recommendation f o r future study As pointed out before, the present e f f o r t s at ob t a i n i n g some a p p r e c i a t i o n as to the physics of the w a l l confinement e f f e c t s at low Reynolds number represent only a modest beginning. There are numerous avenues along which the research program may progress i n f u t u r e . Some of the more important aspects, recommended f o r future s t u d i e s , are summarized below: (i) In the present set of experiments, blockage e f f e c t s on the surface pressure d i s t r i b u t i o n could not be st u d i e d fo r the Reynolds number < 280 , l i m i t a t i o n being imposed by the pressure measuring ins t r u m e n t a t i o n . The surface pressure at lower Reynolds number was found to be so small [0 (10 ^ ) p s i ] t h a t i t presented a problem of measure-ment wi t h an acceptable degree of accuracy and r e p e a t a b i l i t y . Therefore, i t i s suggested t h a t pressure measurements at lower Reynolds number (and higher blockage) should be under-taken to provide a comprehensive p i c t u r e of w a l l c o n f i n e -ment e f f e c t s . This can be accomplished using: (a) more s e n s i t i v e and s t a b l e pressure transducer (e.g., D i g i q u a r t z pressure transducers); (b) a modified d r i v e and the pump system so that higher c o n c e n t r a t i o n of g l y c e r o l - w a t e r s o l u t i o n can be handled. 142 ( i i ) No e f f o r t has been made here t o 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 the sphere, even i n absence o f blockage, i n t h i s range o f Reynolds number remains unrecorded. The i n f o r m a t i o n i s q u i t e important i n comparing r e l a t i v e performance o f d i f f e r e n t p r o s t h e t i c h e a r t v a l v e s , as the parameter d i r e c t l y a f f e c t s deformation, d e s t r u c t i o n and c o a g u l a t i o n o f the red b l o o d c e l l s . ( i i i ) The p r e s e n t study, due to l i m i t a t i o n o f time, was unable to focus a t t e n t i o n on the frequency o f the h e l i c a l v o r t e x shedding. I t would be u s e f u l to e x p l o r e t h i s p e r i o d i c phenomenon i n depth', p a r t i c u l a r l y , when the r e seems to be some q u e s t i o n about i t s v a r i a t i o n with the Reynolds number. Of course, the e f f e c t o f blockage on the S t r o u h a l number has r e c e i v e d no a t t e n t i o n . (iv) Tests should be c a r r i e d out with s p h e r i c a l model under d i v e r s e c o n d i t i o n s of v e l o c i t y p r o f i l e , blockage, t u r b u l e n c e and pressure 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 pr e s s u r e d i s t r i b u t i o n u s i n g prop< d e f i n i t i o n o f the pr e s s u r e c o e f f i c i e n t . (v) Blockage c o r r e c t i o n s f o r b l u f f bodies 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 , sphere, e t c . i n shear flow should p r o v i d e u s e f u l i n f o r m a t i o n . (vi) An important area of i n t e r e s t , which i s p a r t i c u l a r 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 study of p u l s a t i l e flow past b l u f f bodies under wall confinement simulating a t y p i c a l cardiac cycle. This ma; also involve modeling of turbulence character of the b i o l o g i c a l f l u i d flow and e l a s t i c i t y of the aorta. 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 unexplored to date. BIBLIOGRAPHY Torobin, L.B., and Gauvin, W.H., "Fundamental Aspects of Solids-Gas Flow, P a r t I : Introductory Concepts and I d e a l i z e d Sphere Motion i n Viscous Regime," The  Canadian J o u r n a l o f Chemical Engineering, V o l . 37, No. 4, August 1959, pp. 129-141; a l s o "Part I I : The Sphere Wake i n Steady Laminar F l u i d s , " V o l . 37, No. 5, October 1959, pp. 167-176; "Part I I I : A c c e l e r a t e d Motion of a P a r t i c l e i n a F l u i d , " V o l . 37, No. 6, December 1959, pp. 224-236; "Part IV: The E f f e c t s of P a r t i c l e R o t a t i o n , Roughness and Shape," V o l . 38, No. 5, October 1960, pp. 142-153; "Part V: The E f f e c t s of F l u i d Turbulence on the P a r t i c l e Drag C o e f f i c i e n t , " V o l . 38, No. 6, December 1960, pp. 189-200. H e i n r i c h , H.G., Niccum, R.J., and Haak, E.L., "The Drag C o e f f i c i e n t of a Sphere Corresponding to a 'One Meter Robin Sphere' Descending from 260,000 f t A l t i t u d e (Reynolds Nos. 789 to 23,448;Mach Nos. 0.056 to 0.90)," Research and Development of Robin Meteoro- l o g i c a l Rocket B a l l o o n , V o l . I I , Contract AF 19(604)-8034 AD480309, U n i v e r s i t y of Minnesota, Minneapolis, Minn., May 1963. S i v i e r , K.R., "Subsonic Sphere Drag Measurements at Intermediate Reynolds Numbers," Ph.D. Thesis, 1967, The U n i v e r s i t y of Michigan, Ann Arbor, Mich. Z a r i n , N.A., "Measurement o f Non-Continuum and T u r b u l e n c e E f f e c t s on S u b s o n i c S p h e r e D r a g , " Ph.D. T h e s i s , 1969, The U n i v e r s i t y o f M i c h i g a n , Ann A r b o r , M i c h . Ross, F.W.., and Willm a r t h , W.W. , "Experimental Results on Sphere and Disk Drag," AIAA J o u r n a l , V o l . 9, No. 2, February 1971, pp. 285-291. 145 6. B a i l e y , A.B., and H i a t t , J . , "Sphere Drag C o e f f i c i e n t s f o r a Broad Range of Mach and Reynolds Numbers," AIAA J o u r n a l , V o l . 10, No. 11, November 1972, pp. 1436-1440. 7. Goin, K.L., and Lawrence, W.R., "Subsonic Drag of Spheres a t Reynolds Numbers from 200 to 10,000," AIAA  J o u r n a l , V o l . 6, No. 5, May 1968, pp. 961-962. 8. V l n a j i n a c , M., and Covert, E.E., " S t i n g - f r e e Measure-ments o f Sphere Drag i n Laminar Flow," J . F l u i d Mechanics, V o l . 54, P a r t 3, 1972, pp. 385-392. 9. Achenbach, E., "Experiments on the Flow Past Spheres a t Very High Reynolds Numbers," J . F l u i d Mechanics, V o l . 54, P a r t 3, 1972, pp. 565-575. 10. Achenbach, E., "The E f f e c t s o f Surface Roughness and Tunnel Blockage on the Flow Past Spheres," J . F l u i d  Mechanics, V o l . 65, P a r t 1, 1974, pp. 113-125. 11. Maxworthy, T., "Experiments on the Flow Around a Sphere at High Reynolds Numbers," T r a n s a c t i o n s o f the ASME J . of A p p l i e d Mechanics, V o l . 36, No. 3, September 1969, pp. 598-607. 12. M o l l e r , W., " E x p e r i m e n t e l l e Untersuchung zur Hydro-mechanik der Kugel," Phys. Z., V o l . 39, 19 38, pp. 57-80. 13. Cometta, C., "An I n v e s t i g a t i o n of the Unsteady Flow P a t t e r n i n the Wake of C y l i n d e r s and Spheres Using a Hot Wire Probe," Div. Engng, Brown U n i v e r s i t y Tech. Report, WT-21, 1957. 14. Mujumdar, A.S., and Douglas, W.J.M., "Eddy Shedding from a Sphere i n Turbulent Free Stream," Int. J. Heat and Mass Transfer, Vol. 13, 1970, pp. 1627-1629. 146 15. C a l v e r t , J.R., "Some Experiments on the Flow Past a Sphere," Aero. J . Roy. Aero. Soc., V o l . 76, No. 4, 1972, pp. 248-250. Achenbach,E., "Frequency and C o n f i g u r a t i o n of V o r t i c e s Discharged from Sphere at High Reynolds Numbers," Proceedings of the I n t e r n a t i o n a l Symposium on V i b r a t i o n  Problems i n Industry, Keswick, U.K., A p r i l 1973, Paper No. 111. 17. Achenbach, E., "Vortex Shedding from Spheres," J . F l u i d Mechanics, V o l . 62, Part 2, 1974, pp. 209-221. 18. Taneda, S., "Experimental I n v e s t i g a t i o n of the Wake Behind a Sphere at Low Reynolds Numbers," Jo u r n a l of  the P h y s i c a l S o c i e t y of Japan, V o l . 11, No. 10, October 1956, pp. 1104-1108. 19. Magarvey, R.H.,'and Bishop, R.L., " T r a n s i t i o n Ranges fo r Three-Dimensional Wakes," Canadian J . Ph y s i c s , V o l . 39, No. 10, October 1961, pp. 1418-1422. 20. Lee,K., and Barrow, H., "Some Observations on Transport Processes i n the Wake of Sphere i n Low Speed Flow," I n t e r n a t i o n a l J . Heat and Mass Tr a n s f e r , V o l . 8, March 1965, pp. 403-409. 21. Goldburg, A., and Florsheim, B.H., " T r a n s i t i o n and Strouhal Number f o r the Incompressible Wake of Various Bodies," Physics of F l u i d s , V o l . 9, No. 1, January 1966, pp. 45-50. 22. Modi, V.J., and Aminzadeh, M., "Separated Flow Past Spheres at Low Reynolds Number," AIAA 15th Aerospace  Sciences Meeting, Los Angeles, C a l i f . , January 1977, Paper No. 77-134. 23. Stokes, G.G., "On the Theories of the I n t e r n a l F r i c t i o n of F l u i d s i n Motion, and of E q u i l i b r i u m and Motion of E l a s t i c S o l i d s , " Trans. Camb. P h i l . S o c , V o l . 9, Pt. I I , 1851, pp. 8-106. 147 24. Whitehead, A.N., "Second Approximations to Viscous F l u i d Motion," Quart. J . Math., V o l . 23, 18 89, pp. 143-152. 25. Proudman, I . , and Pearson, J.R.A., "Expansion a t Small Reynolds Number f o r the Flow Past a Sphere and C i r c u l a r C y l i n d e r , " J . F l u i d Mechanics, V o l . 2, Pt. 3, May 1957, pp. 237-262. 26. Oseen, C.W. , "Uber Die Stokes 1 sche Formal und liber e i n e Verwandte Aufgube i n der Hydrodynamik," Ark, f o r Mat. A s t r . 0 F r s i k . , V o l . 6, No. 29, September 1910. 27. G o l d s t e i n , S., "The Steady Flow of Viscous F l u i d Past a F i x e d S p h e r i c a l O b stacle at Small Reynolds Numbers," Proc. Roy. S o c , S e r i e s A, V o l . 12 3, No. 791, March 1929, pp. 225-235. 28. T o n i o t i k a , S., and A o i , T., "The Steady Flow of Viscous F l u i d Past a Sphere and C i r c u l a r C y l i n d e r at Small Reynolds Numbers," Quart. J . Mech. Appl. Math., V o l . 3, Pt. 2, 1950, pp. 140-161. 29. Pearcey, T., and McHugh, B., " C a l c u l a t i o n o f Viscous Flow Around Spheres at Low Reynolds Numbers," P h i l o s o p h i c a l Magazine, Ser. 7, V o l . 46, No. 378, J u l y 1955, pp. 783-794. 30. K a w a g u t i , M., "An A p p r o x i m a t e S o l u t i o n f o r V i s c o u s Flow a t Low S p e e d s , " Tokyo I n s t i t u t e o f S c i e n c e and T e c h n o l o g y R e p o r t , V o l . 2, May-June 194 8, pp. 66-71. 31. K a w a g u t i , M., " N u m e r i c a l S o l u t i o n f o r t h e V i s c o u s Flow P a s t a S p h e r e , " Tokyo I n s t i t u t e o f S c i e n c e and  T e c h n o l o g y R e p o r t , V o l . 4, May-June 1950, pp. 154-158. 32. Fox, L., "A Short Account of R e l a x a t i o n Methods," Quart. J . Mech. Appl. Math., V o l . 1, 1948, pp. 253-280. 148 33. Fox, L., and Southwell, R.V., " R e l a x a t i o n Methods A p p l i e d to E n g i n e e r i n g Problems," P h i l . Trans., S e r i e s A, V o l . 239, No. 810, October 1945, pp. 419-460. 34. A l l e n , D.N., de G., and Dennis, S.C.R., "The A p p l i c a t i o n of R e l a x a t i o n Methods to the S o l u t i o n o f D i f f e r e n t i a l Equations i n Three Dimensions," Quart. J . Mech., V o l . 4, P a r t 2, 1951, pp. 199-208. 35. Jenson, V.G., "Viscous Flow Round a Sphere a t Low Reynolds Numbers (<40)," Proc. Roy. Soc., S e r i e s A, V o l . 249, No. 1257, January 1959, pp. 346-366. 36. Hamielec, A.E., Hoffman, T.W., and Ross, L.L., "Numer-i c a l S o l u t i o n o f the Navier-Stokes Equation f o r Flow Past Spheres, P a r t I - V i s c o u s Flow Around Spheres w i t h and without R a d i a l Mass E f f l u x , " J . of The American  I n s t i t u t e o f Chemical E n g i n e e r s , V o l . 13, No. 2, March 1967, pp. 212-219. 37. L e C l a i r , B.P., Hamielec, A.E., and Pruppacher, H.R., "A Numerical Study of the Drag on a Sphere at Low and Intermediate Reynolds Numbers," J . Atmospheric Sciences, Vol. 27, No. 2, March 1970, pp. 308-315. 38. Pruppacher, H.R., LeClair, B.P., and Hamielec, A.E., "Some Relations Between Drag and Flow Pattern of Viscous Flow Past a Sphere and a Cylinder at Low and Intermediate Reynolds Numbers," J . F l u i d Mechanics, Vol. 44, Pt. 4, 1970, pp. 781-790. 39. Dennis, S.C.R., and Walker, M.S., "The Steady Motion o f a Viscous F l u i d Past a Sphere," Aero. Res. C o u n c i l  Report, No. 26105, August 1964. 40. Rimon, Y., and Cheng, S.I., "Numerical S o l u t i o n o f a Uniform Flow Over a Sphere a t Intermediate Reynolds Numbers," P h y s i c s of F l u i d s , V o l . 12, No. 5, May 1969, pp. 949-959. 149 41. Dennis, S.C.R., and Walker, J.D.A., "Calculation of the Steady Flow Past a Sphere at Low and Moderate Reynolds Numbers," J. F l u i d Mechanics, Vol. 48, Part 4, 1971, pp. 771-789. 42. Van Dyke, M., Perturbation Methods i n F l u i d Mechanics, Academic Press, New York and London, 1964. 43. Modi, V.J., and Aminzadeh, M., "Hydrodynamic Per-formance of an A r t i f i c i a l A o r t i c Valve Implant," Proceedings of F i f t h Canadian Congress of Applied Mechanics, Fredericton, May 19 75. 44. Aminzadeh, M., and Modi, V.J., "A Theoretical Approach to the Wedge Shaped Hot Film Probe Performance," Aero. J. Roy. Aero. S o c , Vol.80, No. 791, November 1976, pp. 489-491. 45. Modi, V.J., and Aminzadeh, M., " F l u i d Mechanics of an Aort i c Heart Valve Implant," Proceedings of the 1977 Symposium on Biomechanics, ASME, New Haven, Conn., June 1977, Editor: Richard Skalak, pp. 137-140. 46. Modi, V.J., and Aminzadeh, M., " F l u i d Mechanics of O s c i l l a t i n g Spherical Poppet Used i n Starr-Edwards Heart Valve Prosthesis," Trans. ASME, Journal of  Biomechanics, i n press. 47. Aminzadeh, M., "Hydrodynamic Performance of an A r t i f i c i a l Aortic Valve Implant," Ph.D. Thesis, 1975, The University of B r i t i s h Columbia, Vancouver, B r i t i s h Columbia. 48. Rasmussen, C.G., "The A i r Bubble Problem in Water Flow Hot-Film Anemometry," Pisa Information, No. 5, June 1967, pp. 21-26. 49. Morrow, T.G., "Effects of Di r t Accumulation on Hot-wire and Hot-Film Sensors," F l u i d Dynamic Measurements  in the Industrial and Medical Environments, Proceed-ing of the Disa Conference held at the University of Leicester, Vol. 1, A p r i l 1972, pp. 122-124. 150 50. Pinchon, J . , "Comparison o f Some Methods of C a l i b r a -t i n g H o t - F i l m Probes i n Water," P i s a Information, No. 10, October 1970, pp. 15-21. 51., Kalashnikov, V.N., and Kudin, A.M., " C a l i b r a t i o n o f Hot- F i l m Probes i n Water and i n Polymer S o l u t i o n s , " P i s a Information, No. 14, March 1973, pp. 15-18. 52. B a t c h e l o r , G.K., An I n t r o d u c t i o n to F l u i d Pynamics, Cambridge U n i v e r s i t y P r e s s , 1967, pp. 201-204. 53. B a t c h e l o r , G.K., and Townsend, A.A., "Pecay o f Turbulence i n the F i n a l P e r i o d , " Proc. Roy. Soc., S e r i e s A, V o l . 194, No. 1039, November 194 8, pp. 527-543. 54. Taback, I . , "The Response o f Pressure Measuring Systems to O s c i l l a t i n g P r e s s u r e s , " NACA TN 1819, 1949. 55. I b e r a l l , A.S., " A t t e n u a t i o n o f O s c i l l a t o r y Pressure i n Instrument L i n e s , " J o u r n a l o f Research, N a t i o n a l Bureau o f Standards, V o l . 45, J u l y 1950, pp. 85-108. 56. P'Souza, A.F., and Oldenburger, R., "Dynamic Response of F l u i d L i n e s , " Trans. ASME, S e r i e s D, J o u r n a l o f B a s i c  E n g i n e e r i n g , V o l . 86, No. 3, September 1964, pp. 589-598. 57. Grove, A.S., S h a i r , F.H., Petersen, F.E., and A c r i v o s , A., "An Experimental I n v e s t i g a t i o n o f the Steady Separated Flow Past a C i r c u l a r C y l i n d e r , " J . F l u i d  Mechanics, V o l . 19, Pt. 1, May 1964, pp. 60-80. 58. A c r i v o s , A., L e a l , L.G., Snowden, D.D., and Pan, F., "Fu r t h e r Experiments on Steady Separated Flows Past B l u f f O b j e c t s , " J . F l u i d Mechanics, V o l . 34, Pt. 1, 1968, pp. 25-48. 59. Modi, V.J., and E l - S h e r b i n y , S.E., "A Fre e - S t r e a m l i n e Model f o r B l u f f Bodies i n Confined Flow," Trans. ASME, J o u r n a l o f F l u i d E n g i n e e r i n g , i n p r e s s . 151 60. White, F.M., Viscous F l u i d Flow, McGraw-Hill, New York, 1974, pp. 208-210. 61. Rosenhead, L., Laminar Boundary Layers, Oxford U n i v e r s i t y P r e s s , 1963, pp. 102-109. 152 APPENDIX I CONVENTIONAL PRESSURE COEFFICIENT C~ IN TERMS P OF MEASURED INFORMATION In a low Reynolds number experiment using a l i q u i d tunnel, d i f f i c u l t i e s i n establishing c h a r a c t e r i s t i c reference v e l o c i t y and pressure, P and U , were discussed J c ' CO oo before. However, for comparison and to es t a b l i s h r e l a t i v e effectiveness of the new d e f i n i t i o n of pressure c o e f f i c i e n t , one can calculate the conventional pressure c o e f f i c i e n t — 2 Cp = (P Q - POT) (l/2)pU o o quite r e a d i l y using d i f f e r e n t i a l pressure data measured during the experiment. The x component of Navier-Stokes equation along the stagnation streamline y =0 can be written as 9u 1 3P ± r9 2u ^ 9 2u, dx p 3x L 2 2J 9x 9y Integrating from front stagnation point to minus i n f i n i t y upstream of the sphere y i e l d s , 0 , r0 f0 .2 „2 r ^ 1 9P -i . r3 u , 9 u, , 9u , = - — . • dx + v [ — ~ + ~] dx u 7 7 — dx p 9x „ 2 2J 9x ^ J - o o J - o o 9x 9y ' — C O u 2 u oo 2 153 u + + V 2 2 r 3 U . 3 U n -j L ^ + —=rJ d x 3x dy' P n - P 0 °° 1/2 p U" = 1 + 1/2 u' s 2 r d U 9 U , , l =T + — T - J dx -6 3x 3y r - ^ * 2 1 r 3 u + sr] dx J-<*> 3x 2 3y 2 where 6 i s the boundary-layer thickness. The second i n t e g r a l vanishes because of i r r o t a t i o n a l i t y of the outer flow while in the f i r s t i n t e g r a l usual boundary layer approximation 2 2 3 u 3 u 2 2 3x 3y^ can be introduced. Since 3u 3x 9v 3y at x = 0 0 oo 1/2 p U 2 = 1 - 1/2 U 9u 2~ 3x x = 6 y = 0 " 1 + 5 + where A i s a constant and R i s the Reynolds number. 154 Here the numerical value o f A f o l l o w s d i r e c t l y from the oute r flow s o l u t i o n . Using the p o t e n t i a l flow a n a l y s i s , Homann as w e l l as 57 Grove e t a l . have shown, independently, the value o f A to be 8. Thus, P - P ? = 1 + R + ' 1/2 p IT ' oo i . e . , E - - P60° = ( P 0 " P 6 0 ^ " + ••••)1/2 P U2 . Now P e - p ~ _ p e - p 6 0 ° p o P 6 0 ° + ( 1 + 8 + 1/2 p U 2 1/2 p Uf 1/2 p U 2 R Recognizing t h a t f o r a Reynolds number as low as 300 con-t r i b u t i o n of 8/R term i s l e s s than 3%, the above e x p r e s s i o n reduces to p e " p ~ ( p e - p 6 ( ^ - ( p o - f 6 t f ! _ + ± 1/2 p U 2 1/2 p U 2 155 APPENDIX I I A PROCEDURE FOR DRIFT CORRECTION USING DIRECT MEASUREMENT OF THE DIFFERENTIAL PRESSURE Le t 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 , AP Q = P Q - P between the l o c a t i o n o f i n t e r e s t and 6 9 r the p r e s s u r e a t a r e f e r e n c e p o i n t . L e t 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 I I - l . The diagram a l s o shows the corres 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 ssure AP Q . From the f i g u r e i t f o l l o w s t h a t : AP 0 AP 9 + 61 + 62 AP 6 Ther e f o r e , AP 9 + AP e AP 6 = ( A P e + 6 1 ) + ( A P Q + 6 1 + 6 2 + 6 3 ) 1 3 2 - ( A P g + S1 + A P Q + 6_ + 5-j ; U JL o 156 Figure I I - l A d r i f t correction procedure using d i r e c t measurement of the d i f f e r e n t i a l pressure Now 157 i . e . , A P Q = A P Q + A P Q - A P Q - 6, - 6_ . (II.1) fc) O U 3 O 2 J- J 6 2 = A P e 2 " A p 9 l ' 6 3 = A P e 3 - A p e 2 6 2 + S = A P e 2 " A p e i + A P e 3 " A P e 2 = AP - AP 6 3 9 1 Assuming the d r i f t to be l i n e a r over a sma l l time i n t e r v a l , 6 2 + 6 3 AP„ - A P , Hence the d e s i r e d d i f f e r e n t i a l pressure AP AP - AP 6 3 9 1 AP„ - AP 6 3 6 2 = I A p 0 1 " k A p e 3 1 5 8 T h u s d e t e r m i n a t i o n o f t h e d i f f e r e n t i a l p r e s s u r e A P n i n v o l v e s t h e m e a s u r e m e n t o f A P Q a n d A P Q . I n p r a c t i c e e e i e 3 i t w a s f o u n d t o b e e x t r e m e l y d i f f i c u l t t o a c c o m p l i s h t h e s e m e a s u r e m e n t s w i t h a n a c c e p t a b l e a c c u r a c y . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0094042/manifest

Comment

Related Items