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

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WALL CONFINEMENT  EFFECTS FOR SPHERES IN THE  NUMBER RANGE OF  REYNOLDS  30-2000  by  TOSHINOSUKE AKUTSU B.A. S c . , K a n t o  Gakuin U n i v e r s i t y ,  1970  M.A.  Gakuin U n i v e r s i t y ,  1972  Sc., Kanto  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF:GRADUATE  Department  We  STUDIES  of Mechanical Engineering  accept t h i s  thesis  as conforming t o t h e  required standard  THE UNIVERSITY OF BRITISH COLUMBIA June,  1977  (cP) T o s h i n o s u k e A k u t s u , 1977  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the  requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s under-  stood t h a t p u b l i c a t i o n , i n p a r t o r i n whole, o r the copying o f t h i s t h e s i s f o r f i n a n c i a l g a i n allowed without my w r i t t e n  s h a l l not be  permission.  TOSHINOSUKE AKUTSU  Department o f Mechanical E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver, B r i t i s h Columbia Canada, V6T 1W5  ABSTRACT  This and  thesis  studies i n d e t a i l : formation,  i n s t a b i l i t y o f the v o r t e x r i n g ;  pressure the  distribution;  and  drag  R e y n o l d s number r a n g e o f  ratio  of  3-30%.  solution described  i n the  f o l l o w e d by  t e s t procedures.  so c r i t i c a l definition less is  a t low of  d e p e n d e n t on  functions of The to  the  test  Finally,  approach  facilities the  test  suggest  negligible.  (limit of  support balance  data r e d u c t i o n ,  i s d i s c u s s e d and  and  t h a t the  s h o u l d be  distribution  pressure  a  are analyzed  r a t i o o f the 10  is primarily  30.6%, t h e p r e s s u r e for R  tunnel c a p a b i l i t y  n  to  be  as  Reynolds  at least  new  gradients  number.  model  t o make stem on  confined to  However, f o r t h e model w i t h  r a t i o of  the  program i s b r i e f l y  c o n d i t i o n and  show R e y n o l d s number d e p e n d e n c y  2300  a glycerol-water  t o the  data  in  blockage  I n f l u e n c e o f R e y n o l d s number  surface pressure  highest blockage  the  c o e f f i c i e n t which promises  the confinement results  and  experimental  R e y n o l d s number,  the range R < 1000. • n  to  An  v e r t i c a l stem s u p p o r t  effects  30-2000  instrumentation, drag  the p r e s s u r e  evolved.  f o r a f a m i l y of spheres  an e x p l a n a t i o n o f t h e model  system, p r e s s u r e measuring and  associated surface  In the b e g i n n i n g ,  t u n n e l used  developmen  the  continues  as h i g h  as  for a glycerol-water  iii c o n c e n t r a t i o n used).  In g e n e r a l , the e f f e c t o f Reynolds  number i s to i n c r e a s e the minimum as w e l l as the wake pressure.  On the other hand, the e f f e c t o f an  i n the blockage r a t i o i s j u s t the o p p o s i t e .  increase  The w a l l c o n f i n e -  ment tends t o i n c r e a s e 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 o f s k i n f r i c t i o n on the Reynolds 1/2  number, C-, a R ' , i s maintained. a, r n c  showed inadequacy o f M a s k e l l ' s p a r t i c u l a r l y a t h i g h e r blockage An e x t e n s i v e  The r e s u l t s  vividly  c o r r e c t i o n procedure (S/C > 5%).  flow v i s u a l i z a t i o n  study u s i n g dye  i n j e c t i o n i n c o n j u n c t i o n w i t h high speed photography complements the t e s t program.  iv  TABLE OF CONTENTS Chapter  1.  2.  Page  INTRODUCTION .  1  1.1  P r e l i m i n a r y Remarks  1  1.2  Survey o f L i t e r a t u r e  2  1.3  Sphere Flow F i e l d  12  1.4  The Plan of Study  14  EXPERIMENTAL APPARATUS AND  TEST  PROCEDURES  3.  21  2.1  Glycerol  Tunnel  22  2.2  Hot F i l m Anemometry and  Velocity  Profiles  27  2.3  Models and Support System  39  2.4  P r e s s u r e Measurements  44  2.5  Drag Measurements  59  2.6 Flow V i s u a l i z a t i o n RESULTS AND DISCUSSION  65 70  3.1  Choice o f Reference V e l o c i t y Pressure  3.2  E f f e c t o f Reynolds Number  89  Wall Confinement E f f e c t s  97  •3.3  and  .  71  3.4  Drag C o e f f i c i e n t  114  3.5  Blockage C o r r e c t i o n Using M a s k e l l ' s Theory  122  3.6  Flow V i s u a l i z a t i o n Analysis  126  and Near-Wake  Chapter  Page 3.7  Closing  Comments  . .  138  3.7.1  C o n c l u d i n g remarks  13 8  3.7.2  Recommendation work  141  f o r future  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 P  . .  155  vi  L I S T OF  FIGURES  Figure  1-1  1-2  Page  A summary o f the scope of i n the f i e l d  literature indicating recent important contributions of flow p a s t a sphere . . . . . .  15  S t a r r - E d w a r d s p r o s t h e s i s and i t s e x p l o d e d v i e w : (a) c a g e ; (b) b a l l o r p o p p e t ; (c) s e a t ; (d) s u t u r e r i n g . . . . . . . . .  16  A s c h e m a t i c diagram showing the S t a r r Edwards p r o s t h e s i s o c c u p y i n g 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 b l o c k a g e  18  1- 4  The  20  2- 1  A s c h e m a t i c diagram showing the water s o l u t i o n tunnel  1-3  2-2 2-3  2-4  plan of  study  Calibration plot o r i f i c e meter  f o r the  sharp  glycerol25 edge 26  A p h o t o g r a p h 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 p r o b e : C, c o n s t a n t t e m p e r a t u r e anemometer; D, d r i v e w h e e l ; M, drive motor; P, p r o b e ; R, r o t a t i n g d i s h ; V, d.c. d i g i t a l voltmeter . . . . C a l i b r a t i o n data f o r h o t - f i l m probe the g l y c e r o l - w a t e r s o l u t i o n of C = 55.7%  in .  n  2-5  2-6  2-7  30  32  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 t e m p e r a t u r e on p r o b e ' 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 of d i f f e r e n t concentration  33  I n s t r u m e n t a t i o n l a y o u t used d u r i n g p r o f i l e measurements .  35  velocity  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 station y = 83 cm i n a b s e n c e o f the spherical^models  the 36  vii Figure 2-8  2-9  2-10  2-11  2-12  2-13  Page E f f e c t o f w a l l confinement on the upstream velocity profile: (a) S/C = 7.6%; . . (b) S/C = 30.6%  37 38  V a r i a t i o n o f the c e n t e r l i n e v e l o c i t y as measured by the h o t - f i l m probe w i t h the average v e l o c i t y as given by the flowmeter data  40  A photograph showing s p h e r i c a l models used i n the experimental program c o v e r i n g the blockage r a t i o range of 2.7 - 30.6%  42  E f f e c t o f the stem diameter on measured s u r face p r e s s u r e d i s t r i b u t i o n over a sphere: (a) Sphere diameter = 5.08 cm; • . . (b) Sphere diameter = 6.35 cm  45 46  P l o t s showing s t e m - e f f e c t s to be r e l a t i v e l y independent o f the Reynolds number i n the range 300-1000: (a) D = 5.08 cm, d. = 1.32 mm, d =4. 76 mm; • • (b) D = 5.08 cm, d^ = 4.31mm, d^= 12.70 mm . .  47 48  V a r i a t i o n o f the minimum and base p r e s s u r e s showing the c r i t i c a l v a l u e o f d /D o A schematic diagram o f the B a r o c e l p r e s s u r e transducer  49  3  2-14 2-15 2-16 2-17  2-18  52  A procedure f o r compensation o f the e l e c t r o n i c d r i f t o f the p r e s s u r e measuring system . . .  55  A l i n e drawing o f the i n s t r u m e n t a t i o n set-up used f o r p r e s s u r e measurements .........  58  A schematic diagram showing the s p h e r i c a l model and i t s support system d u r i n g p r e s s u r e measurements  60  An exploded view o f the drag measuring balance: (a) s u p p o r t i n g stem; (b) i n t e r m e d i a t e c o n n e c t i o n ; (c) c e n t r a l suspension b l o c k ; (d) c a n t i l e v e r w i t h s t r a i n gages; (e) needle b e a r i n g s supporti n g the c e n t r a l block 62  viii  Figure 2-19  2-20 2- 21 3- 1  3-2  P g a  (a) D r a g b a l a n c e a s s e m b l y w i t h b r i d g e a m p l i f i e r meter; (b) C l o s e - u p v i e w o f t h e d r a g b a l a n c e Dye i n j e c t i n g visualization  probe used .•  during  e  . . . .  63  . . . .  64  flow 66  A s k e t c h showing t h e e q u i p m e n t l a y o u t during flow v i s u a l i z a t i o n  67  An i l l u s t r a t i o n s h o w i n g 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 t h e velocity profile  75  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 o f 2 p r e s s u r e c o e f f i c i e n t , Cp = (Pg - P ) / ( 1 / 2 ) p u . Note the p r e s s u r e c o e f f i c i e n t i s zero i n t h e v i c i n i t y o f 6 = 60°  78  P l o t s showing s e n s i t i v i t y o f d i f f e r e n t d e f i n i t i o n s f o r 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 ) / ( l / 2 ) p U ;  80  (b) C  81  a  3-3  2  p  ( c )  3-4  c  0  = (P  p  p  =  ( p  0 0  -P  9  ) / ( l / 2 ) pU ; 2  6 Q O  e- 60p  , / ( p  o- 6o«> p  • - '  )  E f f e c t o f R e y n o l d s number on d i s t r i b u t i o n i n t e r m s of:„ (a) C = ( P - P j / ( l / 2 ) pU ;  8 3  surface  pressure 84  2  p  (b) C ( C )  3-5  e  =  p  %  (P  - P j / d / 2 ) pU ,;  ( e- 60°  =  P  85  2  Q  P  ) / ( P  0- 60o P  86  }  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) b a s e d on a v e r a g e v e l o c i t y , C = (P - P ) / (1/2) p U ; P .' . . . °°. . (b) s u g g e s t e d 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 fl  a  s  c  p  =  ^ e - ^ o ^ / ^ o - ^ o ^  •  •  8  87 8  ix  Figure 3-6  Page  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 on s p h e r e s using P^QO reference. Note t h e p l o t s show v e r y l i t t l e d e p e n d e n c e o n (a) v e l o c i t y p r o f i l e ; (b) R e y n o l d s number; (c) b l o c k a g e . . a  3-7  s  S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by R e y n o l d s number a t a s m a l l b l o c k a g e r a t i o o f 4.9%: ( a )  C P  =  (  p  e -  p  6 0 °  )  /  (  p  o - > °  )  ;  (b)  C  (c)  c o m p a r i s o n w i t h r e c e n t d a t a by o t h e r investigators. Note t h e r e s u l t s by Maxworthy and A c h e n b a c h a r e n e a r t h e ^ c r i t i c a l R e y n o l d s number ( R , = 3.7 x 1 0 , R e f e r e n c e 9)  p  = (P - P j / ( l / 2 ) plT;  . .  e  n  3-8  c r  P r e s 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 t o R e y n o l d s number _> 1000 and f o r intermediate values of blockage: (a) C , S/C = 11.0%;  98  '(b)- C , S/C = 15.0%;  • •  99  (c)  . .  100  p  C , S/C = 19.6%;  (d) C^", S/C = 19.6% 3-9  3-10  3-11  96  101  R e y n o l d s number e f f e c t on t h e p r e s s u r e 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%; (b)  C ,  S/C = 30. 6% ;  (c)  Cp, S/C = 30.6%  p  dis102  • .  103  . . . . .  104  R e 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 t o 11% . . . . Pressure  plots  as a f f e c t e d  by h i g h e r  106  blockage:  (a) R  n  i 950;  (b) R  n  = 1250;  108  (c)  n  = 1600  109  R  .  107  X  Figure 3-12  Page E f f e c t o f w a l l c o n f i n e m e n t on t h e minimum and b a s e p r e s s u r e s , 1000 < R < 1600. Note b o t h Cp and C are e s s e n t i a l l y constant up t o "b t h e b l o c k a g e r a t i o o f a r o u n d 13%  110  P l o t s s h o w i n g d e p e n d e n c e o f Cp (and h e n c e Cp, - Cp ) on w a l l c o n f i n e m e n t , e v e n when S/C i s l e s s " t h a n 131, a t R < 1000  112  Condensation o f the base p r e s s u r e d a t a s h o w i n g t h e i n f l u e n c e o f R e y n o l d s number and b l o c k a g e  113  n  p  3-13  m  1  n  3-14  3-15  3-16  3-17 3-18  3-19 3-20  V a r i a t i o n o f t h e measured d r a g c o e f f i c i e n t w i t h R e y n o l d s number and b l o c k a g e : (a) p r e s s u r e d r a g c o e f f i c i e n t ; (b) t o t a l d r a g c o e f f i c i e n t . The d r a g c o e f f i c i e n t i s b a s e d on a v e r a g e v e l o c i t y i n the t e s t - s e c t i o n  116 117  C o m p a r i s o n o f t h e p r e s s u r e and t o t a l d r a g c o e f f i c i e n t s with the standard drag curve and r e c e n t d a t a r e p o r t e d i n l i t e r a t u r e . N o t e t h e r e s u l t s a r e b a s e d on t h e c e n t e r line velocity: (a) p r e s s u r e d r a g c o e f f i c i e n t ; (b) t o t a l d r a g c o e f f i c i e n t  118 119  F r i c t i o n force t o t a l drag  121  as a p e r c e n t a g e  of the  C o r r e c t e d d r a g c o e f f i c i e n t s showing inadequacy o f Maskell's procedure, p a r t i c u l a r l y a t higher blockage  124  Empirical drag  125  correction  formulae  f o r sphere  A t y p i c a l photograph i l l u s t r a t i n g of a v o r t e x r i n g behind a sphere  formation 127  xi Figure 3-21  Page A f l o w v i s u a l i z a t i o n study showing d e v e l o p ment and i n s t a b i l i t y o f v o r t e x r i n g w i t h Reynolds number: (a) R = 3 0 ; (b)  (c) (d) (e) (f) (g) (h)  3-22  3-23 3-24  R R R R R R  n n n n n  = =. = = = =  65; 115; 165; . , . . 221; 265; 280; 280 . .  n T y p i c a l c y c l e o f i n i t i a t i o n , development and shedding o f t h e r i n g v o r t e x a t Reynolds number R = 360 n. P o s i t i o n o f s e p a r a t i o n as a f f e c t e d by Reynolds number and w a l l confinement . . . T y p i c a l photographs showing downstream movement o f t h e s e p a r a t i o n p o s i t i o n due to b l o c k a g e  128 128 128 128 129 129 129 129  133 134 136  xii LIST OF APPENDIX FIGURES Figure II-l  Page A d r i f t c o r r e c t i o n procedure u s i n g d i r e c t measurement of t h e d i f f e r e n t i a l p r e s s u r e .  156  ACKNOWLEDGEMENT  I would l i k e t o take t h i s o p p o r t u n i t y  t o express  my g r a t i t u d e and s i n c e r e thanks t o P r o f e s s o r V.J. Modi f o r the e n t h u s i a s t i c guidance given throughout the r e s e a r c h program and h e l p f u l suggestions the t h e s i s .  d u r i n g the p r e p a r a t i o n o f  H i s help and encouragement have been  invaluable. The  c h e e r f u l a s s i s t a n c e o f the t e c h n i c a l s t a f f i s  g r a t e f u l l y acknowledged.  Their s k i l l f u l  g r e a t l y a c c e l e r a t e d the r e s e a r c h The  assistance  program.  i n v e s t i g a t i o n was supported by the N a t i o n a l  Research C o u n c i l o f Canada, Grant No. A-2181. F i n a l l y , s p e c i a l a p p r e c i a t i o n i s extended t o my parents,  Mr. and Mrs.  Toshi Akutsu, f o r t h e i r encourage-  ment, love and understanding d u r i n g d i f f i c u l t  times.  xi  L I S T OF  cross-sectional theory  SYMBOLS  a r e a o f t h e wake i n M a s k e l l ' s  v e l o c i t y o f sound tunnel cross-sectional  area  F /(l/2)pU S 2  t  p r e s s u r e d r a g c o e f f i c i e n t based on average v e l o c i t y p r e s s u r e d r a g c o e f f i c i e n t based on velocity t o t a l drag c o e f f i c i e n t , 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 centerline velocity —' d  + +  /  centerline  ^ based on based on  t  —'  s k i n f r i c t i o n component o f t o t a l d r a g , based on average v e l o c i t y s k i n f r i c t i o n component o f t o t a l d r a g , based on centerline velocity percentage concentration of g l y c e r o l - w a t e r by w e i g h t  solution  ^e-V^o-V (Pg " P j / ( l / 2 ) (P  -PJ/(1/2) U  2  e  (P  - P )/(l/2)pU  2  e  P  r  base p r e s s u r e c o e f f i c i e n t ,  (P^ - P ) / (Pg - P ) ' r  minimum p r e s s u r e c o e f f i c i e n t , ( P - P ) / ( P Q - P ) m  r  r  2  base p r e s s u r e c o e f f i c i e n t ,  (P^ - P ^ ) / ( 1 / 2 ) p U  XV  D  sphere  diameter  d.  i n s i d e diameter  d  o u t s i d e diameter  1  o  o f stem of stem  f  frequency  F  s e c t i o n a l pressure  F^_  total sectional  k  s i z e o f roughness element  M  n  P P  vortex  sheet  drag  drag  Mach number, U/c static  fl  of shedding  .  pressure  s t a t i c pressure on the s u r f a c e o f the sphere a t an angle 6 from the f r o n t s t a g n a t i o n p o i n t  P  s t a t i c p r e s s u r e of u n d i s t u r b e d  P^  base pressure  P  s t a t i c p r e s s u r e a t r e f e r e n c e tap, i n the present case r = 60° 2 dynamic p r e s s u r e , (l/2)pU  q R ,R  c o l d and o p e r a t i n g r e s i s t a n c e respectively  R  Reynolds number, UD/v  n  stream  o f the hot f i l m probe,  R n, c r  c r i t i c a l Reynolds number  S  diametral cross-sectional  S n  fD/U  T  temperature of s o l u t i o n  U  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 a flow r a t e as given by the o r i f i c e meter  U  centerline  u  1  J  area  velocity  rms value of v e l o c i t y f l u c t u a t i o n s i n the downstream d i r e c t i o n  local  v e l o c i t y as measured by a hot f i l m  d.c. v o l t a g e output of a constant anemometer probe d i s t a n c e from the tunnel  probe  temperature  inlet  l o c a t i o n o f model from the t u n n e l  inlet  v e r t i c a l c o o r d i n a t e with o r i g i n at the bottom of the t e s t s e c t i o n zero d r i f t of the e l e c t r o n i c pressure measuring system angular l o c a t i o n of a pressure to the f r o n t s t a g n a t i o n p o i n t  tap with  reference  angular 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 w i t h r e s p e c t to the r e a r s t a g n a t i o n p o i n t roughness parameter,  k/D  dynamic v i s c o s i t y kinematic  viscosity,  u/p  density component of t u r b u l e n t i n t e n s i t y i n the downstream d i r e c t i o n , u /U 1  s k i n f r i c t i o n , shear differential  s t r e s s a t the w a l l  pressure at o r i f i c e meter  1  1.  1.1  Preliminary  Remarks  Geometrical the  human m i n d  ments  disc'  However, has  bodies  Pyramids/  symmetry  given  fascination for  the c o l o s s a l  at i t s best  special  i s merely  through  i n the form  position  a symmetrical  a r e numerous  of revolution  p a r t i c u l a r operate  Reynolds  number.  submarine  employed  rain  sonars  either  similarity.  fields  of  monustraight  'sun-  i n the  extension  ancient of  belong  systems,  or water  model  and  circle  with  proposed  drying  class  relatively  where  used  platforms  configurations  i n the  system  confined  or through  low  hydrophones  of  studies  chemical  o f problems.  of a given  unintentionally  objects  tanks,meteorological  spray  tunnels  importance  spherical  oceanographic  surveys,  to this  of p r a c t i c a l  or stationary  oil-storage  simulation,  i n wind  i n general  and b a l l o o n s ,  etc.  laboratory  created  detection  drops  industry,  tested  Towed  habitats,  situations  in fluid  i n hydrographic  underwater of  depict  special  space.  where  in  has h e l d  millennia.  been  Sphere  There  in  symmetry  c u r v i l i n e a r symmetry  always  pantheon. in  since  of pharaohs,  lines.  INTRODUCTION  For  i s usually condition  choice  i s  f o r geometric  2  In  the p r e s e n t case, however, a t t e n t i o n on the  s p h e r i c a l geometry was  focussed due  to the i n t e r e s t i n under-  standing and improving hydrodynamic performance prosthetic a o r t i c heart valve.  of the  The Starr-Edwards  when implanted i n an a o r t a , has been observed  prosthesis,  to work under  a h i g h l y c o n f i n e d c o n d i t i o n w i t h the blockage r a t i o i n the range of 15-50%  depending  on the s i z e of the a o r t a and model  of  the p r o s t h e s i s 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 w i t h complex s w i r l i n g component of v e l o c i t y superposed.  The a o r t i c w a l l confinement  complicate the problem. may  U n f o r t u n a t e l y , s u r p r i s i n g as i t  seem, blockage e f f e c t s f o r even a uniform flow past a  sphere a t low Reynolds to date.  number remains  virtually  unexplored  T h i s t h e s i s r e p r e s e n t s a modest step towards b e t t e r  understanding o f t h i s d i f f i c u l t  1.2  would f u r t h e r  problem.  Survey o f L i t e r a t u r e I n t e r e s t i n the behaviour of a sphere moving  a f l u i d goes back to the days of Newton who  i s c r e d i t e d with  the f i r s t recorded measurements on sphere drag. t h i s but p r i o r to 1930,  numerous experiments  f a l l i n g spheres were conducted  Following  on the drag of  and a body of i n f o r m a t i o n —1  generated  f o r Reynolds  through  numbers in. the range 10  6 - 10  .  These  3  data, i n g e n e r a l , show a s i g n i f i c a n t degree of s c a t t e r hence are approximated by the f a m i l i a r curve.  "standard"  drag  I t a p p a r e n t l y a p p l i e s o n l y to smooth spheres  steady motion i n a.non-turbulent, continuum f l u i d of e f f e c t i v e l y theoretical  and p r a c t i c a l  i n f i n i t e extent.  interest  Ever s i n c e ,  i n the s u b j e c t has the c o n t r i b u t i o n s  have been c i t e d by Torobin and G a u v i n  comprehensive review of the  in  isothermal, incompressible  r e s u l t e d i n a l a r g e volume o f l i t e r a t u r e , and up to 1960  1  in their  field.  The drag c o e f f i c i e n t  appears to depend p r i m a r i l y  on the magnitude of the r e l a t i v e t u r b u l e n c e i n t e n s i t y Reynolds number.  and  I n c r e a s i n g turbulence i n t e n s i t i e s cause a  s y s t e m a t i c r e g r e s s i o n o f the t r a n s i t i o n coefficient  and  curve towards lower  r e g i o n of the  drag  Reynolds number together  w i t h a moderate i n c r e a s e 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  Heinrich  supercritical  regions.  In 1963,  2 e t a l . c a r r i e d out sphere drag measurements i n a wind t u n n e l f o r 2 x l O <R < 2 x 1 0 and 0. 078 <M <0.39. T h e i r n n data, however, are s i g n i f i c a n t l y h i g h e r than the standard 3  values.  The  4  d i s c r e p a n c y was  a t t r i b u t e d to the f r e e  stream  3 turbulence. supported  Sivier  spheres  has measured the drag of m a g n e t i c a l l y  i n a wind t u n n e l w i t h a f r e e  stream  t u r b u l e n c e i n t e n s i t y up to 8%, and r e p o r t e d a d e f i n i t e i n c r e a s e i n C_D f o r R n >200,' the i n c r e a s e qrowing with i n ^ -a creasing R .  However, f o r R  <200, he observed  little  or  4  no change i n C level  (- 1%) .  compared to r e s u l t s a t lower t u r b u l e n c e  D  His r e s u l t s are a l s o c o n s i d e r a b l y h i g h e r than  the standard drag v a l u e s . balance system  Zarin  r e f i n e d the  • magnetic  used by S i v i e r and v a r i e d the f r e e  turbulence i n t e n s i t y l e v e l . than 1%,  4  Even a t a t u r b u l e n c e l e v e l  he found, f o r R ' >10  than the standard v a l u e s .  stream  3  less  , drag to be markedly g r e a t e r  However, f o r R  n  <10  3  , the  results  were i n good agreement with the standard v a l u e s .  From t h i s  study Z a r i n concluded t h a t i n the h i g h e r Reynolds  number  3  range  (R  >10  ), a small degree of f r e e stream t u r b u l e n c e  r e s u l t s i n h i g h e r drag v a l u e s . 5 Ross and W i l l m a r t h conducted  drag measurements f o r  spheres moving r e c t i l i n e a r l y through the g l y c e r i n e - w a t e r mixture  f o r 5 <R  n  <10  5  .  T h e i r r e s u l t s agree  with the standard data f o r R  <2 x10  3  fairly  well  but are somewhat g r e a t e r  n for  the Reynolds  3  number exceeding t h i s v a l u e .  r e v e a l e d t h a t the drag on a sphere i s not  The  study  significantly  a f f e c t e d by the v o r t e x shedding (5% v a r i a t i o n ) . On the 6 o t h e r hand, B a i l y and H i a t t c a r r i e d out sphere drag measure5 ments i n the b a l l i s t i c range f o r 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  s t u d i e d subsonic drag on spheres i n the Reynolds range of 200-10,000 using a t e s t range with  number  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 d r a g t o be e v i d e n t f o r the Mach number as s m a l l as  0.2.  Of some i n t e r e s t a r e the r e s u l t s o f V l a j i n a c and C o v e r t 8 i n the l a m i n a r range o f 2 x 1 0 4 - 2 . 6 x 1 0 5 . They found t h a t the c l a s s i c a l wind t u n n e l c o r r e c t i o n s as d i s c u s s e d by P a n k h u r s t and H o l d e r do n o t c o m p l e t e l y 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 .  A l t h o u g h the a u t h o r s do not  s p e c i f y a c t u a l t e s t b l o c k a g e v a l u e s , t h e i r d a t a show cons 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 o t h e r i n v e s t i g a t o r s . In 4  the h i g h e r Reynolds number range of 6  9  5 x 10  <_ R _< 6 x 10 , Achenbach's  cant.  Based on the measured t o t a l d r a g , l o c a l  n  contribution i s s i g n i f i static  p r e s s u r e 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 e s t i m a t e d p o s i t i o n s of the  boundary l a y e r t r a n s i t i o n and s e p a r a t i o n .  Furthermore,  r e s u l t s s u b s t a n t i a t e d a dependence o f the f r i c t i o n  force  on the Reynolds number. In  a n o t h e r study"*"*"*, Achenbach  has i n v e s t i g a t e d the  e f f e c t o f s u r f a c e roughness and t u n n e l b l o c k a g e f o r the f l o w p a s t spheres i n the above range o f Reynolds number.  I t was  o b s e r v e d t h a t an i n c r e a s e i n roughness parameter l e a d s t o a d e c r e a s e 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 d r a g 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 . b l o c k a g e e f f e c t , i n the range of 25-80%, was t o cause an i n c r e a s e i n b o t h the d r a g c o e f f i c i e n t and the  critical  The  6  R e y n o l d s number. lence  to i n i t i a t e  turbulent the  flow.  a slightly  (6x10  a premature t r a n s i t i o n  o f an e a r l i e r  different  4  experiments  These r e s u l t s ,  conclusions  in  Preliminary  i n general,  substantiate  range o f the Reynolds  number  ) , and t h e b l o c k a g e v a r i a t i o n  and c o n f l i c t i n g  separating  a study  laminar'to  i n v e s t i g a t i o n by Maxworthy  I t w o u l d be o f i n t e r e s t  of  from  5 - 2 x10  limited  showed t h e t u r b u -  shear  t o review here  information  layer, although  a s e a r l y as i n 1938.  over  5-25%.  rather  a v a i l a b l e on t h e f r e q u e n c y 12 Moller i n i t i a t e d such  A little  later,  i n 1957,  13 Commetta  extended M o l l e r ' s  study  oT: t h e S t r o u h a l  number  5  v a r i a t i o n w i t h t h e R e y n o l d s number t o R = 5 x 1 0 , b u t 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 t o fl  R  < 4 x10  n  4  .  More r e c e n t l y , Majumdar and D o u g l a s  14  as w e l l  15 as  Calvert  s h e d d i n g from s p h e r e s i n 3 4 R e y n o l d s number r a n g e s o f 5 . 6 x 1 0 <R < 1.16x10 and  the  have r e p o r t e d  4 2x10 the  4 <R  n  <6 x 1 0  base p r e s s u r e  on  R  n  with  , respectively. coefficient  the v a r i a t i o n 4  R  =1.5x10  the  vortex  origin  Calvert's results  showed  t o be s u b s t a n t i a l l y d e p e n d e n t  o f -0.270 t o -0.356  over  4 -6x10  .  o f t h e wake,  The e f f e c t  of trip  wire  was  to s h i f t  l e a v i n g t h e s c a l e unchanged.  must p o i n t o u t o c c a s i o n a l d i s c r e p a n c i e s  One  i n r e s u l t s as  reported  by t h e d i f f e r e n t a u t h o r s . F o r example, M o l l e r 4 m e a s u r e d , a t R =10 , a S t r o u h a l number o f 2 w h i l e Majumdar n J  and  Douglas r e p o r t e d the v a l u e  an o r d e r o f m a g n i t u d e  (S = f D / U = 0 . 2 ) , w h i c h i s t h e v a l u e cylinders  in cross-flow.  turbulent  flow  I t was  t h e r e i s no  typical  also  of  circular  suggested  regular vortex  that i n a  shedding.  t h e o t h e r hand, Commetta d e t e c t e d c o e x i s t e n c e o f shedding  i n two  modes: t h e  lower  h i g h e r mode, a s s o c i a t e d w i t h from  laminar  by A c h e n b a c h 400-5x10^  t o be  '  , i n the  confirms  5 xlO . 3  strong periodic  vortex  low  and  the  of the v o r t e x  sheet  Recent s t u d i e s  R e y n o l d s number r a n g e  lower  critical  In the  fluctuations  shedding  Experimental ation  transition  Moller's results  Beyond t h e u p p e r c r i t i c a l periodic  vortex  17  number, however, t h e found  On  mode a t S = 0 . 2 and  t o t u r b u l e n t , a t S = 0.8 - 1 . 4 . X6  lower  range ^  R e y n o l d s number 6 x10  < R  3  < 3  was x10  5  n  R e y n o l d s number o f was  Reynolds  i n t h e wake f l o w were  observed  3.7 x 1 0 ^  no  detected.  investigation  photographing  a t lower  of  involving  o f t h e wake b e h i n d  R e y n o l d s number r a n g e o f 5 < R  n  - 300  flow  visualiz  a sphere  was  i n the  carried  out  by  18 Taneda  u s i n g a water tank.  critical  R  to  form  n  The  a t which the permanent  i n the r e a r o f a sphere  results  showed t h a t  the  " v o r t e x - r i n g " begins  i s about  24,  size  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 t o t h e l o g a r i t h m o f t h e R , and t h e wake b e h i n d t h e r i n g b e g i n s t o o s c i l l a t e f o r R ~ 1 3 0 . 19 M a g a r v e y and B i s h o p s t u d i e d the t r a n s i t i o n ranges f o r n  t h r e e d i m e n s i o n a l wakes p r o d u c e d an  immiscible l i q u i d  They d i s t i n g u i s h e d with  several  concluded  i n the Reynolds  the observed  number r a n g e  i n each  (as c a n be a n t i c i p a t e d ) t h a t  liquid-liquid  on t h e R e y n o l d s  system  t h e wake p a t t e r n  Furthermore,  i t was  Reynolds  depend on t h e d r o p  deformation.  considered the t r a n s i t i o n  wake p a t t e r n s were l i m i t e d  t o Reynolds  20.  A qualitative  number  interpretation  t r a n s f e r mechanisms i n t h e wake  observed  numbers  However, f o r a l l t h e c a s e s  than  < 2500  n  number r e g a r d l e s s o f t h e  employed.  be o b t a i n e d as t h e y  0 <R  o f t h e c a t e g o r i e s , an  the general values o f the t r a n s i t i o n  cannot  o f a drop o f  wakes a s s t e a d y o r p e r i o d i c  subclassifications  depends e n t i r e l y  that  by t h e m o t i o n  i n the  spread o f l e s s  o f heat  and mass  r e g i o n o f a sphere  i n low 20  speed  flows  (R  < 410) i s p r e s e n t e d  by L e e and B a r r o w  employ m e a s u r e m e n t s o f t h e v e l o c i t y through  flow v i s u a l i z a t i o n  characteristic  flow, a t a v e l o c i t y ocity,  by  point.  The  observed  Taneda's r e s u l t s .  An  o f t h e near-wake i s t h e r e v e r s e d s m a l l e r than  a l o n g the a x i s o f the sphere  stagnation for  much  i n t h e wake  by dye i n j e c t i o n .  flow p a t t e r n s g e n e r a l l y confirmed important  field  who  The wake  the free  stream  vel-  towards the r e a r  transition  and S t r o u h a l number  t h e i n c o m p r e s s i b l e wake o f v a r i o u s b o d i e s was s t u d i e d 21 Goldburg  and F l o r s h e i m  .  B a s e d on t h e e x p e r i m e n t a l  a l ywasc o rs ru e le a st te ed d ftohra ta r and cones r ep sp ur lo tx si ,m a tie t gg tahneg et r ao nfs is tp ih oe nr e sc o u l d be  9  by the Reynolds number based on t o t a l wake momentum t h i c k ness. shedding  F u r t h e r m o r e , i t was  found t h a t f o r r e g u l a r v o r t e x  the d a t a f o r spheres and cones c o u l d be c o r r e l a t e d  w i t h R a y l e i g h - S t r o u h a l f o r m u l a based on the same c r i t e r i o n . B e f o r e moving t o the r e v i e w o f a n a l y t i c a l approaches, i t would be a p p r o p r i a t e t o mention here a r e c e n t and  rather  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  .  It is  p a r t i c u l a r l y r e l e v a n t as the p r e s e n t p r o j e c t r e p r e s e n t s 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 t u n n e l 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 t i o n o f the w o r k i n g  an  concentra-  f l u i d , the a u t h o r s were a b l e to c o r r e l a t e  the p r o g r e s s o f f o r m a t i o n , e l o n g a t i o n , asymmetry 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 w i t h the s u r f a c e  pressure  d i s t r i b u t i o n i n the Reynolds number range o f 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 p r e s s u r e i n the range R  n  = 240 - 275,  which was  found t o be  a s s o c i a t e d w i t h the o n s e t o f i n s t a b i l i t y o f the r i n g l e a d i n g t o i t s p e r i o d i c shedding.  vortex  I n g e n e r a l , Reynolds  number e f f e c t s were c o n f i n e d to the r e g i o n near and downstream o f the minimum p r e s s u r e p o i n t .  An e x t e n s i v e  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 d a t a . T h e o r e t i c a l i n v e s t i g a t i o n o f even a steady i n c o m p r e s s i b l e f l o w p a s t a sphere i s v e r y complex.  viscous I t was  23  f i r s t c o n s i d e r e d by Stokes by many a u t h o r s s i n c e then.  (1851)  , and has been d i s c u s s e d  A l a r g e p o r t i o n o f these s t u d i e s  10  have been concerned w i t h the s o l u t i o n s f o r v a n i s h i n g l y small R . Stokes s o l v e d the problem by n e g l e c t i n g the i n e r t i a 24 of  the f l u i d .  L a t e r , Whitehead  t r i e d to improve  upon  t h i s s o l u t i o n by i n t r o d u c i n g h i g h e r approximations to the flow when the Reynolds number i s not n e g l i g i b l e . i s now  w e l l known, h i s s o l u t i o n i s not v a l i d 25 26  uniform streaming  .  Oseen  But as  i n problems of  s o l v e d Whitehead's  paradox  by assuming t h a t the sphere caused a s m a l l p e r t u r b a t i o n i n the uniform p a r a l l e l  flow and n e g l e c t e d second o r d e r  perturbation v e l o c i t i e s ,  thus t a k i n g the i n e r t i a terms i n t o  account to a l i m i t e d e x t e n t . equationshas been improved 29 and Pearcey e t a l . linearization  Oseen's s o l u t i o n f o r l i n e a r i z e d 27 28 by G o l d s t e i n , Tomotika e t a l . ,  However, as can be  renders these a n a l y s e s inadequate  Of c o n s i d e r a b l e i n t e r e s t are two 30 s o l u t i o n s : one by Kawaguti of  anticipated,  who  satisfied  for R  n  >2.  independent an i n t e g r a t e d  the Navier-Stokes e q u a t i o n f o r f i r s t and second-order  form terms  when expanded by Legendre Polynomials and the o t h e r by 25 Proudman and Pearson who l i n e a r i z e d the Navier-Stokes e q u a t i o n by two approximations, one v a l i d a t a d i s t a n c e from the sphere, and the o t h e r v a l i d near the s u r f a c e of the sphere. 31 Kawaguti  has a l s o developed an a l t e r n a t i v e procedure to  s o l v e the Navier-Stokes e q u a t i o n u s i n g the f i n i t e method.  U n f o r t u n a t e l y , the technique, v a l i d  difference  f o r flow around  11  s p h e r e s up  to R  n  = 20,  proves  t o be  extremely  labourious.  32 33 34 et 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  Fox this  difficulty  by  t r a n s f e r r i n g the technique i n t o a  relaxa-  35 tion  procedure.  On  t h e o t h e r hand, J e n s o n  r e l a x a t i o n method d i r e c t l y vorticity ates 10>  and  stream  to obtain 20,  40.  Hamielec  method, b u t w i t h of  f o r flow around 36 — 38  spheres  have a l s o  g r i d s i z e to obtain  used  Fourier  to solve  number by steady  a sphere  dependent state.  numerical flow  solutions around were  o v e r a wide range o f the Reynolds 39 40 and W a l k e r . Rimon and Cheng derived  solutions  for 1 < R  n  < 1000  by  impulsively  f r o m r e s t w i t h u n i f o r m v e l o c i t y and  i n t e g r a t i o n to carry More r e c e n t l y , •  a family  Dennis  o f Legendre  of  1-40.  used  starta  time  the  s o l u t i o n to the steady 41 and W a l k e r have p r e s e n t e d 42 proposed  functions  Stokes e q u a t i o n s f o r flow around number r a n g e  5,  a similar  f o r the flow v a r i a b l e s  a s e r i e s t r u n c a t i o n method, f i r s t employing  =  n  the problem  Dennis  state  expansions  coordin-  at R  used  the N a v i e r - S t o k e s e q u a t i o n s f o r slow v i s c o u s  spheres.  ing  i n modified spherical  et a l .  finer  the  t o the g o v e r n i n g e q u a t i o n s f o r  function  solutions  applied  spheres  by Van  Dyke  to solve i n the  ,  the N a v i e r -  Reynolds  12  1.3  Sphere  Flow  Field  From t h e b r i e f tant characteristics s p h e r e become c l e a r .  literature  review given  of the flow f i e l d  above,  associated with a  A t a v e r y low R e y n o l d s number,  the f l o w near the sphere i s dominated  by v i s c o u s  resulting  In the range  inertial and  i n the f o r e - a f t effects  symmetry.  increase with increasing  t h e s t r e a m l i n e p a t t e r n no l o n g e r  mentioned  symmetry.  near the r e a r slightly  The f i r s t  stagnation point  greater  than 20(R  n  onset.  by a r e d u c t i o n oscillation  n  number  separation  appears a t a Reynolds  - 24), a l t h o u g h t h e r e  number  i s some  corresponding to  o f the vortex r i n g ,  i n i t ss t a b i l i t y .  o f the bubble ensues  < 0.1,  0.1<R jc24,  The s e p a r a t i o n r e g i o n grows w i t h R e y n o l d s  as s u g g e s t e d by t h e g r o w t h  n  t o t h e above  evidence of flow  n  R  forces  Reynolds  conforms  d i s a g r e e m e n t as t o the p r e c i s e v a l u e o f R its  impor-  accompanied  I n t h e range R and becomes  number,  n  = 130-210,  gradually  stronger. F o r 210 < R is  n  <270, an a s y m m e t r i c a l s e p a r a t i o n  bubble  o b s e r v e d , f o l l o w e d by d i s c r e t e v o r t e x l o o p s s h e d d i n g  periodically  from o p p o s i t e s i d e o f the s e p a r a t i o n bubble i n  t h e r a n g e e x t e n d i n g up t o 700. vortex  shedding begins, often  R e y n o l d s number, s i g n i f i e s  The v a l u e o f R referred  n  t o as lower  the appearance  path o f the free  falling  critical  o f a wake i n w h i c h  t h e f l o w a r o u n d t h e s p h e r e i s no l o n g e r c l o s e d . zag o r h e l i c a l  a t which the  The  zig-  s p h e r e s a t R^ > 210  13  is  directly  is  of  attributed  interest  to t h i s  to note  here  s t a g n a t i o n p o i n t and  the  from a v a l u e  at R  of  zero  c h a r a c t e r o f t h e wake.  t h a t the  angle  number  =24  t o 72°  f o r s p h e r e has been f o u n d  5.6 x 1 0  3  < R  <11.6  between t h e  separation c i r c l e  steadily  = 450. The n a r o u n d 0.2 f o r  t o be  rear  increases  at R  n  It  Strouhal  x10 . 3  n Beyond t h e number, n a t u r e the  o f the  R e y n o l d s number  R e y n o l d s number) o f layer to  value flow  The  No  r e g i o n and  result  condition" sibility,  characteristic shedding  rarefaction,  and  so  heat  i n the  coefficient  R e y n o l d s numbers.  the  f o r both  the  At higher  size  laminar  the of  the  has  been  drag.  recorded  number.  far relates  to the  s u r f a c e roughness, transfer  effects.  i n the p r e s e n t  transition  with  boundary  from of  until  critical  drop i n the  increase i n turbulence i n t e n s i t y r e g r e s s i o n o f the  same  upper  shift  sharp  significant  R e y n o l d s numbers, t o g e t h e r drag  the  Now  frequency  Reynolds  of turbulence,  three parameters are not  systematic  the  i s reached.  i s a rearward  flow d e s c r i p t i o n  devoid  In g e n e r a l ,  5  t o as  a decrease  upper c r i t i c a l  The  Reynolds  s e p a r a t i o n p o i n t changes  well defined vortex  beyond the  critical  remains e s s e n t i a l l y  3.7 x l O  separation point, causing separated  lower  (often referred  upstream o f the  turbulent.  o f the  results  "standard compresThe  study. in a  r e g i o n towards  a moderate i n c r e a s e o f  subcritical  and  last  lower the  supercritical  R e y n o l d s numbers,  surface  14  roughness a f f e c t s early  transition  the  flow  i n the  similar  manner: i t c a u s e s  to a t u r b u l e n t boundary l a y e r  resulting  a r a n g e o f R e y n o l d s number o v e r w h i c h t h e  drag  diminishes  curve  (compared  However, i t may Figure  1-1  butions and  1.4  standard  i n c r e a s e the  briefly  drag  a t low  summarizes t h e  values).  Reynolds  scope of  review  coefficient  numbers.  important  o f the  field  by  contriTorobin  Gauvin"^.  The  Plan of  Study  p o i n t e d out  s t u d y i n g the  fluid  been i n p r o g r e s s  before,  a metal  on  (ii)  high  (iv)  a  at  ball  ranging  of  Primarily,  sewing  noted  corrosion resistance; rubber  f r o m 1.2 - 2.2 called  margin of k n i t t e d  m a i n o b j e c t i v e has  uncoated  a cobalt alloy  silicone  seat normally  has  Starr-Edwards c o n f i g u r a t i o n ,  21,  s t r e n g t h and  a spherical  a metal  the  s i n c e 1969.  cage o f h i g h l y p o l i s h e d  diameter (iii)  aimed  consists of:  casting of S t e l l i t e for  investigation  department  a t t e n t i o n i s focussed  (i)  an  mechanics of p r o s t h e t i c h e a r t v a l v e s  in this  which e s s e n t i a l l y  The  drag  s i n c e a comprehensive  As  the  to the  in  with  cm; orifice;  Teflon  been t o i d e n t i f y  cloth.  factors  causing:  15  Rn 10  10'  10'  10'  10  10'  v  3x10  Wake  Geometry 10.  Wake"  18 Taneda  2.5x10  C  ,0.2<M <3.17  d  n  Heinrich et al. [1965]  4.1 xio  Observation  20 Lee &3Barrow [1965] Sivier [19 6 7] 7 Goin & Lawrence [1968]  4x10  25.  a <8%, M  n  o  2x10'  C  d  ,0.2 <M < 0.98 n  6x10  2x10  Cp,S/C  Maxworthy [l969] 1 1  5x10  4x10  C  d  Zarin  , 0.1 <M <0.57 , a<13% n  5 J x 10  S  n :  a 6x10  5x10  C  ,C  d  , T. , a<0.45°/  p  2xl0  v  o  2.6x10  3  C , 0.02<M <0.31 d  c  n  10  5  n  10 _ 7  [1972] 8 Vlajinac & Covert [1972]  Cpb . S  n  Calvert  'n  6x10  4  C  d  ,S/C  [1972]  15  a  3 xlO  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 o f flow p a s t a sphere.  Achenbach  6xlO  4  4x10  F i a u r e 1-1  9  Bailey & Hiatt  0.1 <M <6  d  [1969]  14 Mujumdar & Douglas [1 9 7 0 ] 5 Roos & Willmarth [1971]  10  20  [1956] 2  [1972]  Achenbach [1974] 1 7  Achenbach  1 0  [1974]  Figure  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 o r poppet; (c) seat; (d) suture r i n g  17  (a)  destruction  (b)  d i s s o c i a t i o n of their the  Suitable  or  valve  at  the on  the  constituents  and  cage r e s u l t i n g i n  i n the  valve  l e a s t minimize valve  geometry t h a t  t h e s e p r o b l e m s and  low  R e y n o l d s number r e p r e s e n t s to approach the  difficulty.  lead  to  p e r f o r m a n c e were a l s o o f i n t e r e s t .  dynamics o f b l u f f  i s forced  would  Although  bodies  in  pulsatile  a challenging  task.  problem i n stages  studies  by  of  Aminzadeh  and  43-47  Modi  '  there  are  have p r o v i d e d a number o f One  of  as  the  poppet of  i m p o s e d by  aortic  position  senting  valve  review revealed itself  number r a n g e o f  problem which  to  the  remain  e f f e c t of  Starr-Edwards valve  the  s i z e of  decided  prosthesis cycle.  that  the  the  p o p p e t may  change i n t h e  i t was  a cardiac  s p h e r e by  the  information,  blockage occupying  _  to a l a r g e  such a heart  the  useful  (Figure. 1 3) .  b l o c k a g e o f f e r e d by  background,  to  them p e r t a i n s  D e p e n d i n g upon t h e  leading  considerable  aspects  unexplored.  as  blood  fluid  increasing  the  cells;  The  Hence one  •22  blood  failure.  improvement i n t h e  flows at  red  deposition  modifications  alleviate an  of  flow  aorta be  substantial  wall  pulsatile  However, a d e t a i l e d corresponding  even i n a u n i f o r m i n t e r e s t remains  prosthesis, 47  character.  to explore i n the  and  flow  With  effects flow  unrecorded.  on  repre-  literature  information i n the  this  for  Reynolds  a  18  Figure  1-3  A s c h e m a t i c diagram showing the S t a r r - E d w a r d s p r o s t h e s i s o c c u p y i n g a o r t i c p o s i t i o n and presenting a large blockage  19  This t h e s i s , therefore, s t u d i e s : ( i ) f o r m a t i o n , development and i n s t a b i l i t y o f vortex ring; ( i i ) associated pressure  distribution;  ( i i i ) d r a g ; and ( i v ) near wake geometry f o r a f a m i l y o f spheres r e p r e s e n t i n g blockage  r a t i o o f 3 - 30%  i n t h e Reynolds number range o f 30 - 2000. In t h e b e g i n n i n g , t h e e f f e c t o f stem used i n s u p p o r t i n g t h e 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 p r o v i d e s a criterion for their selection. detailed  T h i s i s f o l l o w e d by a  study o f the s u r f a c e p r e s s u r e d i s t r i b u t i o n and drag.  An approach t o d a t a 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 i t s m e r i t a s s e s s e d to t h e c o n v e n t i o n a l procedure.  compared,  F i n a l l y , the t e s t  results  a r e a n a l y z e d as f u n c t i o n s o f t h e confinement c o n d i t i o n and Reynolds number.  An e x t e n s i v e f l o w v i s u a l i z a t i o n study i n  c o n j u n c t i o n w i t h s t i l l and h i g h speed movie photography complements t h e t e s t program. p l a n o f study.  F i g u r e 1-4 summarizes t h e  WALL  CONFINEMENT FOR  IN T H E RANGE  S t e m E f f e c t  Mean  Pressure  Distribution  Figure 1-4  EFFECTS  SPHERES  R E Y N O L D S O F  NUMBER  30 —  2 0 0 0  Near  Wake  Configuration  Flow Visualization  The p l a n of study o  21  2.  EXPERIMENTAL APPARATUS AND  T h i s chapter the experimental ployed  introduces  program.  TEST PROCEDURES  the t e s t f a c i l i t i e s used i n  Some of the i n s t r u m e n t a t i o n  c o n s t i t u t e s the standard  equipment i n any w e l l equipped  f l u i d mechanics l a b o r a t o r y and hence needs no On  em-  elaboration.  the o t h e r hand, design and c o n s t r u c t i o n a l d e t a i l s  involved  i n the development o f s p e c i f i c equipments are o f t e n numerous and  hence, though important and  in their entirety. on more s a l i e n t The  The  r e l e v a n t , cannot be  attention i s , therefore,  covered  focussed  features.  t e s t procedures employed are c o n c e p t u a l l y  known but t h e i r implementation o f t e n a t t a i n complexity  well of a  higher order, mainly because of the c h a r a c t e r of the working fluid  (glycerol-water s o l u t i o n ) .  s p e c i f i c experiments make cult.  c e r t a i n measurements q u i t e  Throughout, the emphasis i s on p r a c t i c a l  involved i n executing the  Often p e c u l i a r i t i e s of  the experimental  considerations  programme.  f a c t o r s i n v o l v e d are,seemingly, so t r i v i a l t h a t  would seldom give them a second look.  At  However, a common  of most experimenters i s t h a t r e s o l u t i o n o f  apparently  simple  problems o c c a s i o n a l l y takes days, i f not This i s p a r t i c u l a r l y  where l i q u i d i s the working The  times  one  experience  weeks or months.  diffi-  glycerol-water  true i n the  case  fluid. s o l u t i o n tunnel r e p r e s e n t i n g a  fundamental f a c i l i t y f o r the e n t i r e t e s t program and i t s  22  c a l i b r a t i o n u s i n g the h o t - f i l m anemometry are first. and  T h i s i s f o l l o w e d by an i n t r o d u c t i o n of the models  t h e i r support  pressure  described  system.  transducing  Next, the h i g h l y s e n s i t i v e  system capable  of determining  pressure d i s t r i b u t i o n i s d i s c u s s e d l e a d i n g to the ment used i n drag measurements.  surface arrange-  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 u s e f u l i n o b t a i n i n g p h y s i c a l a p p r e c i a t i o n as to the c h a r a c t e r of the flow, are presented.  Wherever a p p r o p r i a t e ,  procedures employed are e x p l a i n e d and  calibration  corresponding  charts  included.  2.1  G l y c e r o l Tunnel The  t e s t s were conducted i n a g l y c e r o l - w a t e r s o l u t i o n  tunnel designed  to produce Reynolds number i n the range  30 - 6000 (based  on sphere diameter and  v e l o c i t y i n the t e s t - s e c t i o n ) . of the working f l u i d provided  The  r e p r e s e n t a t i v e average  c h o i c e of  a degree of f l e x i b i l i t y ,  o n l y to a c e r t a i n extent, as governed by the of the power u n i t .  but  characteristics  P r i m a r i l y the tunnel c o n s i s t s of  subassemblies: the t e s t s e c t i o n ; the f l u i d and  concentration  three  r e t u r n system;  the power u n i t c o n s i s t i n g of a pump and a d r i v e motor. The  t e s t - s e c t i o n i s b u i l t of four p l e x i g l a s w a l l s  2.44 m (8 f t ) long, 1.9 cm (0.75  in) t h i c k and wide enough to  produce an i n s i d e c r o s s - s e c t i o n of 20 . 32 cm x 20. 32 cm ( 8 i n x 8 in) .  23  Deflection  annular  honeycombs, b r a s s flat  velocity  inside  o f the  port-hole eter, to  x1.27  essentially  cm  i n diam-  the  entrance  a d j u s t models.  tapped  These o p e n i n g s  test-section.  (25x5  1/2x1/2  test-section  i n the  were u s e d  Two  In  comprising  p i p e s and  hose.  A copper a 2.4 mx  annular by  temperature  between t h e end  top w a l l  to support  elbows w i t h  i n ) , recess-mounted flat,  pipe,  15.24  single  cm pass  3 mx  7.62 cm  heat  o f the working  exchanger.  i t was fluid  radiator  photography. test-  section  f l a n g e s and  (10 f t x 3 i n ) , plastic  cloride radiator  in conjunction  pipe  With the  p o s s i b l e to  i n the  homogen-  exchanger, P o l y v i n y l  (8 f t x 6 i n ) PVC  a w a t e r main,  s e c t i o n s o f the  o f the p l e x i g l a s  connecting  a  plates,  system i s the r e t u r n  of heat  1.6cm  t o mount  glass  provided o p t i c a l l y  t h e power d r i v e  (PVC)  and  and  the  via a  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  and  supplied  and  from  p r e s s u r e c o n d u c t i n g T i n e s and  Located  an  and  to  p o r t h o l e , 12.7 cm (5 i n )  p l u g s were d r i l l e d  s i d e s o f the  with  e a c h end  l o c a t e d 0.84m (33 i n )  h o t - f i l m probe i n the  section  three accesses  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  models, take out  eous and  through  The  test-section.  63.5 x 1 4  There are  top.  s e v e r a l s e c t i o n s of  n y l o n w o o l gave e x c e p t i o n a l l y  to reach, p o s i t i o n  i n N-C)  the  and  test-section,  i s judiciously  addition,  of  screens  profiles.  a t the  a d m i t arm  (5/8  vanes t o g e t h e r w i t h  formed coolant  maintain  w i t h i n ± 0.2°C.  PVC  hose p r o v i d e d r e l a t i v e l y  elbows 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 f r e e c o n n e c t i o n between t h e 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 o f a c e n t r i f u g a l pump: A u r o r a type GAPB, 12.7 £/s  (200 g a l / m i n ) , 7.6 mhead, 1750 rpm.  I t i s d r i v e n by a t h r e e horsepower v a r i a b l e speed d.c. The pump i m p e l l e r and h o u s i n g against possible corrosion.  a r e o f c a s t b r a s s t o guard The motor i s e n e r g i z e d by a t h r e e  phase g r i d , t h e v o l t a g e b e i n g a d j u s t e d through former and r e c t i f i e d by s e l e n i u m d i o d e s . o f t h e d.c.  motor.  an a u t o t r a n s -  No f u r t h e r smoothing  o u t p u t was r e q u i r e d .  I t was i m p o r t a n t t o m i n i m i z e the t u n n e l f l u i d .  d i r t contamination of  T h i s was a c h i e v e d by i n c o r p o r a t i n g a  10 u f i l t e r i n a bypass c i r c u i t a c r o s s t h e pump.  The system  f i l t e r s t h e e n t i r e volume a t l e a s t once i n t w e n t y - f o u r of operation.  hours  The t u n n e l i s s c h e m a t i c a l l y shown i n F i g u r e  2-1. Flow r a t e i n t h e t u n n e l was monitored  u s i n g a sharp  edge o r i f i c e p l a t e mounted 0.61m (2 f t ) upstream o f t h e pump inlet.  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 t o make i t s  reading r e l a t i v e l y  independent o f t h e upstream and downstream  d i s t u r b a n c e s i n t h e form o f elbows, change i n s e c t i o n a t t h e pump i n l e t , pump s u c t i o n , e t c .  B e f o r e f i n a l assembly t h e  o r i f i c e p l a t e and a s s o c i a t e d plumbing were c a l i b r a t e d , Under s i m u l a t e d t e s t c o n d i t i o n s , by pumping water from a l a r g e sump i n t o a w e i g h i n g i s presented  tank.  i n F i g u r e 2-2.  The c a l i b r a t i o n p l o t thus  obtained  vent p i p e radiator h o s e  honeycombs  portholes  test section  brass screens  heat e x c h a n g e r  f i c e plate  Figure  2-1  c o o l i n g w a t e r in  A schematic diagram showing the tunnel  glycerol-water  solution  26  Figure  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  27 2.2  Hot F i l m Anemometry and V e l o c i t y  Profiles  As mentioned b e f o r e , an average v e l o c i t y s e c t i o n was the  i n the  test-  determined from the volume flow r a t e g i v e n by  o r i f i c e meter.  mental work i t was  However, d u r i n g the course of the e x p e r i a l s o r e q u i r e d to measure v e l o c i t y  f i l e s arid, i n p a r t i c u l a r , range o f 2.5-15 cm/s.  the c e n t e r l i n e  velocities  Measurements o f f l u i d  proi n the  velocities  at low v a l u e s o f Reynolds number has long been known to be exceptionally  difficult.  Apart from l a s e r - d o p p l e r anemometer, which was i n the e a r l y stage o f acceptance when c a l i b r a t i o n tunnel was  undertaken  (197 3),  still  o f the  a hot f i l m probe appeared to  meet the requirements o f h i g h r e s o l u t i o n i n time and space of flow v e l o c i t i e s q u i t e adequately. wedge shaped p l a t i n u m f i l m probe  1239W)  was  (Thermo-Systerns Inc., model  used i n c o n j u n c t i o n w i t h  temperature anemometric equipment the  Hence a q u a r t z coated  e x i s t e n c e of a comprehensive  the standard c o n s t a n t  (DISA model  55A01).  Despite  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 w i t h the use o f hot f i l m anemometry f o r i n v e s t i g a t i o n o f slow l i q u i d flow.  It is  mainly because o f s e v e r a l d i f f i c u l t i e s i n v o l v e d i n adapting the  anemometer t o use i n water o r o t h e r l i q u i d s : (i) E l e c t r o l y s i s  i s by f a r the worst source of  t r o u b l e c a u s i n g c o r r o s i o n o f the probe, g e n e r a t i o n o f gases and i n s t a b i l i t y i n the  28  electronic control c i r c u i t r y .  This p a r t i c u l a r  problem does not a r i s e i n non-conducting such as d i s t i l l e d water or kerosene. way  liquids,  Another  o f a v o i d i n g s e r i o u s c o r r o s i o n c o u l d be the  use o f h i g h frequency a l t e r n a t i n g c u r r e n t to heat the probe, and/or c o a t i n g the probe to p r o v i d e e l e c t r i c a l i n s u l a t i o n from the (ii)  liquid.  Often the formation of bubbles on the sensor causes i n c o r r e c t and u n s t a b l e o p e r a t i o n o f the 48 probe  .  Bubble  formation can be reduced by  c l e a n i n g the probe i n a s o l v e n t , e.g.,  methyl  a l c o h o l , w i t h the anemometer i n "stand-by" c o n d i t i o n , and/or by adding some s u r f a c e r e a c t a n t s to reduce f l u i d ' s s u r f a c e t e n s i o n , thus preventi n g the formation o f bubbles and t h e i r to the sensor. agent" (iii)  attachment  In the case o f water a "wetting  (Kodak Photo-Flo 200)  can be  used.  Contamination o f the probe by dust p a r t i c l e s o r  .49 o t h e r d e p o s i t s reduces and m o d i f i e s i t s s e n s i t i v i t y To e l i m i n a t e d i r t contamination the s u r f a c e of the l i q u i d should be s h i e l d e d .  Continuous  o f a p a r t o f the c i r c u l a t i n g f l u i d i n m i n i m i z i n g the problem.  filtration  should a l s o h e l p  Both these methods and  frequent c l e a n i n g of the probe were found necessary i n the p r e s e n t s e t o f experiments.  29  There are many ways t o c a l i b r a t e h o t - f i l m probes  '  The c h o i c e o f method depends on the a v a i l a b i l i t y o f a s u i t a b l e standard o f comparison,  the ease o f measurement, and the  d e s i r e d degree o f accuracy. measured by mechanical field  In most cases, the v e l o c i t y  means a t a s p e c i f i c p o i n t i n the f l u i d '  i s compared with the e l e c t r i c a l s i g n a l o f the anemometer.  The degree o f accuracy then depends mainly on the accuracy w i t h which the r e f e r e n c e v e l o c i t y i s known. In the p r e s e n t case the probe was h e l d s t a t i o n a r y in a rotating dish,  30.5 cm diameter  and 25.4 cm h i g h , mounted  h o r i z o n t a l l y on a t u r n t a b l e w i t h i n f i - n i t e l y v a r i a b l e drive  (Figure 2-3).  T h i s arrangement was found  over the v e l o c i t y range o f i n t e r e s t .  speed  satisfactory  S u f f i c i e n t time had to 52  be allowed f o r a q u a s i - s t e a d y s t a t e o f motion t o be set-up The motion o b t a i n e d was very c l o s e l y s o l i d body r o t a t i o n when the probe was not too f a r from the c y l i n d r i c a l o r bottom w a l l s o f the d i s h .  The c i r c u l a r d i s h had t o be s u f f i c i e n t l y  l a r g e t o a l l o w f o r the d i s s i p a t i o n o f v o r t i c i t y  generated  by the probe between s u c c e s s i v e passes; o b v i o u s l y the time 2 constant o f t h i s e f f e c t i s o f the order v / r (r = d i s t a n c e 53 from the probe to the a x i s o f r o t a t i o n )  .  Absolute  c l e a n l i n e s s was found to be e s s e n t i a l i n these t e s t s .  The  complete r i g was kept i n a g l a s s e n c l o s u r e which g r e a t l y reduced  the frequency o f probe c l e a n i n g r e q u i r e d t o produce  repeatable  results.  F i g u r e 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, constant 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  co o  31  Calibration water As  solution of concentration  a n t i c i p a t e d , the  straight The  line  the  from the Since  by  v i r t u e of  cold  f o r the  the the  c h a n g e o f .T . drift  i n the  values  The  weight  glycerol-  (Figure  ratio  results  i s o f the  ancillary  equipment  o f the  true  ratio,  the  a  velocity.  of  0.09 7.  order  from The  scatter  t h a t can  be  alone.  h o t - f i l m temperature T i s kept *• m  during  2-4).  c l u s t e r e d around  maximum d e v i a t i o n  overheat  overheat  resistance)  by  i n the  i s a l e a s t mean s q u a r e f i t  measured d a t a .  experimental  expected  line  out  points  down t o q u i t e low  f i t i s 2.3% the  55.7%  experimental  indicated straight  through the  in  t e s t s were c a r r i e d  a change o f R  constant (probe's  c  measurements w o u l d i m p l y  a  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 overheat  T h i s would g i v e  ratio  induced  by  some a p p r e c i a t i o n as  ambient temperature  t h a t can  be  variations in R . c  t o the  changes i n  tolerated during  a  the  given  test. To fluid  this  dependence o f probe c o l d r e s i s t a n c e  t e m p e r a t u r e was  bath.  Figure  trations of the  end,  2-5  measured  shows t h e s e  using  results  glycerol solution.  same s l o p e  suggesting  the  constant  In the worst case,  observed  about  The calibrate  first  the  step  tunnel,  temperature  for various  A l l the  resistivity. t o be  a constant  curves  on  concen-  have  coefficient  almost  of  t h e maximum d e v i a t i o n  was  1.2%. i n the  test  programme was  i . e . , to obtain  information  to about  the  250  200  (Volt)'  150  100  0  1  7 ,,0.5 .0.5 U ; (cm/sec)  F i g u r e 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 o f C. =55.7% n  Figure  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 water 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  34  boundary l a y e r files  growth as r e f l e c t e d i n the v e l o c i t y  along the t e s t s e c t i o n .  To t h i s end,  pro-  the tunnel  was  f i l l e d w i t h the working l i q u i d of a f i x e d c o n c e n t r a t i o n . All  a i r pockets  and bubbles were removed from the  by c i r c u l a t i n g the t e s t f l u i d ,  with the w e t t i n g agent, f o r  at l e a s t e i g h t hours at around 30°C. a given s t a t i o n was  Velocity  p r o f i l e at  then obtained u s i n g the c a l i b r a t e d  f i l m probe i n c o n j u n c t i o n with a t r a v e r s i n g position  tunnel  hot  gear, which can  the probe with an accuracy o f around ±0.25 mm.  It  should be p o i n t e d out t h a t the probe movement i s c o n f i n e d to the v e r t i c a l d i r e c t i o n i n the c e n t r a l plane o f tunnel.  Step s i z e f o r the probe movement was  the  regulated  a c c o r d i n g t o the v e l o c i t y g r a d i e n t so as to p r o v i d e accurate p r o f i l e near the w a l l .  F i g u r e 2-6  shows  an instrumen-  t a t i o n l a y o u t used d u r i n g the v e l o c i t y p r o f i l e measurements. Velocity 8 3 cm (y  p r o f i l e s were measured a t a  down stream of the entrance  = 83 cm)  station  to the t e s t  section  i n the Reynolds number range of 960 - 3900 based  on the h y d r a u l i c diameter of the t e s t s e c t i o n  and  the average  v e l o c i t y as deduced from the flowmeter data.  Two  distinct  cases were c o n s i d e r e d : t u n n e l without models of d i f f e r e n t blockage ^ Typical  plots  are presented  a model and  with  r a t i o s l o c a t e d at y =100 m i n F i g u r e s 2-7  and  2-8.  i s apparent t h a t 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 at l e a s t over the c e n t r a l  cm.  1  15 cm  of the tunnel h e i g h t .  It flat The  constant  d.c.  temperature annemometer  digital voltmeter .  honeycombs  oscilloscope  •TS hot.film probe , 1 23 9 w , s u p p o r t e d by a traversing test  section  brass screens  CO  Figure  2-6  I n s t r u m e n t a t i o n l a y o u t used d u r i n g  velocity  p r o f i l e measurements  ui  36  20.0 o A  o o o o  1 7.5  15.0  A A A A  o  A  o o  A A  12.5  10.0  Z, cm 7.5  o Yp = 8 3 U,  5.0  2 5  cm  o  c m / s e c  •  3-33  o  6.60  A  10.02  A A  o  A A  o o o O  A A A A  0.0 0  5 U  Figure  2-7  z  ,  10 cm/sec  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 = 8 3 cm i n a b s e n c e o f t h e s p h e r i c a l models p  20-0  < o  37  • <  O  •  o  • o  <  a o n  <  1 7.5  <  o  <  15.0  o n o  i  •  o  .•  o  12.5  •  o  10.0 Z,cm  •  o  •  o  a  o  •  o  7.5  U .cm / s e c  5.0  •  12.70  o  11.30  <  9.80  v  a  •  o o  •  o  • o . •  p = 100 cm  2.5  o  a  o  a •  o •  o < —4  0.0  9-B-  4.0  I  o  •  12.0  8.0  16.0  20.0  U ,cm / s e c z  F i g u r e 2-8  E f f e c t o f w a l l confinement on the upstream profile: (a) S/C = 7.6%  velocity-  38 20.0  o o o o o o o  17.5  15.0  A A A A A  o o 12.5  o  10.0  cm  o  7.5 o Yp = 8 3 U,  cm  o  cm/sec  o o o o  5.0 •  2.5  3.12  o  6-95  A  9.72  '  A  '  0 U 2-8  A  o o o  o.o  Figure  A  z  ,  A A  '  10 cm/sec  I  I  I  I  I  15  E f f e c t o f w a l l confinement on the upstream v e l o c i t y profile: (b) S/C = 30.6%  39 presence o f t h e model does a f f e c t t o some e x t e n t i t s u n i f o r m c h a r a c t e r , however, t h e maximum d e v i a t i o n from the  average v a l u e was found t o be l e s s than 8%.  w i l l see l a t e r , any v a r i a t i o n s  As we  i n t h e v e l o c i t y p r o f i l e can  be accounted f o r by a m o d i f i e d d e f i n i t i o n o f t h e p r e s s u r e coefficient.  F o r t h e purpose o f comparison o f t h e  experimental r e s u l t s with available  information i n the  l i t e r a t u r e , i t was a l s o n e c e s s a r y t o have d e t a i l s o f centerline flow.  v e l o c i t y o v e r t h e o p e r a t i n g range o f t h e mean  T h i s i s shown i n F i g u r e 2-9.  I t i s interesting  t o note t h a t a t a v e r y low v a l u e o f t h e average (based on flowmeter d a t a , U < 3 c m / s ) , velocity  velocity  the c e n t e r l i n e  (U ) i s e s s e n t i a l l y the same as U. However, w i t h c  an i n c r e a s e i n t h e f l o w r a t e , t h e r a t i o U/U  c  gradually  drops and tends t o a t t a i n a u n i f o r m v a l u e o f around 0.74.  2.3  Models and Support  System  A f a m i l y o f seven spheres r a n g i n g i n d i a m e t e r from 3.8-12.7 cm were c a r e f u l l y machined t o l e r a n c e o f 0.0025mm.  from p l e x i g l a s w i t h a  Any d e v i a t i o n from  sphericity  was checked u s i n g two p r o c e d u r e s : (i) (ii)  micrometer; p r o j e c t i n g a photograph o f t h e model on a s c r e e n .  F i g u r e 2-9  V a r i a t i o n of the c e n t e r l i n e v e l o c i t y as measured by the h o t - f i l m with the average v e l o c i t y as given by the flowmeter data  probe  41  Maximum d e v i a t i o n from the mean diameter was l e s s than 0.18%.  The  accuracy was  adequate f o r the t e s t program. t h e i r dimensions  found to be  considered quite  F i g u r e 2-10  and the r e s u l t i n g blockage  shows the models, ratios.  As a g a i n s t c o n s t r u c t i o n 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  system p r e s e n t e d s e v e r a l i n t e r e s t i n g problems dynamics c o n s i d e r a t i o n s . a model i n p o s i t i o n without s t r u c t u r e i n the f l u i d  support  from  I d e a l l y one would l i k e t o h o l d i n t r o d u c i n g any  field.  Although  supporting  such  'non-contact'  magnetic support systems are a v a i l a b l e commercially, tend to be p r o h i b i t i v e l y c o s t l y . to two  fluid  One  they  i s , therefore, forced  t u r n to c o n v e n t i o n a l stem type of support.  T h i s poses  questions: (i)  What i s the d e s i r a b l e o r i e n t a t i o n of the stem r e l a t i v e to the f l u i d  (ii)  field?  What i s the e f f e c t of a stem on the f l u i d To put i t d i f f e r e n t l y , what i s the for  field?  criterion  s e l e c t i o n of the stem s i z e so t h a t i t s e f f e c t  on the f l u i d  f i e l d becomes n e g l i g i b l e ?  A h o r i z o n t a l stem l y i n g i n the wake of the model appears tages.  a t t r a c t i v e , however, i t s u f f e r s from two  disadvan-  Here, to cover the s p h e r i c a l s u r f a c e , pressure  taps would be r e q u i r 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 ,  r  1 D  j  (cm)  S/C (%)  2  3.81  5.08  2.7  4.9  3 7.62 11.0  4  5  6  7  8.89  10.16  11.43  12.70  19.6  24.9  30.6  15.0  43  t h e number o f p r e s s u r e accuracy  o f the p r e s s u r e p l o t .  convey the p r e s s u r e direct  g o v e r n e d by The  suggested  be  used  t u b i n g s , h e n c e i t s s i z e w o u l d be  experiments  t h a t the  the d e s i r e d  stem may  f u n c t i o n o f t h e number o f p r e s s u r e  Preliminary  1.3 mm  taps being  with  inside  tubes  diameter  t o have r e a s o n a b l e  time  taps  of d i f f e r e n t  diameters  s h o u l d be  least  constant  at  (< 5 m i n u t e s ) . t a p s t o be  (this  describe  variations;  inadequate  as w i l l  be  seen  program r e c o r d e d p r e s s u r e well-defined profile), t o be  at least  2.4 cm! sphere would  Obviously has  this  a diameter  the  attractive  diameter  3.8cm.  the  stem  items  of  of the  alternative.  A  stem s u p p o r t  single pressure  the e n t i r e  horizontal  plane  the e n t i r e  s u r f a c e o f the  pressure  signal  e v e r , we  still  smallest  study.  o t h e r hand, a v e r t i c a l  t a p , can  around  Furthermore,  about the v e r t i c a l  to the  of  a  stem  the  t h e near-wake, one  meridional section.  of the  when  a systematic rotation  tal  local  experimental  i s unacceptable  of  fifteen  at t h i r t y "locations to provide  leads to i n s i d e  i n the present On  actual  i . e . , the o u t s i d e diameter  i n t e r f e r e with  interest  an  2 cm,  to p r e c i s e l y later,  a  used.  C o n s i d e r i n g a minimum number o f p r e s s u r e i s grossly  to  and  sphere  axis,  hence, through i f l o c a t e d on  Furthermore,  serve  as a c o n d u i t  t o an  externally  the  presents  tap, can  through cover  symmetry, the h o r i z o n -  stem, i f c o n n e c t e d  for transferring  the  located transducer.  need t o answer t h e q u e s t i o n c o n c e r n i n g  Howan  44  a p p r o p r i a t e s i z e of the stem t h a t would not d i s t u r b the pressure  field. To a r r i v e at a c r i t e r i o n  f o r stem s i z e , an  t e s t program with four stems of equal i n s i d e (d^ = 1.32mm) but v a r y i n g o u t s i d e diameter 6.35, of  and  12.70mm) was  diameters  5.08  i n F i g u r e s 2-11  and  diameter  (d =1.58, 4. 76 ,  undertaken u s i n g two  6.35 cm.  to 2-13.  The  The  extensive  s p h e r i c a l models  r e s u l t s are  presented  e f f e c t of o u t s i d e stem diameter  on pressure p r o f i l e s at a given Reynolds number as p l o t t e d i n F i g u r e 2-11  c l e a r l y shows t h a t  a stem diameter < 4.76  does not a f f e c t the data s u b s t a n t i a l l y .  Furthermore,  mm 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 i n v e s t i g a t e d (Figure 2-12). The  r e s u l t s on minimum pressure  and  the base p r e s s u r e  when p l o t t e d a g a i n s t the o u t s i d e stem diameter to sphere diameter r a t i o  (d /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 Q  as shown i n F i g u r e 2-13.  Note t h a t f o r d /D Q  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 .  <_ 0.1  the  In the present  program, depending on the s i z e of the sphere,  d /D Q  value  test-  varied  i n the range 0 . 0 1 2 5 - 0 . 0 8 3 .  2.4  Pressure  Measurements  The mean pressure component, being extremely  small  2 (of  the order of 0.6898 N/m  instrumentation  ) demanded a h i g h l y s e n s i t i v e  f o r i t s measurement.  T h i s was  accomplished  45  o • A  Rn= 9 6 0 D  2  o •  =  5.08  cm  dj'•=  1.32  mm  d  1.58  mm  0  o •  4.76  a  6.35  A  12.70  O  o  o •  • A  • A  ° a a A A  0  3 0  6 0  A  • A  a A  9  • A  •  •  •  A  A  a  A.  9 0  120  150  180  0°  Figure  2-11  E f f e c t o f the stem diameter on measured s u r f a c e 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  Rn=  960  D  6.35  cm  1.32  mm  1.58  mm  di do  9  o  =  o •  4.76  °  6.35  A  12.70  o ft  fi  sr • • •  Q  o 8  *  *  ft  fi  •  •  D  a  A  A  A  A  o  o  a A  A  ® A A A  0  30  60  90  120  150  180  0° Figure  2-11  E f f e c t o f the stem diameter on measured s u r f a c e pressure d i s t r i b u t i o n over a sphere: (b) Sphere diameter = 6.35 era  47  D  =  5.08  cm  dj  =  1-32  mm  4.76  mm  d  0  =  Rn = )•  607 9 5 5  A  X  A A * A A «  0  3 0  6 0  A  •  # A  9 0  1 2 0  1 5 0  1 8 0  9° F i g u r e 2-12  P l o t s showing s t e m - e f f e c t s to be r e l a t i v e l y independent o f 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 0  *  .8  D  =  5.08  c m  dj  =  4-31  mm  do =12.70  mm  o  314  o  Rn= {• 6 6 4  * 967 .4 Cp 9  .2  A  8  .0  O o  3 A  A  .2  A  %  .  A  o  -.4  0  30  F i g u r e 2-12  60  ft©  A  0  o ,° .S88««.»« 8  e  o  °  A ^ A £ °  •O  0  90  e°  120  150  180  P l o t s showing s t e m - e f f e c t s to be r e l a t i v e l y independent o f the Reynolds number i n the range 300-1000: (b) D = 5.08 cm; d = 4.31 mm, d = 12.70 mm o ±  49  r  - . 3  o  -  .2  f  Cn  -  .  o«  0  5  Rn C p , min  0  Cpb .05  1.  1.5  =  9  6 0  •  D = 5.08  c m  o  D = 6-35  c m  •  D = 5 0 8  c m  •  D = 6-35  cm  2.  D F i g u r e 2-13  V a r i a t i o n o f the minimum and base p r e s s u r e s showing the c r i t i c a l value o f d /D • o 3  -i  2.5 x 1 0  50  using  a " B a r o c e l Modular P r e s s u r e  developed The  type  Transducing  by D a t a m e t r i c s l n c .  o f Watertown,  550-5 B a r o c e l s e n s o r  i s designed  System"  Massachusetts. to operate  with 2  fluids The  over  unit  divider, steel  the pressure  i s a high p r e c i s i o n ,  the  applied pressure.  with degassed transmitting  fluid  and s e n s o r  both  and as a d i e l e c t r i c .  The  to the s i l i c o n e  pressure  system, the The volume  diaphragm i s f i l l e d  o i l which serves  from t h e e x t e r n a l l i q u i d  bellows  the e x t e r n a l  metallic bellows.  isolator  silicone  prestressed  capacitor plates.  diaphragm-capacitance  uses h i g h l y s e n s i t i v e  between t h e b e l l o w s ,  ).  p r o p o r t i o n a l l y t o the magnitude o f To i s o l a t e  medium f r o m t h e s e n s o r  signal  (68.98 kN/m  stable capacitive voltage  d i a p h r a g m p o s i t i o n e d between f i x e d  diaphragm d e f l e c t s  the  psia  the v a r i a b l e element o f which i s a t h i n  The  unit  range o f 0 - 1 0  as p r e s s u r e pressure  medium i s t r a n s m i t t e d by  o i l which i n t u r n d e f l e c t s the  d i a p h r a g m t o p r o d u c e t h e 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  stationary  capacitor plates,  mines an o u t p u t capacitance plates. ing  v o l t a g e a t 10 Hz i s a p p l i e d t o t h e and a b r i d g e  circuit  v o l t a g e d e p e n d e n t on t h e r a t i o  deter-  of the  o f the diaphragm t o each o f the s t a t i o n a r y  The c a r r i e r  voltage  to the input pressure.  i s t h e r e f o r e modulated The u n i t  sensitivity  accord-5  i s 10  2 psi  (0.07 N/m  lated  ) provided  the pressure  from e x t e r n a l sources  was i m p e r a t i v e  sensor  of vibration  i s fully  and n o i s e .  isoIt  t o ensure removal o f a l l t r a c e s o f a i r  51  pockets  from t h e p r e s s u r e  Barocel  i s accurately calibrated  Figure  2-14 p r e s e n t s  ducting  for satisfactory f o r steady  a schematic  operation.  pressures.  diagram o f the p r e s s u r e  transducer. It  was i m p o r t a n t  temperature  excursions  was a c h i e v e d large The  to minimize  by m o u n t i n g t h e t r a n s d u c e r  arrangement v i r t u a l l y  fluid  on a h e a t  a complete  removal o f a i r bubbles  center  46 cm  downstream o f t h e l a s t  d u c t i n g was f i l l e d  with  to a Barocel pressure  polyethylene  and M y l a r  tubings  The p r e s s u r e  read  i n the no-flow  zero output  screen.  Next, the  fluid  transducer  and was  v i a a setof  removal o f a i r  pockets  s e n s i n g u n i t was b a l a n c e d t o condition.  W i t h t h e pump  a t a p r e s e l e c t e d speed t o g i v e a d e s i r e d Reynolds  number and t h e t e s t t h e mean p r e s s u r e meridional  The  from t h e  section with i t s  the t e s t  after  from the l i n e .  velocity  inside.  eliminated the i n f l u e n c e of  a model was p o s i t i o n e d i n t h e t e s t  operating  sink, a  circulating  fluid,  connected  This  transients.  After  pressure  o f ambient  on t h e B a r o c e l ' s p e r f o r m a n c e .  aluminum b l o c k w i t h w o r k i n g  temperature  the e f f e c t  fluid  held a t a constant  distribution  around the h o r i z o n t a l  c r o s s - s e c t i o n was m e a s u r e d .  profile  temperature,  u p s t r e a m of. t h e s p h e r e  For each run the was a l s o  h o t - f i l m p r o b e , mounted on a t r a v e r s i n g g e a r ,  positioned  25 cm  was r e p e a t e d  over  upstream o f the sphere.  The  a r a n g e o f mean f l o w r a t e s .  recorded. was  procedure  .52  Power input  P  2  Modulated output  Stationary Diaphragm capcitor plates  Figure  2-14  A schematic diagram of the B a r o c e l transducer  pressure  53  A point concerning s i z e of the p r e s s u r e systematic time  large  diameter  time  several  constant  of  d e p e n d on  fluid  1.6mm  (>20min) . and  Of  a number o f p a r a m e t e r s  volume i n c l u d i n g  around  o f 4.6 - 6.4 mm 3-5  tubes  as  the  including  the  of the  time  dynamic  diameter  fluid,  inline  c h a r a c t e r of  convenient  i n the  by  would  F o r t h e mean p r e s s u r e  resulting  with  suggested  constant  the t r a n s d u c e r ' s c a v i t y , etc.  and a s s o c i a t e d  s t u d i e s on  time  ments u n d e r c o n s i d e r a t i o n , i t was lines  course,  tubings, v i s c o s i t y  signals,  A  t o have an e x c e s s i v e l y  the  4  size  showed t h e  experimental  lines"*  l e n g t h o f the  pressure  than  the  emphasized here.  of d i f f e r e n t  state pressure  less  theoretical  response  and  steady  appropriate choice of  t u b i n g s must be  study w i t h tubes  to reach  internal  an  measure-  t o use  fluid  constant  (T) o f  minutes.  To  insure accuracy  the measured d a t a ,  i t was  and  compensate  f o r any  and  associated electronic  as w e l l as r e p e a t a b i l i t y o f utmost importance  drift  of the pressure  circuitry.  to  of minimize  transducer  Minute c h a r a c t e r of  -4 the p r e s s u r e l o n g time all  signals  (10  psi) together with  i n v o l v e d i n r e a c h i n g the  t h e more n e c e s s a r y .  Chart  steady  times  as  l a r g e as  well defined pattern. involved  50%  o f the The  relatively  s t a t e made  this  r e c o r d i n g s o f the d r i f t  p e r i o d s o f 2 4 - 4 8 h o u r s showed them t o be at  the  actual  drift  quite  signal,  compensation  t h r e e s u c c e s s i v e measurements a t e q u a l  over  significant, but  of  no  procedure intervals of  54  time  corresponding  This  i s explained i n detail Let  to the  time  of  the sphere  sure of  diagram a l s o  differential r  = P  r  - P  location, case.  w  'a' and  Thus, Now,  as  t o be  w  and  a  c  taken  P  corresponds  indicated  AP  r  c  , where AP  c  the  a  surface pres-  zero  of =P  2-15. the  a  at a  drift  -P  w  and  suitable  t u n n e l w a l l i n the  present  the d e s i r e d P from  -P =AP - AP . a r a r 2-15,  Figure AP  AP  (AP ) r  1 +  + 6 , , 1  r  + 6 , + 6  a  (AP ) r  3  (AP  r  + 6 ) + (AP 1  2  2  AP  Hence,  the  to the  drift  represents pressure on  on  i n Figure  shows t h e c o r r e s p o n d i n g  Here P  differential  L e t the a r b i t r a r y  s y s t e m be  p r e s s u r e s AP  .  t o measure a  represents pressure  at a reference point.  the e l e c t r o n i c  The  AP  at point  system.  below.  t h e o b j e c t i v e be  p r e s s u r e P ^ - P ^ , where P^  c o n s t a n t o f the  r  +  5, + 1  r  + c% + 6 1  2  +  6) 3  55  Figure  2-15  A procedure f o r compensation o f the e l e c t r o n i c d r i f t o f the p r e s s u r e measuring system  56  (AP J (AP  P^) r ' l, +' (—A r ' 3  a'2  =  -  ( A P  a  )  1  - A P  6 P - P  a  r  2  -  r  - 5  6 1  -  2  +  6  3  -63  +  Assuming t h a t the e l e c t r o n i c zero s e t t i n g  drifts  l i n e a r l y d u r i n g the i n t e r v a l marked by the p r e s s u r e measurements ( A P ) , ,  ( A P ) ~ then  r 1  r 2  6  - 63  2  =  0  ,  i.e., (AP  )  + (AP )  Thus d e t e r m i n a t i o n o f the d i f f e r e n t i a l pressure i n v o l v e d the measurement of ( A P ) , , ( A P ) and ( A P ) i n t h a t r x a £ r o 0  order.  The procedure  gave data t h a t can be reproduced w i t h i n  an accuracy o f ±2%.  Note, the e v a l u a t i o n o f the d i f f e r e n -  t i a l pressure P ^ -  a t some d i f f e r e n t l o c a t i o n on the sphere  would f o l l o w the same procedure.  Thus (AP  P  b  " r P  -  (AP  b>4 "  )  + (AP ) .  '  57  A q u e s t i o n may t i a l pressure P  a  -  a r i s e : why  directly.  not measure the  differen-  As shown i n Appendix I,  although t h i s can be done, i t i n v o l v e s n e c e s s a r i l y more measurements.  Furthermore, the r e s u l t i n g formula  i s more  i n v o l v e d and l a c k s r e c u r s i o n c h a r a c t e r l e a d i n g to a subs t a n t i a l i n c r e a s e i n time and Furthermore, any  effort.  f l u c t u a t i o n i n the l i n e v o l t a g e  would be r e f l e c t e d on the pump speed and hence on the s i g n a l s from the model. t o r e d through  The  pressure  speed f l u c t u a t i o n s were moni-  v a r i a t i o n s i n the o r i f i c e meter data.  The  output v o l t a g e from the p r e s s u r e t r a n s d u c e r was  damped using  a DISA type 550  with a r-c  d i g i t a l d.c. v o l t m e t e r equipped  damping c i r c u i t to p r o v i d e an a d j u s t a b l e time constant of up to 100  seconds. A schematic  l a y o u t i s shown i n F i g u r e The  diagram of the 2-16.  t e s t s were conducted  ranging i n diameter range of 280-2200.  instrumentation  on a f a m i l y of  from 3.8 -12.7cm,  spheres,  i n the Reynolds number  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 t u b i n g , which a l s o served as a p r e s s u r e conducting was  line.  Its outside  d i c t a t e d by the r e l a t i v e s i z e of the sphere  i n f l u e n c e on the pressure f i e l d . i n s i d e diameter  was  diameter  and the stem  On the o t h e r hand, the  governed by the time constant to reach  the steady s t a t e pressure as d i s c u s s e d b e f o r e .  The  adopted f o r the experiments had an i n s i d e diameter  stem  of 1.32  mm  u  air supply toflush iquid  d.c. d i g i t a l v o l t m e t e r  osci  u-v. recorder  lloscope  _  Figure  2-16  filter  1  A l i n e drawing o f the i n s t r u m e n t a t i o n used f o r p r e s s u r e measurements  set-up  in  line  59  and an o u t s i d e diameter  of 1.83mm  r e s u l t i n g i n D/d  Q  in  the range of 22.8-66.6. The p r e s s u r e measurements were c o n f i n e d to 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 of the model. connected  the stem through  i n the body of the sphere t a l plane was  a groove  A 1/16 i n . pressure  tap  (1/16 in.dia.) d r i l l e d  (Figure 2-17).  The  entire horizon-  covered by a c o n t r o l l e d r o t a t i o n o f the stem  i n a step s i z e t h a t v a r i e d between 4° to 10° depending upon the g r a d i e n t of the p r e s s u r e p r o f i l e . g e n e r a l were c o n f i n e d to o n l y one  The measurements i n  s i d e of the sphere,  except  f o r o c c a s i o n a l checks to c o n f i r m flow symmetry.  2.5  Drag Measurements The balance used f o r drag measurements e s s e n t i a l l y  c o n s i s t s o f three components: (i) (ii)  removable stem s u p p o r t i n g the s p h e r i c a l model; c e n t r a l suspension o f needle  (iii)  b l o c k supported  by a p a i r  bearings;  i n t e r c h a n g e a b l e c a n t i l e v e r type s e n s i n g u n i t w i t h s t r a i n gages a f f i x e d near i t s r o o t . The  stem s u p p o r t i n g the model i s attached to the  c e n t r a l b l o c k by a thread and nut arrangement, which to be q u i t e convenient  i n changing  i n g the r e s t o f the balance.  the models without  Interchangeable  proved affect-  c h a r a c t e r of  60  to  p r e s s u r e  t r a n s d u c e r  horizonta I meridional section  1/1  6in.dia.  Figure  2-17  A schematic diagram showing the s p h e r i c a l model and i t s support system d u r i n g p r e s s u r e measurements  61  the  s e n s i n g e l e m e n t was  purposely introduced  desired  degree of accuracy i n a given  Several  beams o f v a r y i n g  constructed  flexural  to achieve a  range o f d r a g .  rigidity  and  l e n g t h were  f o r measurement o f d r a g i n t h e r a n g e o f  0.1-15  -3  grams w i t h t h e s e n s i t i v i t y ment i n s e n s i t i v i t y gain of a bridge  o f 10  c a n a l s o be  amplifier  gram. attained  meter.  TWO  on e a c h s i d e o f t h e beam, were u s e d sation. shaped  The  support.  vertical of  t i p o f t h e beam r e s t e d  strain  against  for elimination  arrangement  local  contribution  any c o n t r i b u t i o n  deviation  unit.  carried  2-18  stem  of the i n i t i a l  The  actual i n the  tunnel.  compensate  arrangement  while Figure  from v e r t i c a l .  2-19  This  a l i g n m e n t a s s u r e d by  o f t h e stem,  shows t h e a c t u a l  to can  t h e wedge  no m a t t e r how  i s schematically  cali-  test  f r o m t h e w e i g h t o f t h e s p h e r e due  o f the f l e x i b i l i t y be.  out under  important to e f f e c t i v e l y  o f the s u p p o r t i n g  arise in spite  Figure  was  accuracy of  w i t h a s p h e r i c a l model l o c a t e d  is particularly  may  wedge  c h a r a c t e r o f t h e beam demanded t h a t  b r a t i o n o f t h e b a l a n c e be  because  compen-  a fixed  o f any  one  (0.001 i n ) . Sensitive  This  the  gages,  t h e model w e i g h t t o t h e d r a g , t h e wedge s u p p o r t  0.025 mm  it  by a d j u s t i n g  A l i g n m e n t o f t h e stem w i t h t h e  being c r i t i c a l  improve-  f o r temperature  mounted on a m i c r o m e t e r w i t h p o s i t i o n a l  for  Further  small  shown i n assembled  Figure  2-18  An exploded balance: mediate block; (d) (e) needle  view of the drag measuring \ (a) s u p p o r t i n g stem; (b) i n t e r connection; (c) c e n t r a l suspension c a n t i l e v e r with s t r a i n gages; bearings s u p p o r t i n g the c e n t r a l block  F i g u r e 2-19  (a) Drag balance assembly w i t h b r i d g e a m p l i f i e r meter  65  2.6  Flow V i s u a l i z a t i o n To b e t t e r a p p r e c i a t e the p h y s i c a l c h a r a c t e r of  fluid  f i e l d a s s o c i a t e d with s p h e r i c a l models under c o n f i n e d  c o n d i t i o n , flow v i s u a l i z a t i o n was glycerol-water  o f the model. food c o l o u r .  undertaken.  The  The  i n j e c t e d approximately  dye  employed was  Appropriate  10 cm  volumes of the dye  the same d e n s i t y as t h a t of the t e s t f l u i d . probe c o n s i s t i n g of seven #23  p l a c e d 0.5 -1.0  cm  and  "Intramedic"  r a t e o f i n j e c t i o n was  A dye  tubings  supporting any  manifold.  convenient  i . e . , to  s t u d y i n g the near wake  to i n j e c t dye d i r e c t l y  through  tap l o c a t e d on the s u r f a c e o f a sphere v i a the  stem.  I t was  now  p o s s i b l e to i n t r o d u c e dye  d e s i r e d l o c a t i o n on the sphere.  vortex.  provide  2-21.  The  at  procedure proved  q u i t e e f f e c t i v e i n i d e n t i f y i n g s e p a r a t i o n p o s i t i o n of ring  con-  A schematic diagram  i s shown i n F i g u r e  At times, p a r t i c u l a r l y while  the pressure  was  suspended from the  above the i n j e c t i o n l e v e l .  geometry, i t was  (0.38mm)  c o n t r o l l e d with brass needle v a l v e s .  s u f f i c i e n t head, the supply b o t t l e was  of the complete set-up  inject-  (0.6 mm i n s i d e  to a  To ensure adequate flow through each needle,  ceiling 4 m  s o l u t i o n of  support,  diameter) were used to connect the needles The  pure  s y r i n g e needles  a p a r t on a s t r e a m l i n e d  (Figure 2-20).  upstream  an i m i t a t i o n c o c h i n e a l  g l y c e r i n were mixed to produce a g l y c e r o l - w a t e r  structed  dyed  s o l u t i o n of the same c o n c e n t r a t i o n as t h a t  of the t e s t f l u i d was  ing  the  the  66  F i g u r e 2-20  Dye i n j e c t i n g probe used d u r i n g flow  visualization  F i g u r e 2-21  A sketch showing the equipment l a y o u t d u r i n g flow  visualization  68  I t would be a p p r o p r i a t e  to p o i n t out here the type  of l i g h t i n g system used as i t played photographing process. i n t e n s i t y photo f l o o d s  a c r i t i c a l r o l e i n the  A combination of three  variable  (maximum 5 0 0 watts, 3 4 0 0 ° K ) back-  i l l u m i n a t e d the s u b j e c t through the t u n n e l g l a s s window. e l i m i n a t e hot spots, the l i g h t beam was evenly  d i f f u s e d by  masking the t e s t s e c t i o n w a l l w i t h a t r a c i n g paper. of t r i a l runs helped a r r i v e a t the a p p r o p r i a t e  A set  aperture  s e t t i n g and exposure time f o r the type of f i l m used high  To  (Kodak  speed Ektachrome type E H B - 1 3 5 ( s t i l l ) or E F - 7 2 4 2 ( m o v i e ) ,  tungsten, 3 2 0 0 ° K , A S A 1 2 5 , f i l t e r  81A) .  During the course o f v i s u a l i z a t i o n study, i t was discovered fluid  t h a t i n s p i t e o f the l a r g e volume o f the t e s t  ( 4 0 U.S. g a l l o n s ) , a r e l a t i v e l y small amount o f dye  ( 8 f l u i d oz) was s u f f i c i e n t t o p o l l u t e the working f l u i d to the p o i n t t h a t no c l e a r photographs c o u l d be taken.  This  presented a r a t h e r s e r i o u s problem i n terms o f time,  effort  and  c o s t i n v o l v e d i n r e p l e n i s h i n g the working  fluid.  C l e a r l y , i t was necessary t o f i n d an agent t h a t would n e u t r a l i z e the dye without a t t a c k i n g 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 p h y s i c a l p r o p e r t i e s o f the t e s t f l u i d . no  such agent has been reported  considerable  Unfortunately,  i n the l i t e r a t u r e .  A  amount o f p a t i e n t t e s t i n g w i t h numerous  o x i d i z i n g agents l e d t o sodium h y p o c h l o r i t e which has a l l the d e s i r a b l e a t t r i b u t e s .  Only 3 0 0 cc o f the agent was  69  s u f f i c i e n t to completely n e u t r a l i z e the dye.  To keep the  c o n c e n t r a t i o n of the t e s t f l u i d constant, s u f f i c i e n t amount of g l y c e r i n  was  periodically  added thus o f f s e t t i n g  d i l u t i n g e f f e c t of the dye removing  agent.  the  70  3.  RESULTS AND  DISCUSSION  With some a p p r e c i a t i o n of background to the problem, instrumentation adopted, we  used and  The  amount of experimental  and data  i s r a t h e r enormous, thus d i c t a t i n g a compromise i n  p r e s e n t a t i o n between conciseness The  procedures  are ready to look i n t o the t e s t r e s u l t s  their interpretation. obtained  the experimental  and  comprehensibility.  g u i d i n g p r i n c i p l e has been to i n c l u d e only those  which have immediate relevance  to the study  help i n e s t a b l i s h i n g d e f i n i t e t r e n d s .  i n hand,  In g e n e r a l ,  sequence i n which the r e s u l t s are presented  results and the  a l s o denote  the c h r o n o l o g i c a l order of the t e s t s .  To begin with,  approach to data r e d u c t i o n , so c r i t i c a l  at low  an  Reynolds  number, i s d i s c u s s e d . T h i s i s f o l l o w e d by p r e s e n t a t i o n of the s u r f a c e pressure d i s t r i b u t i o n r e s u l t s as a f f e c t e d by Reynolds number and blockage. drag values are analyzed and  Next, measured sphere  as f u n c t i o n s of system parameters  compared with i n t e g r a t e d p r e s s u r e  skin f r i c t i o n contribution.  drag to e s t a b l i s h  F i n a l l y , near-wake s t r u c t u r e  i s s t u d i e d u s i n g flow v i s u a l i z a t i o n i n c o n j u n c t i o n still  and  16 mm  movie photography.  Available results  from l i t e r a t u r e are i n c l u d e d when a p p r o p r i a t e comparison and of  blockage.  with  for  to a s s i s t i n emphasizing the i n f l u e n c e  71  3.1  Choice  of Reference  Before of the  test  Velocity  proceeding  results,  one  and  w i t h p r e s e n t a t i o n and  must a d d r e s s  tal  q u e s t i o n s which are p a r t i c u l a r l y  low  R e y n o l d s number  immersed ambiguity and of  i n an  flow s t u d i e s .  unbounded u n i f o r m  concerning  stream  analysis  t o s e v e r a l fundamensignificant  Clearly, stream  i n the  with  constant  f a r away from  velocity  the model.  and  For  velocity  pressure  low  Reynolds  number f l o w i n a t u n n e l , however, t h e  fluid  and  the a x i s o f  pressure vary  test  section,  layer and  significantly  the w a l l s .  some compromise i s i n d i c a t e d  Grove e t a l . pressure d i r e c t l y the  model problem.  in selection  of  with  o f t h e pump, as  t o be  acceleration  have s u g g e s t e d  use  of  below the c e n t e r l i n e o f t h e i r pressure  t h e model a b s e n t  and but  the c h a r a c t e r i s t i c  a small blockage prove  58 '  reference static  velocity,  with  to boundary  of the the  the  parameters. 57  may  Presence  a s s o c i a t e d wake w o u l d o n l y a c c e n t u a t e  these  velocity  e v e n i n a b s e n c e o f t h e model due  growth a l o n g  Obviously  as  along  a model  t h e r e i s no  reference or c h a r a c t e r i s t i c  p r e s s u r e : I t i s the the  Pressure  this  model  centerline  a t the  same  velocity.  setting  For  choice of reference  adequate but w i t h  of the  the  the  models  pressure  a l a r g e r blockage,  f l o w a t t h e model l o c a t i o n ,  the  due  to  r e f e r e n c e pressure i s indeed a f f e c t e d function To put  of w a l l confinement  i t d i f f e r e n t l y , the  as suggested above has in i t .  becomes a  (besides other parameters).  c h o i c e of r e f e r e n c e p r e s s u r e  a degree of optimism i m p l i c i t  I t assumes t h a t e f f e c t s of the upstream adverse  pressure gradient exactly  and  created  c a n c e l s the  by presence of the  influence  of a c c e l e r a t i o n  the model l o c a t i o n thus g i v i n g One would be  possible  model  the d e s i r e d  improvement i n the  i n gaps a t  P . T O  choice  of  to take i t as the pressure a t the model l o c a t i o n  (but without the model) w i t h o p e r a t i n g c o n d i t i o n t u n n e l kept the position.  the  same as t h a t used w i t h the model i n  However, t h i s s t i l l  cannot account f o r  changes i n v e l o c i t y p r o f i l e from s e c t i o n a given tunnel,  of  and  the  to s e c t i o n  between tunnels used by  in  different  investigators. U s e f u l n e s s 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 a l s o poses s e v e r a l g e n e r a l , the by  v e l o c i t y p r o f i l e s are  questions.  substantially  affected  l o c a t i o n , boundary l a y e r growth, screen's mesh s i z e ,  blockage, pump speed and  the  total circuit  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 e t a l . can  hardly  be  c  resistance.  proposed by  considered a s u i t a b l e  Another p o s s i b l e  Grove  reference.  compromise would be  to take  uniform p o r t i o n o f the v e l o c i t y p r o f i l e f a r upstream use  In  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  and  distance would, and  i n v o l v e d to account  i n general,  f o r boundary l a y e r  depend upon t h e  tunnel  effects  u s e d , model  i t s location. A  presenting  rather significant data  investigators,  p o i n t t o keep i n mind i n  i s to ensure i t s r e p e a t a b i l i t y using d i f f e r e n t  test  i n mind and  by  facilities,  to  With t h i s  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 a b o v e a com-  test  s e c t i o n b a s e d on  T h i s a p p r o a c h has a t e s most o f  the  unchanged. of  average v e l o c i t y  model l o c a t i o n ,  u s e d and  size  type  o f the  problems o f p r e s s u r e would  facilitate  being  more p r e c i s e l y  instrumentation. does n o t  with  d i s t a n c e and  to  location  defined. and  flow  s e c t i o n but  gradient  and  the  a l s o overcomes  Furthermore,  resulting  question  straighteners  The  , reference  However, i t must be  model.  model)  test-section  choice  velocity i t s measure-  conventional emphasized  c o r r e c t f o r changes i n v e l o c i t y  of the  adopted.  without  blockage.  involves only  hence t h e  the  It elimin-  does i t e l i m i n a t e t h e  of tunnel,  test  and  as w i t h  i n the  d u p l i c a t i o n of R  ment i s q u i t e s i m p l e  this  only  (U), was  in  above.  conducted with  same meter s e t t i n g  Thus, not  rate  advantages.  the problems mentioned  (but a t the  would l e a v e  t h e mean f l o w  tests  consider-  average v e l o c i t y  s e v e r a l obvious  Obviously, model  velocity,  careful  permit  comparison.  promise c h a r a c t e r i s t i c  after  other  pressure  that  profile effects  due  T h i s b r i n g s us to t h a t ing P .  e l u s i v e task of  As d i s c u s s e d e a r l i e r , the P  Grove e t a l . has  advocated by  l i t t l e meaning here i n view of the l a r g e  blockage presented  by the model.  view o f r e p e a t a b i l i t y pressure  From the p o i n t o f  and comparison of data, the use  at a s p e c i f i e d  of  tap on the s u r f a c e of the model  as r e f e r e n c e appears q u i t e a t t r a c t i v e .  Although  cannot account f o r l o c a l v a r i a t i o n s due effects  select-  this  to blockage  (from p o i n t to p o i n t at the s u r f a c e o f the model),  i t could e f f e c t i v e l y  compensate f o r i t i n an  average  fashion. Thus one way efficient P  r  to p r e s e n t p r e s s u r e  form would be as C  p  = (P. -P  corresponds to the pressure  9  r  data i n co-  )/(pU H  2 /2) '  at a s p e c i f i e d  where  tap on  the  s u r f a c e o f the poppet and U as c a l c u l a t e d from the average flow r a t e o f 20. 32 cmx still  (average flow r a t e / t e s t - s e c t i o n area  20. 32 cm) .  However, t h i s d e f i n i t i o n i s  s u s c e p t i b l e to e r r o r s i n t r o d u c e d by  of the v e l o c i t y  profile  (at a pressure  non-uniformity  tap and  the  r e f e r e n c e l o c a t i o n ) p a r t i c u l a r l y because the denominator remains u n a f f e c t e d by t h i s change.  One  way  to v i r t u a l l y  e l i m i n a t e t h i s shortcoming i s to express pressure i e n t as e x p l a i n e d below (Figure  3-1).  coeffic-  75  F i g u r e 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 non-uniformity of the velocity profile  Let e r r o r s i n pressure due of the v e l o c i t y p r o f i l e be P  .  Expressing  pressure  at P Q , e  Q  at P  Q  and  ( P  gives  9  (P  +  0  £  9  +- ) &Q  )  at  between t h a t a t a tap i n  the s t a g n a t i o n p o i n t with r e s p e c t to  reference pressure,  P  non-uniformity  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 p r e s s u r e s , q u e s t i o n and  to  -  ( P  r  (P  +  r  £  +  r  }  e) r  the  where P , P , P„ correspond H r u Q  velocity profile.  (  Note t h a t e - e n  0  uniform  Thus  P. - P =  to p r e s s u r e s w i t h  0  r  1 +  (e -e )/(P -P )  1 +  (e -e )/(P -P )  e  0  r  e  r  r  0  r  and e_ - e are l i k e l y t o be very s m a l l . 0 r  r  On the o t h e r hand, P  -P r  D  o  and P  A  u  - P represent r e l a t i v e l y r  l a r g e q u a n t i t i e s compared t o the r e s p e c t i v e e r r o r differentials.  Therefore, 0 - r P -P £  -  E.  6r  . and  £  —  —  Q  -  0" r =r— P -P  £  e_ Or  r  are l i k e l y t o be v a n i s h i n g l y s m a l l .  =  £  =  Q  r  Consequently, the  term 1 +e  0r  1  +  £  -.  Q  "  1  , a  n  d  0r  C  p P  „ P  P  Q  0  -P  0  - P r  r  Both numerator and denominator b e i n g s e n s i t i v e , the proposed  d e f i n i t i o n o f the p r e s s u r e c o e f f i c i e n t  promises  to p r o v i d e adequate compensation f o r 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  profile.  The r e f e r e n c e l o c a t i o n was taken t o be a t 0 = 60°, The c h o i c e was prompted by the t e s t data which showed C  77  to  reach  i.e.,  Pggo  location The  zero  i n the  ~  ( F i g u r e 3-2).  o f the  pressure  definition  general v i c i n i t y  reference pressure  data presented  P  =  in this  -  P  -  P  P  0  i s easy  course,  i n general,  chapter  arbitrary,  use  the  as  60°  9  P  6 = 60°,  is entirely  of pressure c o e f f i c i e n t  C  It  Of  of  60°  to r e c o g n i z e the  term  P Q - P^QO  as  an  approximation  2 of  (1/2)pU^ .  However, now  we  are  the e r r o r s  i n t r o d u c e d by  profile.  T h u s , i n summary, t h i s  advantages: blockage and  i t tends  effects,  drifts  electrical  drift  in  in  was  comparison with  t o compensate  this  new  has  several  sphere  diameter),  similar  test  d a t a by  2).  by  (the Furthermore,  ( b a s e d on  i t promises other  gradient,  profile  sensing system  t h e R e y n o l d s number  for  velocity  f o r the p r e s s u r e  d i s c u s s e d i n Chapter  average  to  assist  investigators  facilities.  A q u e s t i o n may culty  coefficient  o f the p r e s s u r e  and  using d i f f e r e n t  of the  i n p r e s s u r e measurements c a u s e d  conjunction with  flow v e l o c i t y  non-uniformity  to account  i r r e g u l a r i t y o f the v e l o c i t y  possible errors  electrical  likely  arise  definition  may  as  to the p o s s i b l e  cause  i n comparing  diffitest  78  D  1.0  2  S/C  6 0.5  5.08  .  =  Rn  6  4.9 •  302  o  640  A  953  cm %  ©  a  2  0.0  o 6  -  o  AD 6  o  0.5  . • . . • -  1.0  -  1.5  -  2.0  2.5  0  30  60  120  90  180  150  6°  F i g u r e 3-2  T y p i c a l pressure 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 p r e s s u r e c o e f f i c i e n t , 2 Cp =  ( P Q - Pa,) / (1/2) p u  c  .  Note the p r e s s u r e  e f f i c i e n t i s zero i n the v i c i n i t y o f 9 = 60°  co-  79  data w i t h other published i n f o r m a t i o n . does n o t p r e s e n t  any problem.  conventional pressure  Fortunately,  this  As shown i n Appendix I ,  coefficient C  can be w r i t t e n i n  P  terms o f measured i n f o r m a t i o n as  cr P  =  P  e" " P  ( p =  " ^ P "  e- 60" p  )  ( p  o- 6o° p  l/2pu  2  )  +  1  2  w i t h an e r r o r o f <3% i n t h e Reynolds number range i n v e s t i gated h e r e .  ~  Representative i n Figures  surface pressure  d a t a as  presented  3-3 t o 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  o f t h i s new d e f i n i t i o n o f t h e p r e s s u r e  coefficient.  The e f f e c t o f d i f f e r e n c e s i n v e l o c i t y  profiles,  as e n c o u n t e r e d by a model l o c a t e d a t d i f f e r e n t s t a t i o n s i n t h e t e s t - s e c t i o n i s shown i n F i g u r e 3-3. pressure  The  conventional  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 a l m o s t P  over t h e e n t i r e s u r f a c e e x c e p t f o r a s m a l l r e g i o n i n t h e v i c i n i t y of the stagnation  ( F i g u r e 3-3a).  Although  f a i r s b e t t e r i n t h e r e g i o n 8 > 60°, t h e p l o t s a r e c o n s i d e r a b l y d i s t o r t e d i n t h e upstream d i r e c t i o n ( F i g u r e 3-3b).  S u r p r i s i n g l y , t h e new p r e s s u r e  coefficient  remains q u i t e i n s e n s i t i v e t o the v e l o c i t y changes over  80  1.Q  0.5  0.0  o  h  -  0.5  -  1.0  D  =  8.89  S/C  =  15.0 %  Rn  =  1003  Model  Location  Tunnel o  cm  from  Inlet  •  100  cm  o  163.  cm  • -  •  •  1.5  *•••* -  •  ooooo  ° o  o  °  o  2.0 °  o  o  o o -  2.5  0  30  60  o  o  o  90  120  150  180  0° Figure 3-3  P l o t s showing s e n s i t i v i t y o f 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 profile: „ (a) C  p  =  (P  e  - P o o ) / ( 1 / 2 ) plT  81 3.0 • o  2.5 [  D  =  S/C  =  Rn  =  2-0  8.89  cm  15.0 1 0 °  Model  % 3  Location  Tunnel  from  Inlet  •  100  cm  o  163  cm  1.5 A  Cp  |  O  1.0  O 0.5  0.0 \  o < * » © « . 2 » ®  • o  9  •o 0.5h  9  0  Figure  30  3-3  60  *o  h  90  120  150  180  P l o t s showing s e n s i t i v i t y o f d i f f e r e n t d e f i n i t i o n s f o r pressure c o e f f i c i e n t to changes i n v e l o c i t y profile: (b) C = ( P Q - P o ) / ( l / 2 ) p U 2 p  6 0  the e n t i r e s u r f a c e even w i t h t h e e n l a r g e d s c a l e used i n p l o t t i n g the data  (Figure  3-3c).  F i g u r e 3-4 summarizes e f f e c t o f t h e Reynolds number on t h e s u r f a c e p r e s s u r e f o r t h r e e d i f f e r e n t d e f i n i t i o n s o f the pressure c o e f f i c i e n t . the c o n v e n t i o n a l d e f i n i t i o n C  Note t h a t both  as w e l l as i t s P  modification  as g i v e n by t h e use o f 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 t o t h e i n f l u e n c e o f Reynolds number ( F i g u r e 3-4a,b). definition  On t h e o t h e r hand, t h e proposed  ( F i g u r e 3-4c) shows o n l y s l i g h t  i n and near t h e wake r e g i o n .  sensitivity  Note t h a t t h e s c a l e used  i n F i g u r e 3-4c m a g n i f i e s d e v i a t i o n s by a f a c t o r o f 2.5. Thus t h e 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 t h e p r e s s u r e d i s t r i b u t i o n v i r t u a l l y  independent  o f t h e Reynolds number i n t h e range i n v e s t i g a t e d .  The  r e l a t i v e independence o f t h e 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  F i g u r e 3-5 makes t h e proposed d e f i n i t i o n  extremely  attractive. As p o i n t e d o u t b e f o r e , t h e c h o i c e o f r e f e r e n c e pressure i s quite a r b i t r a r y .  However, e f f e c t i v e com-  p e n s a t i o n o f e r r o r s i n t r o d u c e d through  various  sources  mentioned e a r l i e r w i l l indeed depend on a g i v e n P f o r 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  D  =  8.89  S/C  =  15.0 %  Rn  =  1003  Model  o  .6  Location  Tunnel  .4  cm  from  Inlet  •  1 0 0  cm  o  1 6 3  c m  o a  •  Cp .2  .0  O  6 9  9  «  ®  ®  o  o • ° _ o« -.2  .4  0  30  60  90  120  150  180  6°  F i g u r e 3-3  P l o t s showing s e n s i t i v i t y o f 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 profile: ( C )  C P  =  <P -P .>/<P -P o> e  6 0  0  6 0  84  1.0  o  D  =  5.08  S/C  =  4.9%  cm  A  0.5  Rn  9 0.0  •  302  o  640  A  953  6  o  -0.5  A  o A  Cp  fi -1.0  A  A 6 0  2  o  2  2  2  .o  2  6  ri  A°  a S  A  AO  °  ^  • •  -1.5  • •  • •• -2.0  -2.5  0  30  60  90  120  150  9° F i g u r e 3-4  E f f e c t o f Reynolds number on s u r f a c e d i s t r i b u t i o n i n terms o f : (a) C  p  = ( P - P j / ( l / 2 ) pU e  2  pressure  180  85  D  1.0  2  S/C  6 0.5  -  5.08  =  4.9  Rn 6  2  0.0  ©  Q  6 e,  A  %  •  302  o  640  A  953  ©  fi  cm  A  Q  6  AO 6 A 6  O  ©  O  ©  6  0.5  1.0  • •• -  1.5  -  2.0  2  "  5  0  30  60  90  120  150  6°  F i g u r e 3-4  E f f e c t o f Reynolds number on s u r f a c e d i s t r i b u t i o n i n terms o f :  pressure  180  86  1.0  D  =  5.08  S/C  =  4.9%  o  a  .8  Rn .6  cm  •  302  o  516  A  640  •  953  .4 Cp  i  .2  •  8 §  .0 ° AO o O A D A O « • AO • D  N  2 - . 2  Q A A  »° -  -  .4  0  30  D  A  O •  o • o •  e  60  6  120  90  150  o  Figure  3-4  E f f e c t o f Reynolds number on s u r f a c e d i s t r i b u t i o n i n terms o f : (c) C  ^ 9 ~ 60°^ ^ ^ 0 P  P  P  P  60  o )  pressure  180  87  Rn 825 948 994 999  •  • o A  •  D.cm S / C % 3.81 2.7 7.62 11.0 10.16 19.6 12.70 30.6  A  a  6 o •o ° ° o ° ° o  o  • o • o o , • o o  o A  O  •  ,  o  0  A A  o o  A  o  A  o  o  o  A  A  Q  A  •  A A  A  •  •  •  •  0  30 3-5  60  D  90  120  150  9° 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 of 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 pressure c o e f f i c i e n t (C ) based on average v e l o c i t y , C  ==  (P  e  - P j / d / 2 )  pU  2  P  180  88  i.Or  .8  6 3  Rn  D.cm  S/C  %  •  825  3.81  2.7  o  7.62  11.0  ^  948 994  10.16  19.6  •  999  11.43  24-8  •  999  12.70  30.6  A  .6  O  1  .4  O  Cp A  .2  °  ^XD • • Oo »  .0  *:^. •  cj  9  n~  o  ° -  .2  r*  fl  A  •  30  60  »  s . : A  ft  *.  O  A  i  4  •  O  D  90  120  150  0° F i g u r e 3-5  R e p r e s e n t a t i v e pressure 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 of 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 : (b) suggested pressure c o e f f i c i e n t d e f i n e d as C  p  =  < e- 6op  p  ) / { p  o- 60 R  o )  6  A  • •  .4 0  _  A A  •  6  •  A  •  -  A  0  •  geometry o f t e s t model, e t c .  Hence, although  d e f i n i t i o n i s l i k e l y to be l e s s dependent on  the proposed fluid  dynamical parameters compared to the c o n v e n t i o n a l degree of v a r i a t i o n  may  reference pressure.  indeed  depend upon the chosen  Data reduced using 0 = 3 0 ° ,  90° as r e f e r e n c e s s u b s t a n t i a t e a q u e s t i o n concerning pressure  arises.  60°,  this observation.  the optimum c h o i c e of the  In g e n e r a l , i t would be  50° -120°  Thus  impossible However,  r e c o g n i z i n g the f a c t t h a t f o r most b l u f f bodies  the  between the minimum and base p r e s s u r e s  r e l a t i v e l y constant,  remains  r e f e r e n c e 6 i n the range of around  i s l i k e l y to l e a d  to good r e s u l t s .  Figure  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  PgQ  as r e f e r e n c e .  0  Note t h a t p l o t s  remain  u n a f f e c t e d except f o r a s m a l l r e g i o n i n the of the  3.2  and  reference  to i d e n t i f y an optimum r e f e r e n c e f o r a l l cases.  difference  C , the  3-6  using  essentially vicinity  stagnation.  E f f e c t of Reynolds Number F i g u r e s 3-7  prehensive  through 3-9  summarize a r a t h e r com-  s e t o f data on the s u r f a c e pressure  b u t i o n f o r a sphere as a f f e c t e d  distri-  by the Reynolds number  f o r a given blockage i n the range 4.9 - 30.6%.  Results  90  1.0'  Q  \  9  D  =  8.89  .cm  S / C  =  15.0 %  Rn  =  1003  o  Model Tunnel  Location  from  Inlet  • 1 0 0  c m  °  c m  1 6 3  2\  a  # c  v* »»* a  6  • *  9  ft  o  -.2  0  30  60  90  120  150  9°  F i g u r e 3-6  Surface pressure d i s t r i b u t i o n on spheres using PgQo as r e f e r e n c e . Note the p l o t s show veryl i t t l e dependence on;(a) v e l o c i t y p r o f i l e  180  91  1.0  D  =  5.08  S/C  =  4.9 %  cm  A  .8  Rn  •6 o  •  3 0 2  o  640  A  953  A  o A  •  O A  2 \  2.  o 6 2  .0  -  .2  t«  *.  6«  a  #22  o A  o A  o  o  o A  2S  30  0  Figure  3-6  60  90  120  150  Surface pressure d i s t r i b u t i o n on spheres using PgO° as 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 o n ( b ) Reynolds number :  180  92  D  O  •  •  Rn  D.cm  •  825  3.81  o  994  •  999  S/C % 2.7  O  • a o  o  • o  8  <3 W o"  cfl  30  3-6  60  90  120  150  0° 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 PgQo as 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:(c) blockage  180  93  by  other  investigators  when a v a i l a b l e . sented  C  P  also  definition  are a l s o  included to i l l u s t r a t e  by b o t h At  approaches  t h e o u t s e t one  limited  ratio  of  can  say  except  n  and  c o n f i n e d to the  point  together with  the  location  even here i t  of  Reynolds  (as i n d i c a t e d  uniform pressure  r e g i o n o f t h e wake) t e n d  upstream.  the  front  p r e s s u r e p o i n t , the e f f e c t t o be  just  w i t h an  the o p p o s i t e ,  the  to s h i f t  to note  s t a g n a t i o n and  i . e . , the p r e s s u r e  the a  little  t h a t i n the the  o f R e y n o l d s number  i n c r e a s e i n the Reynolds Figure  of  pressure  the b e g i n n i n g o f  I t i s o f some i n t e r e s t  r e g i o n bounded by  t h e wake  approximate l o c a t i o n by  blockage  o f t h e minimum  separation point  of  region  very high  In g e n e r a l , the e f f e c t  Furthermore,  3-8d).  t h a t the e f f e c t  f o r the  of  trends  number i s t o i n c r e a s e t h e minimum as w e l l as pressures.  coefficient  i n terms  identical  z e r o p r e s s u r e p o i n t and  to R <1000,  30.6%.  results  ( F i g u r e s 3-7b  R e y n o l d s number i s e s s e n t i a l l y downstream o f t h e  information i s pre-  of the p r e s s u r e  b e f o r e , however, t y p i c a l  predicted  is  i n c l u d e d f o r comparison  I n most c a s e s , t h e  u s i n g t h e new  discussed  are  zero appears  decreases  number.  3-7(c) compares t h e p r e s e n t d a t a  with  t h e h i g h e r R e y n o l d s 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  at R  n  =  5848 t e n d  to s u b s t a n t i a t e e a r l i e r  observation  94  1.0  D  =  5.08  S/C  =  4.9%  o  S  .8  Rn  •  3 0 2  o  516  A  640  •  953  cm  .4 Cp  e •  .2  .0  • • A  • o  2 - . 2  • A  •  •  D  A  0  30  •  °  •  Q  A  • ••  .4  ® O  • AO * • AO • O o.^.b• ^••A • •°o o°  -  0  •  e  60  90  120  150  o  F i g u r e 3-7  S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by Reynolds number a t a s m a l l blockage r a t i o o f 4.9%: ( a )  C P  -  ( p  e- 60° p  ) / ( p  o- 60° p  }  180.  95  1.0  °  •  0.5  D  =  S/C  =  A  \  cm  4.9%  Rn  o  0.0  5.08  6  •  302  o  640  A  953  o  -0.5  A  o A  2  A  2  -1.0  A 6 0  o  2  2  2  A  J>  6  6°  AO S  ,  ©0°"  •  -1.5  ••• -2.0  -2.5  0  30  60  9 Figure  3-7  120  90  150  o  S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d R e y n o l d s number a t a s m a l l b l o c k a g e r a t i o of 4.9%: (b)  C  p  =  (P  0  180  - Pj/(l/2)  pU  2  by  96  „ o  D  =  5.08  S/C  =  4.9%  Rn  •  cm  3 0 2  •  5848  |  4  ?  Aminzadeh  • °  g  S  5  1  o  6  A  640  •  9 5 3  137X10  3  Maxworthy 162 x 1 0  3 g  Achenbach •  •  198 X 1 0  ^  Maxworthy ^ • • •  •  •  with Trip  •  •  Wire  © •  o oo • cr> O  ^  30  3  60  • A  s A  o  t°  °  O  •  90  120  150  6° S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by R e y n o l d s number a t a s m a l l b l o c k a g e r a t i o o f 4.9%: (c) c o m p a r i s o n w i t h r e c e n t d a t a by o t h e r investigators. Note t h e r e s u l t s by Maxworthy and A c h e n b a c h a r e n e a r c r i t i c a l R e y n o l d s number (R = 3.7 x 10^, R e f e r e n c e 9) n,cr '  180,  97  concerning  independence of  However, n o t e t h e approaches the R  n  =137  3  x10  for R >  value  pressure  profile  R e y n o l d s number  point.  and  This  an  1000.  n  as  one  (Maxworthy"'""'",  T h e r e i s a sudden i n c r e a s e  minimum p r e s s u r e separation  pressure  change i n t h e  critical  ).  the  in  upstream s h i f t  i s associated with  the  in the  the familiar 3  slight  increase  However, a t  i n the  the  drag i n t h i s  critical  region  R e y n o l d s number  (R  =6x10  (R  5 -2x10). 5 v 2x10 )  n,critical there  i s a sudden r i s e  s t r e a m movement o f classical  reduction At  3-8) for  number  begins (Figure  Modi and  i n drag  3-9). 59  study with Wall  11-19.6%  i s maintained.  i n the  a little  of  down-  wall  (Figure However,  confinement the  s e n s i t i v e to the  also observed c y l i n d e r of  the  same t r e n d  35.5%  base  Reynolds  I t i s i n t e r e s t i n g to note here  a circular  in  that their  blockage.  Confinement E f f e c t s Figures  3-10  and  3-11  summarize r e s u l t s on  i n f l u e n c e o f b l o c k a g e o f f e r e d by It  the  n  same t r e n d  t o be  and  p o i n t r e s u l t i n g i n the 11 3 (Maxworthy , R =198 x 1 0 ).  blockage r a t i o s  the  Sherbmy  base p r e s s u r e  separation  further increase  pressure  3.3  higher  essentially any  the  i n the  must be  attainable  recognized  that  s p e e d s i n any  the  liquid  the  the  s p h e r i c a l models.  minimum and tunnel  are  maximum fixed  by  98  1.0 6  D .8  =  7.62  S/C = 1 1 . 0  6  cm  %  594  Rn  o  948  ©  A1294  •4  o A  o  Cp  A  .2 A  o .o\  A<oa£a6S*6  © A  2 6  '6  A  o  *  *  0  AO A° O •  -.2  - .4  0  30  60  90  120  150  180  0° F i g u r e 3-8  Pressure 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 i n t e r mediate values o f blockage: (a) C , S/C=11.0% P  99  1.0  .8  D  =  S/C  =  Rn  15.0 •  cm %  1002  o1283  2  .6  8.89  A  1585  .4  .2  ^fi ! S • g  .0 6  A  ©  &  *  o  $  AO  A <5  o*  - . 2 1  .4 0  30  60  90  120  150  180  0° F i g u r e 3-8  Pressure 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 i n t e r m e d i a t e values o f blockage: (b) C , S/C = 15.0%  100  1.0  .8  £  D  =  10.16  S/C  =  19.6  Rn  •  cm %  994 o 1 2 8 5  e A A  1589  2 . 4  Cp  .0  A $ $ ° « A •  8 8  0  30  60  *  4  8  g  ®2 6  .2  «  i  #  A «  *$9  .  90  120  150  180  0° F i g u r e 3-8  Pressure 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 j> 1000 and f o r i n t e r m e d i a t e values o f blockage: (c)~C , S/C = 19.6%  101  1-0  ! 0.5  ft  D  =  10.16  S/C  =  19.6  %  994  Rn  •  0.0  c m  Q1285 A  -0.5  1589  A  o  Cp  -1.0 2 -1.5  5  . 8 8 ° *  8  ©*  4  8  -2.0  A  -2.5  0  30  60  2  8  § 2 '  5  5  A  A  A£®  90  120  150  180  9° F i g u r e 3-8  Pressure 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 2^1000 and f o r i n t e r m e d i a t e values o f blockage: (d) C , S/C =19.6%  102  1.0 2  .8  6  D  =  11.43  cm  S/C  =  24.6  %  Rn  •  6  .6  999  O1282 A2014  O  .4  2[  .0  8  e  -.2  6  A  w  0  .4 0  30  60  90  120  150  9° F i g u r e 3-9  Reynolds number e f f e c t on the p r e s s u r e t i o n a t h i g h e r blockage r a t i o s : (a) C , S/C P  =  24.6%  distribu-  180  103  1.0  .8  I  D  =  12.70  cm  S/C  =  30.6  %  Rn o  .6  •  e  •  999  o  1602  A  2006  •  2291  .4  .2  .0 o o •  • •  n  a  a  o  6  9 A  O A  9 ^  8  2~ • -  .4  0  30  60  °  90  120  150  9° F i g u r e 3-9  Reynolds number e f f e c t on the p r e s s u r e b u t i o n a t higher blockage r a t i o s : (b) C , S/C P  =  30.6%  distri-  180  1.0  104  0.5 h  0.01-  o  °  D  =  12.70  S/C  =  30.6  A  Rn  $  •  %  •  999  o  1602  A  2291  -0-5  cm  -1.0  -1.5  A O -2.0 ° Q  A  o  2.5  A  g  2 2  Q  o  o  A  A  eA  ° A  ° A  •  ° A •  O  A  O  ©  Og  A  2  o  -3.0  0  30  60  90  120  150  0°  F i g u r e 3-9  Reynolds number e f f e c t on the p r e s s u r e b u t i o n a t higher blockage r a t i o s : (c)  C , S/C  =  30.6%  distri-  180  105  design  considerations.  For the present  facility  were 0.5 cm/s and 15 cm/s, r e s p e c t i v e l y . given blockage, desired led  n  r a n g e o f R e y n o l d s number  to unavoidable  > 600.  (300-2000).  gaps i n t h e r e s u l t s are reasonably  From F i g u r e  tendency (Figure  3-10 i t i s a p p a r e n t  The minimum p r e s s u r e  rearward  shift.  t h a t f o r up  are e s s e n t i a l l y  t o r e d u c e t h e minimum a n d b a s e  distinct  has d e f i n i t e pressures  p o i n t shows a  S i m i l a r downstream movement  o f t h e s e p a r a t i o n p o i n t c a n a l s o be d i s c e r n e d it  i s not quite d i s t i n c t .  described can  later  be e x p e c t e d  effect  ( S e c t i o n 3.'6) c o n f i r m e d from t h e p r e v i o u s  pressure  trend.  t h e same f o r R  earlier  >1000.  (Figure  3-12 shows v a r i a t i o n  pressure  3-4d).  o f the average  and t h e minimum p r e s s u r e w i t h b l o c k a g e .  t o S/C o f a r o u n d 12 - 15% t h e b a s e p r e s s u r e t h e minimum p r e s s u r e  As  n  i n terms o f c o n v e n t i o n a l  c o e f f i c i e n t were p r e s e n t e d Figure  this  study  d i s c u s s i o n , the blockage  J  results  although  A flow v i s u a l i z a t i o n  remains e s s e n t i a l l y  Corresponding  here,  well established for  b u t beyond t h a t the blockage  3-11).  T h i s has  presented  t o a r o u n d 11%, t h e c o n f i n e m e n t e f f e c t s negligible  Hence, f o r a  i t wasn't always p o s s i b l e t o c o v e r t h e  however, t h e t r e n d s R  they  remains e s s e n t i a l l y  however, b e y o n d t h a t t h e r e i s a d i s t i n c t  base  Up  as w e l l a s  constant, reduction i n  106  1.0  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  o  .8  W  A •  •6  2 o 4  .4 Cp  e  •0  •  A - «5 O  - .2  Q*  a  60  AP'  90  120  150  0° F i g u r e 3-10  6  *5  5  30  A  A *  A  0  S  cfi  *  - .4  &  # 0  R e 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%  180  tor  Rn  D.cm  •  825  3.81  2.7  °  948  7.62  11.0  994  10.16  19.6  •  999  11.43  24-8  •  999  12.70  30.6  9  m 6  •  A  D  •  S/C  %  o  I o  A  o • °  ^ f t C ^ A  B  A  • •  0  30  60  .  A  A  A  A A  A A •  90  •  •  120  150  9° F i g u r e 3-11  Pressure p l o t s as a f f e c t e d by h i g h e r (a) R  n  =  950  ^  blockage:  1*  108  1.0  Rn  s  S/C,%  •  1227  6.35  7.6  o  1  283  8.89  1 5.0  A  1  285  1 0.1  •  •  1  282  •  •  1278  .8  .6  D,cm  6  19.6  1  1.43  24.8  1  2.70  30.6  • 11  o  g  .2  •  .0 •  A  o  -  0* «o A i O'' <tfO A •  -2  0o  •  _° •  o0  0  2  A  " •  *  •  •  0  •0 0• 0 •0 • A  n  n D  DI  -  -4  0  30  60  90  120  150  9° F i g u r e 3-11  Pressure p l o t s as a f f e c t e d by h i g h e r blockage: (b) R  n  =  1250  180  109  1.0  I  Rn  8 •  •-  .  8  n  D,cm  S/C,%  •  1585  8.89  1 5.0  o  1589  10.16  19.6  A  1582  1 1.43  24.8  •  1602  1 2.70  30  6  •  • A  • 4  l  •  •  Q  6  2\  .o i  •  . • #  .  °  a  ' 0 4  30  60  A  A  A  O  Q  A A  O  •  o  Q A  • °  A  D  •  o • o  o  A  A  o  A  D  °  •  .  ° A  °  A  D  n  =  90  1600  °  •  120  P r e s s u r e p l o t s as a f f e c t e d by higher '(c) R  A  •  150  9° F i g u r e 3-11  •  o °  blockage:  180  110  1000<  Rn  <1600  ° -Cpb •  ~Cpm Cpb  A  -  Cpm  A  I  I  8  S  j. 10 gure 3-12  20  30  S/C % 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 < 1 6 0 0 . Note both C and C are e s s e n t i a l l y constant up to the blockaVige r a t i o o f around 13% D  D  m  the p r e s s u r e s w i t h blockage. a critical effect  Thus t h e r e a p p e a r s  value of the blockage  of wall  confinement  tends  ratio  to  above w h i c h  be  the  t o become s i g n i f i c a n t .  This represents information of considerable p r a c t i c a l significance. Interestingly  the d i f f e r e n c e  C  - C  p  b a measure o f t h e p r e s s u r e r i s e s u s t a i n e d by layer of  prior  to s e p a r a t i o n , remains  the blockage  this  throughout.  q u a n t i t y from  relative  blockage.  free  However, a t l o w e r  change.  this to  (Figure  remain  Although  drop  - C p  Useful  Figure effects  on  The  b  (S/C o f 30.6%  n  and  to  this  =600  trend  confirmed  there i s in a  . p  m number  and  base p r e s s u r e i s p r e s e n t e d i n results  suggest  are c o n f i n e d to the range ratios  at R  to blockage,  t h a t Reynolds R  3  blockage  from  number, due  condensation o f Reynolds  effects  3-14.  local  i n t h e minimum p r e s s u r e r e s u l t i n g  increase of C  blockage  to the  the base p r e s s u r e c o n t i n u e s  relatively insensitive  a definite clear  3-13).  of  suggests  would expect  Measurements c o n d u c t e d  boundary  independent  resulting  Reynolds one  the  independence  effects  velocity  dominance o f v i s c o u s f o r c e s , to  near  o f the boundary-layer stream  m  virtually  the confinement  insensivity  changes i n the  The  , which i s p  R  n  <1000 f o r a l l n  >1600 f o r t h e h i g h e r  i n the p r e s e n t c a s e ) .  number  blockage  112 Rn  =  600  o  Cpb  •  -Cpm Cpb -  A  Cpm  .2  .1  o  •0  0  5 S / C  F i g u r e 3-13  o  10 %  P l o t s showing dependence of C  (and hence m ) on w a l l confinement, even when S/C n  p  C  n,  ~ C  n  i s l e s s than 13%, a t R  < 1000 n  o  D,  S/C%  cm  o  6.35  7.6  •  8.89  15.0  A  10.16  19.6  •  11.43  24.8  •  12.70  30.6  3  4  5  6  7  Rn X 1 0 Condensation of the base p r e s s u r e data showing number and blockage 2  F i g u r e 3-14  the i n f l u e n c e of  Reynolds  114  3.4  Drag  Coefficient Pressure  having  the  surface of a  i n t e g r a t e d value of drag  d e p e n d e n c e on the  direct  on  b e e n e s t a b l i s h e d , the next l o g i c a l step was  the p r e s s u r e  for  distribution  skin  blockage. friction  and  measurement o f s k i n  On  friction  cannot  account  t h e o t h e r hand, t h e  i s c o n f i n e d to  4 high  to obtain  assess i t s  This, of course,  contribution.  sphere  the  6  R e y n o l d s number r a n g e o f 5 x 1 0  - 6 x10  , by  9 Achenbach  .  The  corresponding  information at  lower  4 R e y n o l d s number this  (R  as b a c k g r o u n d ,  ment o f t h e described  total  <5 x 1 0  ) is totally  i t was  decided  drag  ful  i n checking  (in  absence o f b l o c k a g e ) .  influence the with  skin  available  The  drag  primarily  would expect  helped  the  beyond R  Since  skin  as  comparison  of  to  be  the  distribution friction  the p r e s s u r e p r o f i l e s  =1000, the p r e s s u r e d r a g  the  literature.  location  minimum p r e s s u r e p o i n t , t h e p r e s s u r e  tribution.  as w e l l  coefficient  g o v e r n e d by m a g n i t u d e and  the  establish  their in  use-  investigators  total  found  the drag  downstream o f i t , as w e l l as  measure-  balance  other  components and  several empirical relations One  by  I t also on  gage  With  i n f o r m a t i o n proved  results  of w a l l confinement friction  to undertake  using a strain  i n S e c t i o n 2.5.  missing.  do  not  coefficient  conchange for a  given blockage (Figure  i s expected  3-15a).  t o remain  essentially  However, t h e t o t a l  d r o p w i t h an i n c r e a s e i n R e y n o l d s  d r a g w o u l d show a  number.  This i s  precisely  t h e t r e n d shown i n F i g u r e 3 - 1 5 ( b ) .  expected,  the e f f e c t  coefficient velocity. 30%  because  o f blockage of local  average  and R e y n o l d s  velocity  i n the t e s t  velocity. in  or i n free  being based  condition.  here  on t h e  results,  plots  on m e a s u r e d  values  using the c e n t e r l i n e  most o f t h e r e s u l t s  The f i g u r e  either  recorded towed  shows p r e s e n t  experimental data together with the standard drag results  by S i v i e r  3  , Zarin  4  It  the drag  are obtained u s i n g spheres  fall  100%.  section-.  d r a g and r e d u c e d  literature  by more t h a n  with available  T h i s i s because  stream  approximately  number a r e b a s e d  F i g u r e 3-16 a r e i d e a l  of the t o t a l  by  t o point out that both  For comparison in  i n c r e a s e i n the free  changes the drag c o e f f i c i e n t  coefficient  As c a n be  i s to i n c r e a s e the drag  N o t e , a change i n b l o c k a g e  w o u l d be u s e f u l  constant  , Ross and W i l l m a r t h  5  curve  , and  60  the e m p i r i c a l heartening  relation  t o note  present r e s u l t s their  accuracy.  suggested  by W h i t e  a rather excellent  at smaller blockage  .  I t was  correlation  thus  of the  substantiating  D, c m  2.5  2.0  C  d  p  1.5  S/C  o  3.81  2.7  •  6.35  7.6  o  7.62  11.0  •  8.89  15.0  6  19.6  A  10.1  •  11-43  24.8  •  12.70  30.6  X  Aminzadeh S / C  %  < 4.9%  1.0 X  X  O xx  o o  •  'O X  0.5  •  5  6  7  8  Rn F i g u r e 3-15  20  9 10  ,  30  x10'  V a r i a t i o n of the measured drag c o e f f i c i e n t w i t h Reynolds number and blockage: (a) pressure drag c o e f f i c i e n t  D,cm 2.5  2.0  •  •  mo  AA  o  3.81  2.7  •  5.08  4.9  o  7.62  1 1.0  A  10.1  6  •  12.70  %  19.6 30.6  o  Cd,t1.5  o  S/C  Q  A A A A A  1.0  A  o °o<5>. :  2  A  A  A  O  0.5  i  5  6  7  8  3-15  i  910 Rn  Figure  i  i  i  i  11  2 0 X 1 0  30  2  V a r i a t i o n o f t h e measured d r a g c o e f f i c i e n t w i t h R e y n o l d s number and b l o c k a g e : (b) t o t a l d r a g c o e f f i c i e n t . The d r a g c o e f f i c i e n t i s b a s e d on a v e r a g e v e l o c i t y i n t h e t e s t - s e c t i o n  i—•  o  D, c m S/C % 3.81 2.7  •  6,3 5  o A  •  7.62 1 0.1 6 12.70  7.6 11.0 19.6 30.6  —Standard Drag Curve; Qd t = Sivier, Zarin ; C^t 5 x Ross &Willmarth j C ^ t  4 Figure  3-16  5 6 7 8910  20 Rn  30  40 50 6070 * 1 0  2  Comparison o f 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 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 oo  D,  1.0  2  5  10  20  30  40  cm  o  3.81  •  6.3  o  7.62  A  1 0.1  •  12.70  S/C  %  2.7 5  7.6 11.0  6  50 6070  19.6 30.6  xio  Rn F i g u r e 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 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 : (b) t o t a l drag coefficient  W i t h p r e s s u r e and t o t a l it  was c o n v e n i e n t t o p l o t  with  drag information  variation  R e y n o l d s number and b l o c k a g e .  of skin  a t hand  friction  Corresponding results  9 by A c h e n b a c h  ( u n c o n f i n e d flow) near c r i t i c a l  end o f t h e  R e y n o l d s number r a n g e and e m p i r i c a l  relations  by  included f o r  R o s e n h e a d ^ and W h i t e ^ ^  comparison the  (Figure  3-17).  are also  The r e s u l t s  as s u g g e s t e d  tend to confirm  classical  d e p e n d e n c e o f s k i n f r i c t i o n on t h e R e y n o l d s n-1/2 number, C-, ^ R , however, t h e i n f o r m a t i o n i s n o t e x t e n s i v e enough t o 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 a  for  the blockage e f f e c t .  critical  Achenbach's  results  R e y n o l d s number and t h e p r e s e n t d a t a i n a  relatively  l o w e r R e y n o l d s number r a n g e c a n be  quite well  along the l i n e  6.08 R  which  On t h e o t h e r hand, W h i t e  -0.5  - 6 . 0 8 ) and C  tends  number.  =0.  J  d,p  and R o s e n h e a d ' s p r e d i c t i o n s  c o n s i d e r a b l e d i s c r e p a n c y which i n the Reynolds  fitted  n  c o r r e s p o n d s t o C, _ = 2.4 32/(R d,f n  increase  near the  show  t o i n c r e a s e w i t h an  Figure  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  ISO  i—*  122  3.5  Blockage  C o r r e c t i o n Using M a s k e l l ' s Theory  M a s k e l l developed  a theory f o r blockage  correction  based on momentum balance between the u n d i s t u r b e d  flow  upstream o f the body and t h a t downstream where e f f e c t i v e wake reaches  i t s maximum width, B.  For a square  plate  normal t o the flow, and assuming p r e s s u r e i n the wake to be uniform and equal t o the base p r e s s u r e , P^, the drag c o e f f i c i e n t i s given by  C, a  =  where m = B/S.  m [K - (1 - m S/C) 2  _  1  ]  Here K r e p r e s e n t s r a t i o o f the v e l o c i t y  on s e p a r a t i n g s t r e a m l i n e to the f r e e stream v e l o c i t y .  By  h y p o t h e s i s , he d e r i v e d an e x p r e s s i o n f o r the e f f e c t o f blockage on the wake width as  JL  =  1 _  d " d  C  . / \  C  S  n  ( K - l ) (K 2  -lAc/  2  where the s u b s c r i p t c stands f o r c o r r e c t e d v a l u e s .  This  gives the c o r r e c t i o n formula as  K K  c  _ =  i i d 1 + K -1 c 2  .. +  r/  S,2,  Q [ () ]  C  2 where the terms o f order small.  (S/C)  were c o n s i d e r e d n e g l i g i b l y  With t h i s the c o r r e c t i o n f o r drag and p r e s s u r e  coefficients  are d i r e c t l y g i v e n by  1 - C P_ 1 - C p  c  I t s h o u l d be mentioned here t h a t the t h e o r y c o n s i d e r s i n v a r i a n c e o f the s e p a r a t i o n p o i n t under c o n s t r a i n t , hence M a s k e l l doubted i t s a p p l i c a b i l i t y t o bodies.  However, the c o r r e c t i o n procedure  well-rounded has been  q u i t e p o p u l a r w i t h i n d u s t r i a l a e r o d y n a m i c i s t s , who  have  applied i t to situations 'totally unrelated with that c o n s i d e r e d i n the t h e o r y . validity present  I t was  decided to assess  o f M a s k e l l ' s c o r r e c t i o n procedure  i n the  case. F i g u r e 3-18  shows v a r i a t i o n o f p r e s s u r e  inte-  g r a t e d as w e l l as t o t a l measured drag as f u n c t i o n s o f blockage  a t R^ = 1000.  The  corrected values using  M a s k e l l ' s approach are a l s o p r e s e n t e d . t h a t t h i s c o r r e c t i o n procedure particularly  It is  i s quite  at higher blockage r a t i o s .  apparent  inadequate The e r r o r was  found t o v a r y from 8% t o 84% over the b l o c k a g e  ratio  range o f 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 p r e s s u r e coefficients convenience  are summarized i n F i g u r e 3-19.  For the  o f a p p l i c a t i o n , the r e l a t i o n s are  i n terms o f average as w e l l as c e n t e r l i n e  drag  specified  velocities.  124 1.8  1.6  •  Measured  o Corrected  Total  Drag  Drag  1.4 Cd.t 1-2  1.0  o  0.8  0  o  1.4 • 1.2  Pressure  o Corrected  Integrated  Drag  Drag  1.0 Cd,p 0.8 o o  0.6  0 Figure  3-18  10  20 S/C,%  30  C o r r e c t e d drag c o e f f i c i e n t s showing inadequacy of M a s k e l l ' s procedure, p a r t i c u l a r l y a t h i g h e r blockage  3.6  Flow V i s u a l i z a t i o n To  stantiation  provide of the  measured d a t a , visualization diameter  certain  behaviour  program.  A  - 12.7  o f 54%  o f the  s e t o f spheres  cm  were u s e d  the  The  use  o f dye  vortex  ring  earlier, these ring  the  proved  increments  and  I t was  also  concerning  i t s movement.  explained  in  typical  o f asymmetry  t u r b u l e n t shedding The  vortex  ring  R e y n o l d s number.  pictures illustrating  o f the  vortex  are presented  e x i s t e n c e o f an f o r low  at  Only  formation,  and  detail  i n achieving  I t showed f o r m a t i o n  o f the  main  the a s s o c i a t e d  Numerous p h o t o g r a p h s were t a k e n  elongation, onset by  The  i n a r a t h e r s p e c t a c u l a r f a s h i o n as p r e s e n t e d 3-20.  in  glycerol-  some i n d i c a t i o n  procedure,  flow  development  data.  t o be q u i t e e f f e c t i v e  objectives.  Figure atic  and  separation position injection  the  ranging  i n the  formation,  t h e measured p r e s s u r e  o f the  sub-  to undertake extensive  hoped t h a t t h i s w o u l d p r o v i d e location  e x h i b i t e d by  c o n c e n t r a t i o n by w e i g h t .  to observe  instability  i n f l u e n c e on  b e t t e r a p p r e c i a t i o n as w e l l as  decided  water s o l u t i o n  and  Near-Wake A n a l y s i s  i t was  f r o m 0.95  o b j e c t i v e was  and  axisymmetric,  R e y n o l d s number i n a  system-  a few  of  symmetric  instability i n Figure  in  followed 3-21.  stable stream,  c F i g u r e 3-21  d  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 o f v o r t e x r i n g w i t h Reynolds number: (a) R =30; (b) R =65; (c) R =115; (d) R =165 n n n n 3  M  to 00  130  essentially Figures  free of macroscopic  3-21(a) t o 3 - 2 1 ( e ) .  above a c r i t i c a l  value  formation of a stable separate  from  immediately from  the  such at  ring, ^'  the  the  which v o r t i c i t y  main s t r e a m .  R  As  form  sphere.  this  p o i n t s was earlier  also  (Figure  streamlines  a closed region  A single  The  i s generated the  stream  size  o f the  and  the  suggested  the  by  the  and  disturbed rate  by  into  the the  flow d i r e c t i o n  to  separation points  stagnation point  movement o f t h e s e p a r a t i o n the p r e s s u r e p l o t s  presented  state  an  asymmetry  motion w i t h i n the v o r t e x sheet  asymmetry  The  is  3-7).  a corresponding itself  long  rate  dissipated  i n the  front  This forward  circulatory  ring  to a ring  F o r R e y n o l d s number between 1 7 0 - 2 3 0 in  emerges  R e y n o l d s number i s i n c r e a s e d  e q u i l i b r i u m , and  3-21).  first  an e q u i l i b r i u m between t h e  move u p s t r e a m t o w a r d s t h e (Figure  number  closed region extending  v o r t e x r i n g becomes e l o n g a t e d maintain  Reynolds  - 24) , t h e '  n  sphere.  as t o m a i n t a i n  i s shown i n  ( c o r r e s p o n d i n g to the  the v e r t e x o f the  distance behind  For the  s u r f a c e and  behind  turbulence  i n the  a resultant  circulatory shift  of unsymmetrical  further  i n c r e a s e i n the  at which v o r t i c i t y  i s diffused  from  but  produces  motion i n  the  steady  centerline. wake i s  R e y n o l d s number. from  the  sheet  into  The  t h e main body o f t h e but  the  vortex  ring  creates unstable  Basically,  r e l e a s e , but through  an  no  the process  i n t h e end  wake a b o u t t h e  s t r e n g t h o f the  sudden motion o f the  ring  o f the  shape.  sheet  oscillation  frequency  and  a x i s o f symmetry.  When  the  reaches  value,  a critical  ring  symmetrical. vorticity  concentrated  motion o f the  In the  generated on  discharged  diametrically  ejection  opposite The  g r e a t e s t are  sheet  the 3-21f-h)  number, higher  sheet  ceases  and  release,  s i d e s o f the  flow  alternately  fluid.  i s carried  to  becomes  s e c t i o n s i n which  t h e main body o f t h e  a p o r t i o n o f the  in  assumes  boundary l a y e r  sheet.  s t r e n g t h i s the into  ring  a  position  Reynolds  cycle of build-up i n the  and  (Figures  c i r c u l a t i o n w i t h i n the  a x i s w i t h i n the v o r t e x vortex  250 - 300  vortex  a  sheet, which i n  to i t s o r i g i n a l  f u r t h e r i n c r e a s e i n the  the  during the  T h i s phenomenon a p p e a r s t o o c c u r  the o s c i l l a t o r y  the  escapes  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  With  the  vortex  and  of  R e y n o l d s number r a n g e o f a b o u t  be  ring  the  vortex  of build-up  r i n g d i s t u r b s the  consequent r e t u r n o f the and  i s one  T h i s i n t u r n causes the  asymmetrical  turn  constant,  c o n d i t i o n w i t h i n the  s i z e a b l e p o r t i o n o f the  opening  cycle.  vortex  remains p r a c t i c a l l y  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  sheet.  the  fluid  With away.  each The  132  v o r t e x element d i s c h a r g e d the d i s p e r s e d l i q u i d Figures  3-22  into  cycle of i n i t i a t i o n ,  of the r i n g  As m e n t i o n e d b e f o r e , results provide the  plotted that  layer.  n  ratio  other  To t h i s  as shown  o f 2.7%, o v e r  i n Figure  v a l u e b a s e d on h i s own Hamielec e t a l .  vortex  available  represents  as w e l l as t h o s e 4  on p o s i t i o n  3-23.  number results  spheres  i n Figure  R e y n o l d s number,  by J e n s e n  The f i g u r e  3-24.  different  i n general,  downstream.  determination  , also  of the separating  the w a l l confinement tends  separation position the v i s u a l  offering  by  an a v e r a g e 35  T y p i c a l photographs o f the vortex  associated with  Note  Here t h e l i n e  and, Rimon and C h e n g ^ .  o f blockage  sheet.  are presented  data  position  by as much as 20°  the Reynolds  For comparison,  38 t o Pruppacher e t a l .  shows e f f e c t  location of  end t h e p h o t o g r a p h s  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 .  attributed  develop-  visualization  t h e s e p a r a t i o n p o i n t moves f o r w a r d  range o f 1 0 0 - 6 0 0 .  that  the flow  s y s t e m a t i c a l l y and t h e s e p a r a t i o n  as a f u n c t i o n o f R  f o r blockage  with  vortex.  useful information concerning  s e p a r a t i n g shear  were a n a l y z e d  interacts  t o f o r m a r e g u l a r wake p a t t e r n .  shows a t y p i c a l  ment and s h e d d i n g  the stream  I t must be  ring  blockage f o r a given t o move t h e emphasized  of separation point i s ,  133  Figure  3-22  T y p i c a l c y c l e o f i n i t i a t i o n , development and shedding o f the r i n g v o r t e x a t Reynolds number R = 360 . n  o •  D,cm 3.81 7-62 8.89 11.43  S/C,% 2.7  11.0 15.0 24.8  Aminzadeh  47  Taneda ^ 1  Pruppacher 5  6  7  8  910  15  38 etal.  20  Rn Figure  3-23  P o s i t i o n o f s e p a r a t i o n as a f f e c t e d by Reynolds number and w a l l confinement  x10  2  135  b F i g u r e 3-24  T y p i c a l photographs showing downstream movement o f the s e p a r a t i o n p o s i t i o n due t o b l o c k a g e : (a) R = 170, S/C = 2.7%; (b) R = 170, S/C = 30.6% n  -24  T y p i c a l photographs showing downstream movement o f the s e p a r a t i o n p o s i t i o n due to b l o c k a g e : (c) R = 290, S/C = 2.7%; (d) R = 290, S/C = 30.6%  at b e s t , approximate.  C o n s i d e r i n g t h i s and the unstable  c h a r a c t e r o f the p r o c e s s , s c a t t e r results i s surprisingly  small.  i n the  experimental  138  3.7  C l o s i n g Comments It  can  be  said with  a measure o f c o n f i d e n c e  the e x p e r i m e n t a l  programme a c h i e v e d  than  objectives.  its initial  been a n a l y z e d assessing  only with  the blockage  considerable  scope  To  for further  o f the d a t a which would y i e l d variety  of aspects  m e c h a n i c s a t low was  provided  an  throughout  and  and  tives  has  inquiry  left  me  The  because  sophisticated and,  a  fluid  experience  the  project  experimental  of p a r t i c i p a t i o n  in a  awareness o f b r o a d e r that a a  perspec-  scientific  beginning.  appropriate to  results  and  express  review a  few  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  likely  Concluding Important  r e s u l t s may  interpretation  more i m p o r t a n t l y ,  i t w o u l d be  which are  and  be  t o be  profitable.  remarks c o n c l u s i o n s based  summarized  of  is a  total  Before  on  goal  The  T h i s i s merely  thoughts  immediate  fundamental  i s unending. closing  have  the  humble f o r I r e a l i z e  some o f t h e more s i g n i f i c a n t  3.7.1  analysis  satisfying  a feeling  s e a r c h f o r knowledge.  results  better appreciation of  procedures  t h e r e was  the  However, t h e r e  R e y n o l d s number.  exposure to the  instrumentation  date  associated with  indeed e x c i t i n g  c o n s i d e r a b l y more  r e f e r e n c e to the effects.  that  on  as f o l l o w s :  the  experimental  139  (i) The use o f average v e l o c i t y  i n the t e s t - s e c t i o n  on the mean flow rate) as a r e f e r e n c e v e l o c i t y  (based  together  w i t h the p r e s s u r e c o e f f i c i e n t d e f i n e d as  promises to promote 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 used.  errors  r e g a r d l e s s o f the t e s t  T h i s approach tends  effects,  and comparison o f data facilities  to compensate f o r blockage  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  i n p r e s s u r e measurements caused  by e l e c t r i c a l  drift  of the p r e s s u r e measuring system.  (ii)  A v e r t i c a l stem s u p p o r t i n g the s p h e r i c a l  negligible  model has  i n f l u e n c e on the p r e s s u r e 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 g r e a t e r than 10.  (iii)  F o r p r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a sphere,  the e f f e c t o f Reynolds number i s e s s e n t i a l l y  c o n f i n e d to  the r e g i o n downstream o f zero pressure p o i n t and even here it  i s limited  blockage  to R  n  < 1000, except  r a t i o o f 30.6%.  f o r the very h i g h  In g e n e r a l , the e f f e c t o f  Reynolds number i s to i n c r e a s e the minimum as w e l l as the wake p r e s s u r e .  Furthermore, l o c a t i o n s  o f the minimum  p r e s s u r e and s e p a r a t i o n tend to s h i f t l i t t l e  upstream.  140  (iv)  For pressure  the confinement a r o u n d 11% tendency  distribution  effects  blockage.  to reduce  on  the  surface of a  are e s s e n t i a l l y But  negligible  b e y o n d t h a t i t has  t h e minimum and  a  (v)  i n c r e a s e i n the  Drag c o e f f i c i e n t  distribution results^  data  and  thus  o b t a i n e d by  agrees  base p r e s s u r e s .  tends  the  Furthermore,  . thus  gage b a l a n c e .  Results also  velocity.  1 / 2  total  friction  on  the  of  drag  .  in  coefficient  rise  show t h e  Reynolds  .  confidence  local  the  blockage  In g e n e r a l , the d r a g  free  dependence o f s k i n •  the  reinforcing  because o f the  •C„d, f « K "n  Aminzadeh's  .  increases with blockage stream  shift  pressure  the r e s u l t s , a t s m a l l  investigators  strain  The  to s u b s t a n t i a t e r e l i a b i l i t y  favourably with  other  integrating  rather well with  3-5 by  to  blockage.  measuring instrumentation. compares  up  definite  minimum p r e s s u r e p o i n t shows a d i s t i n c t r e a r w a r d w i t h an  sphere,  in  the  classical  number,  f  (vi)  Maskell's  inadequate  (vii) as  correction  t o compensate  Flow v i s u a l i z a t i o n  to the  physical  showed t h e  an  increase i n  data i s effects.  provided better appreciation flow  i n terms  of  i n s t a b i l i t y o f the v o r t e x  separation location blockage.  f o r drag  f o r h i g h e r confinement  c h a r a c t e r o f the  f o r m a t i o n , e l o n g a t i o n and It  procedure  t o move downstream  ring. with  141  3.7.2  Recommendation f o r f u t u r e study As p o i n t e d o u t b e f o r e , t h e p r e s e n t e f f o r t s a t  o b t a i n i n g some a p p r e c i a t i o n as t o the p h y s i c s o f t h e w a l l confinement  e f f e c t s a t low Reynolds number r e p r e s e n t o n l y  a modest b e g i n n i n g .  There a r e numerous avenues a l o n g  which t h e r e s e a r c h program may p r o g r e s s i n f u t u r e . o f t h e more i m p o r t a n t a s p e c t s , recommended  Some  f o r future  s t u d i e s , a r e summarized below: (i)  In the present s e t o f experiments,  blockage  effects  on t h e s u r f a c e p r e s s u r e d i s t r i b u t i o n c o u l d n o t be s t u d i e d f o r t h e Reynolds number < 280 , l i m i t a t i o n b e i n g imposed by t h e p r e s s u r e measuring i n s t r u m e n t a t i o n .  The s u r f a c e  p r e s s u r e a t lower Reynolds number was found t o be so s m a l l [ 0 (10 ^ ) p s i ]  t h a t i t p r e s e n t e d a problem o f measure-  ment w i t h an a c c e p t a b l e degree o f a c c u r a c y and r e p e a t a b i l i t y . T h e r e f o r e , i t i s suggested  t h a t p r e s s u r e measurements a t  lower Reynolds number (and h i g h e r blockage)  s h o u l d be under-  taken t o p r o v i d e a comprehensive p i c t u r e o f w a l l c o n f i n e ment e f f e c t s . (a)  T h i s can be a c c o m p l i s h e d  using:  more s e n s i t i v e and s t a b l e p r e s s u r e  transducer  (e.g., D i g i q u a r t z p r e s s u r e t r a n s d u c e r s ) ; (b)  a m o d i f i e d d r i v e and the pump system so t h a t higher concentration of glycerol-water solution can be handled.  142  (ii)  No e f f o r t  distribution, t h e wake. sphere,  has b e e n made h e r e  turbulence  In f a c t  to  evaluate  c h a r a c t e r and s h e a r  shear  pressure  stress i n  s t r e s s on t h e s u r f a c e o f t h e  even i n absence o f b l o c k a g e ,  i n this  range o f  R e y n o l d s number r e m a i n s u n r e c o r d e d .  The i n f o r m a t i o n i s  quite  performance o f  important  different directly  i n comparing r e l a t i v e  p r o s t h e t i c h e a r t v a l v e s , as t h e p a r a m e t e r affects  of the r e d blood  (iii)  helical this  periodic  there  the  (iv)  study,  to focus  vortex  d e s t r u c t i o n and c o a g u l a t i o n  cells.  The p r e s e n t  was u n a b l e  the  deformation,  due t o l i m i t a t i o n  a t t e n t i o n on t h e f r e q u e n c y  shedding.  phenomenon i n depth',  seems t o be some q u e s t i o n  R e y n o l d s number.  o f the  I t w o u l d be u s e f u l t o e x p l o r e particularly,  when  about i t s v a r i a t i o n  Of c o u r s e ,  the e f f e c t  with  o f blockage  on  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 .  Tests  s h o u l d be c a r r i e d  out with  under d i v e r s e c o n d i t i o n s o f v e l o c i t y turbulence universal definition  (v)  o f time,  and p r e s s u r e  cylinder,  o f the pressure  provide  flat  profile,  gradient to firmly  c h a r a c t e r o f the p r e s s u r e  Blockage  spherical  establish u s i n g prop<  coefficient.  sphere,  useful information.  blockage,  distribution  c o r r e c t i o n s f o r b l u f f bodies plate,  model  e t c . i n shear  such flow  as c i r c u l a r should  (vi)  An  important  significant  area of i n t e r e s t ,  in biological  study o f p u l s a t i l e  which i s  particular  f l u i d mechanics, would be  flow past b l u f f bodies  the  under w a l l  confinement s i m u l a t i n g a t y p i c a l c a r d i a c c y c l e .  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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. 589598.  57.  G r o v e , A.S., S h a i r , F.H., P e t e r s e n , F . E . , and A c r i v o s , A., "An E x p e r i m e n t a l I n v e s t i g a t i o n o f t h e S t e a d y Separated Flow P a s t a C i r c u l a r C y l i n d e r , " J . F l u i d M e c h a n i c s , V o l . 19, P t . 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 P a n , F., " F u r t h e r E x p e r i m e n t s on S t e a d y S e p a r a t e d F l o w s P a s t B l u f f O b j e c t s , " J . F l u i d M e c h a n i c s , V o l . 34, P t . 1, 1968, pp. 25-48.  59.  M o d i , V . J . , and E l - S h e r b i n y , S.E., "A F r e e - S t r e a m l i n e M o d e l f o r B l u f f B o d i e s i n C o n f i n e d F l o w , " T r a n s . ASME, Journal of Fluid Engineering, i n press.  151  60.  W h i t e , F.M., V i s c o u s F l u i d Y o r k , 1974, pp. 208-210.  61.  Rosenhead, University  F l o w , M c G r a w - H i l l , New  L., Laminar Boundary L a y e r s , P r e s s , 1963, pp. 102-109.  Oxford  152  APPENDIX I CONVENTIONAL PRESSURE COEFFICIENT C~ IN TERMS P  OF MEASURED INFORMATION  In a low Reynolds number experiment u s i n g a l i q u i d tunnel, d i f f i c u l t i e s  in establishing characteristic  r e f e r e n c e v e l o c i t y and p r e s s u r e , P J  before.  and U , were d i s c u s s e d  CO  '  c  oo  However, f o r comparison and t o e s t a b l i s h  e f f e c t i v e n e s s o f the new d e f i n i t i o n o f pressure one can c a l c u l a t e the c o n v e n t i o n a l p r e s s u r e —  relative coefficient,  coefficient  2  Cp = ( P - P ) ( l / 2 ) p U Q  OT  oo  quite r e a d i l y using  differential  p r e s s u r e data measured d u r i n g the experiment. The x component o f Navier-Stokes s t a g n a t i o n s t r e a m l i n e y =0 9u dx  1 3P p 3x  r  along the  can be w r i t t e n as  9 u ^ 9 u, 2 2 9x 9y 2  ±  equation  2  L  J  I n t e g r a t i n g from f r o n t s t a g n a t i o n p o i n t t o minus i n f i n i t y upstream o f the sphere y i e l d s ,  r  0  ^ 9u , u 9x dx 7 7 —  '  —CO  2 u  2  oo  , 0 1 - — r  =  p  ^  J-oo  9P -i . . • dx + v 9x  f  0  .2 „2 3 u , 9 u, , [ — ~ + ~] dx „ 2 9x 9y 2  r  J  J-oo  u  153  u  +  P  n  +  r  L  V  3  2  U  3x  2 3 Un  .  r-^  =  *  s  1 +  1/2 u'  dx  dy'  - P  0 °° 1/2 p U"  -j  ^ + —=rJ  r  2  d  9  U  -6 l 3x =T  1  J  2  +  sr] dx  3y  2  where 6 i s the boundary-layer  thickness.  vanishes because o f i r r o t a t i o n a l i t y  2 3 u 2 3x  The second  approximation  2 3 u 2 3y^  can be i n t r o d u c e d .  3u 3x  Since  9v 3y  oo  U  = 2  x = 0  at  1 -  1/2 U 2~  9u 3x  integral  of the o u t e r flow while  i n the f i r s t i n t e g r a l usual boundary l a y e r  1/2 p  ,  dx  2  r 3 u -<*> 3x  0  U ,  —3y T-J  +  x =6 " y =0  1  +  5  +  where A i s a constant and R i s the Reynolds number.  154  Here  the numerical  outer  flow  value of A follows d i r e c t l y  from the  solution.  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 . to  be 8.  have shown, i n d e p e n d e n t l y ,  the value o f A  Thus,  P  - P ?  =  1  R  +  1/2 p IT '  '  +  oo  i.e., -  E  - 60° P  =  ( P  0" 60^ " P  +  ••••)1/2 2 P U  .  Now  P - ~ 1/2 p U  _  p  e  Recognizing tribution reduces  2  that  p  e- 60° 1/2 p Uf p  p  o 60° 1/2 p U P  +  ( 1  +  2  term  i s less  than  3%, t h e above  to  e" ~ 1/2 p U  + R  f o r a R e y n o l d s number as low as 300  o f 8/R  p  8  p  ( p 2  e- 6(^- o-f6tf!_ 1/2 p U p  ( p  2  +  ±  con-  expression  155  APPENDIX I I  A PROCEDURE FOR  DRIFT CORRECTION USING DIRECT  OF THE DIFFERENTIAL  Let pressure,  the p r e s s u r e drift II-l. the  PRESSURE  t h e o b j e c t i v e be t o measure a  AP  = P  6  Q  9  Q  - P  at a reference point.  o f the e l e c t r o n i c  of interest  L e t the a r b i t r a r y  and  zero  s y s t e m be as i n d i c a t e d i n F i g u r e  The d i a g r a m a l s o shows  differential  differential  between t h e l o c a t i o n  r  MEASUREMENT  pressure  AP  Q  the corresponding .  From t h e f i g u r e  drift  of  i t follows  that:  AP  AP  AP  0  9  +  6  1  +  6  2  6  Therefore,  AP  9  1  +  AP  e3  AP  6  2  =  (AP + 6 )+(AP e  -  1  S  (APg +  AP  Q  U  +  1  6_ JL  Q  +  +  + 5-j o  ;  6 +6 +6 ) 1  2  3  156  Figure I I - l  A d r i f t c o r r e c t i o n procedure using d i r e c t measurement o f the d i f f e r e n t i a l pressure  157  i.e.,  AP  =  Q  AP  fc)  + AP  Q  O  - AP  Q  U  3  Q  O  2  - 6,  - 6_  J-  J  .  (II.1)  Now  6  6  6  2  =  3  2  =  +  e  A P  A P  e  S  "  2  -  3  =  A P  9 l  A p  e  =  e  2  "  2  A p  +  A P  e i  -  AP 6  Assuming  '  A p  3  "  A P  e  2  AP  3  the d r i f t  e  9  1  t o be  linear  over a small  time  interval,  6  Hence  2  the desired  6  +  -AP,  AP„ 3  differential  pressure -  AP 6  3  AP  AP 9  1  6  =  I  A p  0 1  " k  A p  e  3  - AP  AP„  3  6  2  158  Thus AP  involves  n  determination the  of  measurement  the of  e i t  differential AP  was  found  measurements  to  be  with  extremely an  and  Q  e  i  d i f f i c u l t  acceptable  AP  to  accuracy.  .  Q e  pressure In  practice  3  accomplish  these  

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