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The effect of twist on flexible hydrofoils Milne, John Gordon 1974

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THE EFFECT OF TWIST ON FLEXIBLE HYDROFOILS  by JOHN GORDON MILNE B.A. S c . , University of B r i t i s h Columbia, 1972  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering  We accept t h i s thesis as conforming to the required standard  Marchi 1974  In  presenting  for  thesis in partial  an a d v a n c e d d e g r e e  agree  that  the  and s t u d y . of  this  this  Library  I further  thesis  for  my D e p a r t m e n t o r  at the  Civil  my w r i t t e n  M  a  r  c  h  '  1 9 7 4  for  requirements  It  is  for  extensive  may be g r a n t e d  permission.  Columbia  the  available  permission  thesis for  Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, B . C . , Canada  Date  this  of  of B r i t i s h Columbia,  freely  representatives.  copying or p u b l i c a t i o n of  Department o f  that  s c h o l a r l y purposes  by h i s  be a l l o w e d w i t h o u t  University  s h a l l make i t agree  fulfilment  reference copying  by t h e Head  understood  financial  I  of  that  gain shall  not  ii  ABSTRACT  Flexible investigate  the e f f e c t  and i n p a r t i c u l a r , with stiffeners  measured o v e r  are  in  and t h a t attack.  the  angles  to produce  to  twisting  lift  is  to  characteristics  Flexible  models  and d r a g  set  forces  were  attack. c a n be u s e d as an  e l l i p t i c loading. in  Twisting of  t h e c a s e s t e s t e d and t h e theory  of  i s more b e n e f i c i a l  t w i s t e d wing  tested  and r i g i d m o d e l s w i t h a  twisting  the simple  were  and d r a g  force.  and t h e  of  that  t o be b e n e f i c i a l  contradiction  was f o u n d t h a t  flume  lift  drag  positions  was c o n f i r m e d  to tapering  w i n g s was shown  induced  in a water of  hydrofoils  t w i s t on t h e  the high  a range  It  of  at various  t w i s t were t e s t e d  ative  low a s p e c t r a t i o  less sensitive  alterntapered benefits  e l l i p t i c loading.  than  the simple  t o changes  It  theory  in angle  shows of  i i i  ACKNOWLEDGEMENTS  The a u t h o r w i s h e s t o t h a n k for  h i s a s s i s t a n c e and encouragement  project.  the k i l l e r  National  their  Research Council of  M.C.  Quick, this  the Vancouver P u b l i c  help which allowed the author  whale f l u k e .  Dr.  throughout the course o f  The a u t h o r a l s o w i s h e s t o t h a n k  Aquarium f o r of  his supervisor,  to obtain  measurements  F i n a l l y , t h e author wishes to thank Canada f o r  their financial  support.  the  iv  TABLE OF CONTENTS Chapter  Page  LIST OF FIGURES  vi  LIST OF PLATES  viii  1.  INTRODUCTION  1  2.  THEORY OF LIFT AND DRAG ON AEROFOILS  6  2.1  L i f t and Drag  6  2.2  Circulation  8  2.3  Vortex System  10  2.4  Induced Drag  12  2.5  Expression for Induced Velocity  14  2.6  Expressions for L i f t and Induced Drag  16  2.7  E l l i p t i c Loading  19  2.8  The Effect of Twist  19  2.9  The Effect of Yawing  20  3.  4.  APPARATUS AND INSTRUMENTATION  21  3.1  Water Flume  21  3.2  Strain Gauge Balance  3.3  Models  27  3.4  Electronic Equipment  29  .  24  EXPERIMENTAL PROCEDURE  33  4.1  Calibration Methods  33  4.2  Test Procedure  33  4.3  Errors  36  V  Chapter 5.  Page DISCUSSION OF RESULTS  38  SUMMARY OF CONCLUSIONS  50  LIST OF REFERENCES  51  APPENDIX A - TABLE OF RESULTS  52  APPENDIX B - METHOD OF SOLUTION  57  APPENDIX C - MOLDING AND CASTING  61  vi  L I S T OF  FIGURES  Figure  1.  Page  Definition  Diagram  for  Lift,  Drag,  Angle  of  A t t a c k and C i r c u l a t i o n  2.  P l o t of  Lift  to  Attack  for  Ratio,  A = 1,3,6  3.  Horseshoe  4.  Definition  7  Drag R a t i o V e r s u s A n g l e  Unsymmetric A e r o f o i l s  of  of  Aspect 9  V o r t e x System  Diagram  for  11  Induced  Velocity  and  Reduced A n g l e o f A t t a c k  5.  Definition Spanwise  Diagram  for  13  Integration  15  Schematic Layout of Experimental  7.  S t r a i n Gauge B a l a n c e  8.  Dimensions  9.  P l o t of  10.  Plot of Attack  K in  Direction  6.  Attack  of  of  Lift for  the Three  to  Equipment  26  Models  32  Drag R a t i o V e r s u s A n g l e  of  the Three Epoxy A e r o f o i l s  L i f t t o Drag R a t i o V e r s u s A n g l e for  23  the Aluminum A e r o f o i l  39  of  at Various  o f T w i s t t o Reduce A n g l e o f A t t a c k  Angles 41  vii Figure  11.  Page  P l o t of L i f t to Attack  for  Angles of  Drag  R a t i o Versus Angle  the Aluminum A e r o f o i l Twist to Increase  at  of  Various  the Angle  of  Attack  12.  Plot of Attack  13.  43  Lift for  Plot of Yaw  for  t o Drag  t h e Whale  L i f t t o Drag  Ratio  Versus Angle  of  F l u k e Models  R a t i o Versus Angle  t h e Aluminum A e r o f o i l  44  of 49  vi i i  L I S T OF PLATES Plate  Page  1.  Overall  View o f  Flume  •.  2.  B a l a n c e and R e c o r d i n g Equipment  25  3.  Epoxy A e r o f o i l  28  4.  Whale  5.  S t r a i n Gauge B a l a n c e  34  6.  Sample Output  37  Model  F l u k e Model  .  from C h a r t Recorder  22  31  1  CHAPTER 1  INTRODUCTION  When a f l u i d produce  a variety  direction addition  flows past objects  of e f f e c t s .  of motion  It  in  large  shapes  a steady force  lateral  the d i r e c t i o n o f motion.  object are subject to o s c i l l a t i n g l a t e r a l further  different  may p r o d u c e  o r i t may p r o d u c e  to the f o r c e  of  it in  components  the in  Some s h a p e s  or i n - l i n e  can  of  f o r c e s and a  c a t e g o r y e x p e r i e n c e i n s t a b i l i t i e s termed g a l l o p i n g  or  divergence. This study which are  c o n c e n t r a t e s on a p a r t i c u l a r g r o u p o f  c l a s s i f i e d as f l e x i b l e  low a s p e c t a e r o f o i l s .  is  an i m p o r t a n t f a c t o r w i t h s u c h a e r o f o i l s  to  i n v e s t i g a t e whether  drag component. with tip  chord,  tends  to produce  To r e d u c e  elliptic the  are  d r a g components  span  to approximate  Such m i n i m i z a t i o n  with a f l e x i b l e aerofoil the  lift.  small  compared wing  drag.  t h e i n d u c e d d r a g t o a minimum f o r  i s concerned with  out  are  because o f  a given  the optimum,  c a n a l s o be a c h i e v e d  t h i c k n e s s t o c h o r d r a t i o a n d by t w i s t i n g t h e a e r o f o i l .  thesis  of  short  drag  aerofoil  component and a r e l a t i v e l y  higher  to as induced  usually tapered  loading.  lift  the  sets  Typically,aerofoils  Low a s p e c t r a t i o , w h i c h means  losses, referred  aerofoils  a high  Induced  study  c o n t r o l l e d spanwise t w i s t i n g of  c a n be u s e d t o r e d u c e t h e i n d u c e d d r a g . o b j e c t s which produce  and t h i s  shapes  twisting of  the a e r o f o i l  i n w h i c h t h e amount o f  and  so by  lift, called, reducing This  particularly  twist is a  function  2 In  three dimensions  three separate parts: friction  the drag of  f r i c t i o n drag,  drag o r s k i n f r i c t i o n  surface of  the w i n g .  the shape o f  d r a g w h i c h depends  mainly  dimensional  profile  foil.  In  three  the  also of  rotation  lumped  trailing  originating and t h u s  the  the s e c t i o n .  rotates  the  c a l l e d the  component  induced drag.  vortex  pattern,  the  in total  drag.  compressibility and.cavitation Hydrofoils  higher  are,  s t r e s s e s produced  low a s p e c t r a t i o  The a s p e c t r a t i o generally  independent  induced drag  c a n be  and t h e s p a r r o w ;  of  drag of  the  lift  upon t h e  having  increases  lift  away  resulting may  i n c r e a s e the  a strong of  length  vortex  the t i p  of span.  this  vortices  This  thesis  and s t u b b y  Hydrofoils,  thesis  will  hydrofoils  lift  aerofoil  a high  due t o then,  structural to drag  is less  t h i s comes f r o m a c o m p a r i s o n o f  the a l b a t r o s s  the wing  identical with  to c o u n t e r a c t the higher effect  aero-  ignored.  by w a t e r l o a d i n g .  a low a s p e c t r a t i o o r s h o r t  good d e m o n s t r a t i o n  two  profile  and t h e r e b y m a k i n g a  of necessity, short  has a g r e a t  to  In a d d i t i o n  which are  friction  The i n d u c e d d r a g  The k i n e t i c e n e r g y  induced drag are almost  reduction  In  of  which c o n s i s t s of  from each w i n g t i p .  on  The  two d i m e n s i o n a l  The the  and c a l l e d the  induced downflow near  c o n c e n t r a t e on l o w s p e e d a e r o f o i l s  have  together  The h o r i z o n t a l  c o n c e n t r a t e on r e d u c i n g  significant  if  thicker profiles.  be t h o u g h t o f a s t h e w o r k done on t h e f l u i d  the  will  is  of  of  r e s i s t a n c e depends  d r a g makes up t h e t o t a l  dimensions  from the p e r p e n d i c u l a r .  for  on t h e s h a p e o f  the a n g l e o f a t t a c k and t h e r e b y  from t h i s  on t h e s m o o t h n e s s  The f o r m d r a g o r e d d y  d r a g and form d r a g a r e g e n e r a l l y  i s made up  f o r m d r a g and i n d u c e d d r a g .  i s dependent  t h e w i n g and i s g r e a t e r  theory  an a e r o f o i l  the  must  loading. ratio  and  efficient. the  aspect r a t i o ,  A  albatross is a  very  3 efficient gliding  b i r d , w h i l e the h i g h l y manoeuverable  h a v i n g a low a s p e c t r a t i o , must c o n t i n u a l l y in  the a i r .  it  is  Since the  have a p r o p o r t i o n a l l y in for  the  for  short  greater  spans  than  effect for  (short a e r o f o i l s ) ,  The minimum p o s s i b l e obtained  by e l l i p t i c  span and t h e m i n o r Elliptic form o r  In  to  stay  independent  of  span  l o n g o n e s as  it  will  the s h o r t e r  induced drag, which c o n s t i t u t e s a large  hydrofoils  along  beat i t s wings  induced drag i s almost  f a r more i m p o r t a n t  loading,  will  part  force  the major a x i s of  axis being  the magnitude  its  c a n be b r o u g h t a b o u t length  e f f e c t the  of  for  the  of  the  the induced drag  for millions  use o f  the Wright B r o t h e r s , method o f first  obtaining  recognized  consisted of  a higher  In  copying lateral  control the  providing  birds  flight  control  powered f l i g h t .  c o u l d be t w i s t e d t o wing t i p .  to help  over  of  the  their  the p i l o t w i t h c o n t r o l  than  tip.  nature  In  adapted  by  is.reduced.  1903  t w i s t as a  c a l l e d wing  f l e x i b l e wings  the angle  of  t h e w i n g on t h e o t h e r  the  warping,  attack at  t h i s way t h e w i n g o n o n e s i d e c o u l d be made t o  or lower l i f t  plan  p l a n e w h i c h made  of  the  vortices  force  in f l i g h t . birds,  is  aerofoil  been u s e d i n  This method,  increase or diminish  in  done on t h e f l u i d  t w i s t i s n o t new a n d has  of years  lift  force.  trailing  these v o r t i c e s with the r e s u l t  drag  being  is elliptic  o r work  that  total  a given  a t t a c k towards  the s t r e n g t h  reduction  beneficial.  lift  and s o d e c r e a s e s t h e k i n e t i c e n e r g y  The  the  by t w i s t i n g a r e c t a n g u l a r  to decrease the angle of  t w i s t decreases  of  A  the e l l i p s e  l o a d i n g c a n be a c h i e v e d by a w i n g t h a t it  span.  t h e n be q u i t e  induced drag  sparrow,  which the  produce  s i d e and t h e  lift  4 differential axis.  In  because o f  would cause the plane to r o l l  foils  its  longitudinal  l a t e r a e r o n a u t i c a l w o r k w i n g w a r p i n g was r e p l a c e d by  ailerons  t h e i r p r a c t i c a l i t y a n d b e c a u s e i t was r e c o g n i z e d t h a t  twist-  i n g c a u s e d f l u t t e r and c o n t r o l aerofoils  about  r e v e r s a l , so t h a t t w i s t i n g o f  came t o be r e g a r d e d a s h a z a r d o u s .  or h y d r o f o i l s ,  however,  Twisting of  short  showed t h e s p a r r o w t o be a r a t h e r to drag r a t i o .  manoeuverability,  However,  even f o r  the sparrow, which  member o f t h e d o l p h i n  family.  w h a l e was c h o s e n was t h a t from c l o s e q u a r t e r s  to the forward lift  tends  us some i d e a s t o w a r d  it  but  unit  or  tail  of  reducing  i s very  c o u l d be r e a d i l y o b s e r v e d  edge.  the f l u k e  the centre of  stiff  so t h a t the c e n t r e o f  The s w e p t b a c k s h a p e means t h a t further is quite  whale, flukes  the main reason the  at the Vancouver P u b l i c Aquarium.  t o move a f t ,  l e a d i n g edge,  An e x a m i n a t i o n  s t u d y was t h e k i l l e r  The t h r u s t  to the wings of the sparrow,  the f l u k e  nature  force.  The c r e a t u r e c h o s e n f o r  edge o f  of  requires  and s o i s b e s e t w i t h l o w a s p e c t r a t i o w i n g s ,  such a c r e a t u r e as t h e s p a r r o w s h o u l d g i v e  similar  sparrow  i n e f f i c i e n t c r e a t u r e i n terms  seems t o do i t s u t m o s t t o i n c r e a s e t h e e f f i c i e n c y .  the induced drag  aero-  may be q u i t e b e n e f i c i a l .  P r e v i o u s l y a c o m p a r i s o n o f t h e a l b a t r o s s and t h e  lift  long  away f r o m t h e r o o t .  and The  largest are  killer  photographed leading  twist is  near  the c e n t r e Apart from  f l e x i b l e and t h i s , combined w i t h  effect  of  t w i s t b e i n g ahead o f  brings  about a r e d u c t i o n i n the angle of  the centre of  attack.  of the the  pressure,  5 The k i l l e r the t y p i c a l a e r o f o i l A model  of  this  w h a l e f l u k e has a r a t h e r  shape,  b e i n g much t h i c k e r f o r  s h a p e was i n v e s t i g a t e d a s p a r t o f  e f f e c t of yawing the a e r o f o i l to determine  t w i s t i n g as a method o f  then,  w i t h o u t reducing the l i f t  t h e same  length.  the study.  The investigated  force.  the purpose of  improving  reducing the induced drag.  s e c t i o n from  f o r w a r d a n d b a c k w a r d was a l s o  i t s i n f l u e n c e on t h e d r a g I n summary,  angle of  different  this  t h e s i s i s to  the e f f i c i e n c y of  hydrofoils  study by  The i n d u c e d d r a g c a n n o t be d e c r e a s e d force,  so t h e p r o b l e m i s  t w i s t f o r which the advantage  o u t w e i g h e d by t h e d i s a d v a n t a g e o f  of  to f i n d  the  decreasing the drag  decreasing the  lift.  optimum is  not  6  CHAPTER  2  THEORY  L i f t and  2.1  Drag  When a b o d y i s s u b j e c t it  experiences  of motion  in  force is  is  the  is  force which l i e s  and a e r o f o i l s very  nearly  are the  class  perpendicular  term used to d e s c r i b e the  a definition  diagram  for  of attack i s  line of  Some  direction  bodies  to the  body f o r w h i c h t h e  to the  direction resultant  d i r e c t i o n of motion. of  the r e s u l t a n t  and d r a g  and d r a g . w h i l e drag  i s the  Lift is is  Lift  force  term  Figure  describing  1  the useful  compon-  t h e f o r c e w h i c h must  have  zero  zero.  o f motion  lift  The a n g l e of  which are d e a l t w i t h e x c l u s i v e l y force of  the f l u i d  when  attack is  the  be  in  angle  the angle  and t h e . c h o r d ,  drawn between t h e s h a r p - t r a i l i n g  the  the  force  thrust.  paper,  direction  of  component  lift  Symmetric a e r o f o i l s , this  resultant  along  the d i r e c t i o n o f motion.  ent which c a r r i e s the aeroplane by  lies  a t some a n g l e  to the d i r e c t i o n of motion  r e s i s t a n c e force along  overcome  the  t h e same d i r e c t i o n as . t h e f l u i d m o t i o n .  perpendicular the  flow,  i s a r e s i s t a n c e which generally  experience a resultant of motion  to a f l u i d  between  the  the chord being  edge and t h e c e n t r e o f  the  curvature  nose. The q u a l i t y  ratio of  lift  to drag.  or e f f i c i e n c y of Practical  an a e r o f o i l  aerofoils  i s measured  have a h i g h  lift  to  by  the  drag  FIGURE  I 7  Lift  Starting  DEFINITION A N G L E  DIAGRAM  OF ATTACK  FOR LIFT,  Vortex  D R A G ,  A N D C I R C U L A T I O N  .  8 r a t i o and t h e s e a r e very  nearly  drag  r a t i o depends  perpendicular  aspect r a t i o . span s q u a r e d ratio  of  aspect  to the d i r e c t i o n o f m o t i o n .  to a large  e x t e n t on t h e a n g l e  to the area o f the a e r o f o i l ,  span to chord l e n g t h  unsymmetric  which the net r e s u l t a n t  The a s p e c t r a t i o may be d e f i n e d  shows a p l o t o f  clearly  the ones f o r  lift  to drag  aerofoils  of  t h e dependence  for  various  of  lift  versus  to the  as t h e r a t i o o f  the  aerofoil.  ratio  and  the  Figure  2  for  demonstrates  on a n g l e  of  attack  and  ratio.  Circulation  2.2  The f o r m a t i o n theory. certain on the above  of  lift  c a n be e x p l a i n e d i n  When a b o d y i s s u b j e c t t o l i f t positive top.  pressure  c a n be o b t a i n e d  s t r e a m v e l o c i t y on t o p o f  the r i g h t  So t h a t then  if  than  wise around the  r e a r edge.  velocity  on t h e f l o w p a s t t h e  so as t o  i n c r e a s e the  body a n d d e c r e a s e  a lift  pressure  stationary  This body,  fluid  i t on t h e b o t t o m  and t h e f l u i d  flow  f o r c e t h e c i r c u l a t i o n must be  is  of to  clock-  body.  When an a e r o f o i l sharp  the  the  to a  the v e l o c i t y below the body.  by s u p e r i m p o s i n g  a body i s  to develop  circulation  i t m u s t be s u b j e c t  From B e r n o u l l i ' s e q u a t i o n we c a n s t a t e t h a t  a c i r c u l a t i o n around i t , which flows  body.  then  terms o f  on t h e b o t t o m a n d a c e r t a i n s u c t i o n  t h e body m u s t be g r e a t e r  condition  the  lift  o f a t t a c k and  angle of attack  aspect ratios  to drag  The  is  t h i s may be r e d u c e d t o  a rectangular  ratio  force  This  s t a r t s from r e s t a v o r t e x  vortex,  c a l l e d the s t a r t i n g  is  formed  vortex,  is  at  the  always  FIGURE 2  9  25  4  -2  0  2  4  6  8  10  12  14  16  A n g l e of A t t a c k ( degrees)  PLOT OF LIFT TO DRAG RATIO VERSUS ANGLE OF ATTACK FOR UNSYMMETRIC AEROFOILS OF A S P E C T RATIO,A= 1,3,6 . ( B A S E D ON DATA F R O M = T H E O R Y OF WING A B B O T A N D V O N D O E N H O F F ).  SECTIONS,  10 a s s o c i a t e d w i t h t h e same p a r t i c l e s o f washed downstream as the a e r o f o i l circulation  a r o u n d any  large  t h e s t a r t i n g v o r t e x must  zero,  c a n be e x p l a i n e d by t h e  fact that  wing equal  and o p p o s i t e  to the  that other  vortices  fact  continues  in motion.  then the generation  starting  vortex.  This  shed as the a e r o f o i l from the  the c i r c u l a t i o n which would o t h e r w i s e circulation  a r o u n d an a e r o f o i l  2.3  System  line  vortices.  Since a linear  a fluid,  then  decay.  Figure  v o r t i c e s along the span.  but must c o n t i n u e trailing  v o r t i c e s , are  downstream w i t h the  as f r e e  length  of  the  vortex cannot  terminate  perpendicular  flow.  Generally  the w i n g as a s h e e t ,  single  spring  line  originating vortex  only  v o r t e x and the  trailing  f r o m each wing t i p .  system,  i s shown i n  Figure  in maintain the  in at  the  the wing  3.  bound  t o t h e w i n g and a r e these v o r t i c e s are  The b o u n d v o r t e x  is  of  tips  vortices, called  v o r t i c e s are also l i n e  This  of  interior  the forced  shed  however a s i m p l e r s i t u a t i o n  f r o m the wing t i p s .  say  a n d i s made up  t h e c i r c u l a t i o n K a r o u n d t h e w i n g h a s c o n s t a n t v a l u e and t h e vortices  the  These a r e c a l l e d the  These f r e e  and  vortex.  cannot terminate  vortices.  turned  fluid  the  to  to  1 shows  is  circulation  along,  t r a i l i n g edge  and t h e s t a r t i n g  t h e bound v o r t i c e s  of  i s not  moves  The c i r c u l a t i o n moves w i t h t h e a e r o f o i l a series of  Since  t h e r e m u s t be a v o r t e x a r o u n d  v o r t i c e s are shed c o n t i n u o u s l y  Vortex  b r e a k s away a n d  contour which includes the a e r o f o i l  remain  are not  f l u i d so i t  is  along when  trailing now a  vortices  s i t u a t i o n , c a l l e d the  horseshoe  FIGURE 3  HORSESHOE  VORTEX  11  SYSTEM  12 The o p e n end o f which remains tinually  the vortex r e c t a n g l e  at  the s t a r t i n g p o i n t w h i l e  increase in  terms o f  bottom to the  it  top around  the w i n g t i p s .  along the bottom o f  c e n t r e on t h e t o p .  2.4  Induced  the  called  to f l o w from  the the  causes an outward  flow  t h e span and a f l o w t o w a r d  an a e r o f o i l  f r i c t i o n drag,  upon  around  the w i n g t i p s  the  from  vortices.  upon  the  c a n be d i v i d e d  smoothness  the shape o f  of  v e l o c i t y w h i c h i s due t o t h e is  directed generally  is  small  compared  being  This  to the v e l o c i t y V of  of  greater  lumped  depend m a i n l y  influence  downwards.  Figure  are  c a n be e x p l a i n e d  the induced v e l o c i t y i s e q u i v a l e n t a t t a c k a s shown i n  three  of  and  the a e r o f o i l .  the  the  induced  vortices  velocity,  (3)).  thicker  on t h e s h a p e o f  i n terms  to a r e d u c t i o n  for  form  together  the t r a i l i n g  normal  4 (from Glauert  The  t h e s e c t i o n and t h e  the p r o f i l e ,  drag s i n c e they  The i n d u c e d d r a g  into  form d r a g and i n d u c e d d r a g .  The f r i c t i o n d r a g a n d f o r m d r a g  the p r o f i l e  section.  This  explained  p r e s s u r e below  fluid  two d i s t i n c t t r a i l i n g  drag of  f r i c t i o n d r a g depends  sections.  con-  Drag  separate parts:  depends  for  The f l o w w h i c h r o l l s  The t o t a l  drag  Due t o t h e h i g h e r  i s a tendency  to t o p forms  the t r a i l i n g v o r t i c e s  t h e t r a i l i n g v o r t i c e s c a n a l s o be  there  toward the w i n g t i p  bottom  of  spanwise f l o w s .  w i n g than above  vortex  length.  The f o r m a t i o n in  i s c l o s e d by t h e s t a r t i n g  and  denoted  w,  The e f f e c t  in the angle  of  of  FIGURE 4  DEFINITION AND  DIAGRAM  REDUCED ( FROM  FOR  ANGLE  13  INDUCED OF  VELOCITY  ATTACK.  GLAUERT )  14 Due t o t h e  induced v e l o c i t y the o r i g i n a l  a i s now r e d u c e d by t h e amount w / V s o t h a t is  a  Q  = a -  w/V.  The l i f t  force  is  a n g l e w / V a n d so t h e r e  i s a component  w/V times  the o r i g i n a l  lift  component  of  the  The  lift  force  to  term " t i p  loss"  i n c r e a s e the  the spanwise  new a n g l e o f  incidence  refers  length  to the  of  two d i m e n s i o n a l  the  lift).  2.5  Expression for  this  a vortex w i l l  Induced  This  done o n  pattern.  f o r c e and o r i g i n a t e s  the  trailing  s e c t i o n the value be c o m p u t e d .  for  induced v e l o c i t y  Generally  the  along  t h e span  vortex of  this  of  by  "  ( 1 )  circulation K varies  f o r w becomes more c o m p l i c a t e d . a s t r i p dy o f  has shown t h a t  adjacent  vortex  -  Prandtl  the  downward v e l o c i t y i n d u c e d a d i s t a n c e h away f r o m a l i n e a r given  from  Velocity  normal  K is  from  vortices.  the  strength  The  the f l o w around the w i n g t i p  t o l o w p r e s s u r e was t h e c a u s e o f  to  force.  t r a i l i n g vortex  induced drag  f l o w e x p l a n a t i o n where  an  i n the drag d i r e c t i o n equal  induced drag  high  to  attack  i n d u c e d d r a g may a l s o be t h o u g h t o f a s t h e , ; w o r k  the f l u i d  In  of  now t i l t e d b a c k w a r d s a t  (the  i s c a l l e d the  the  angle  the v o r t e x  across the  The a b o v e  band a s i n  Figure  result 5.  t h e c i r c u l a t i o n d e c r e a s e s by ^ strength  springs  s p a n and t h e  from element  expression  c a i i be a p p l i e d  to  B e t w e e n y and y + dy dy dy.  and a  trailing  The n o r m a l  induced  FIGURE  DEFINITION OF  K  D I A G R A M IN  5  FOR  S P A N W I S E  ( FROM  15  INTEGRATION DIRECTION.  G L A U E R T  )  16 v e l o c i t y a t a n y p o i n t y^ m u s t be o b t a i n e d  f r o m t h e sum o f  ing v o r t i c e s o r i g i n a t i n g along  of the wing.  a half  So t h a t  for  Glauert  (3).  of attack for  aerofoil  L  =  a  o  a  1  -  y)  Drag  for this  section originates  from  by  o  ( 3 )  the curve of  the a e r o f o i l  lift  c o e f f i c i e n t versus  angle  i n two d i m e n s i o n a l m o t i o n , w h i c h may be  t o ir i n m o s t c a s e s .  The c i r c u l a t i o n K a r o u n d  the  is  L  c  V  =  i s the chord l e n g t h .  induced v e l o c i t y w are be f o u n d  4Tr(y  s  the m a t e r i a l  K = k  where-c  _  L i f t and I n d u c e d  i s the slope of  taken equal  (2)  The l i f t c o e f f i c i e n t i s g i v e n  k  Q  dy *  =  Expressions for  Most o f  fdy  /•s J  where a  length  trail-  span s  w(yj  2.6  the  the  from  a  Q  c (Va -  w)  (4)  Once t h e c i r c u l a t i o n K a n d t h e  known t h e n t h e  lift  L and i n d u c e d d r a g  normal D can  17  / s D  =  *  U  -s  (5)  (6)  p w K dy  i n which p i s the f l u i d  density.  For purpose of  assume K c a n be e x p r e s s e d by t h e  4 s V  Z  n=l  where  0 i s the c i r c u l a r  along  the a e r o f o i l ,  Substituting we  the  A  Fourier  s o l u t i o n we c a n  series  s i n ne  (7)  c o - o r d i n a t e w h i c h r u n s f r o m 0 t o IT ( - s  s o t h a t y i s r e l a t e d t o 6 by y = - s c o s  Fourier expression for  K into  to  +s)  0.  the expression f o r w  get  =  w(e,)  fir c o s n0  and s i n c e  then  Since  cos  0  J  0  de  cos 0 -  =  sin  sin  n0  sin  e  n  s i n n6  (8)  1  n6  TT  cos  n  e  sin  V E n A  4s V £ A  cos  d0  cos 6 -  =  n0  D  =  w(e.,)  K  I  ., f[ I n A -—  e  1 1  1  (9)  1  =  a c(Va Q  w)  18  sin Then  4s V I A  or  Z A  for  s i n n0  =  a c(Va -  s i n ne ( n u + s i n 6)  the general  This equation  point  6,  =  VEn A, n sin  n6  (10)  6  u a sin 0  (ID  where  c a n now be u s e d t o f i n d  the values  of  the c o e f f i c i e n t s  A , so t h a t t h e c i r c u l a t i o n K and hence t h e v a l u e s o f n found.  The l i f t  a n d i n d u c e d d r a g c a n be e x p r e s s e d i n  L a n d D c a n be terms o f  the  coefficients  p V K dy  L -s  =  D  2  2T\ S^ p  r  2  A  (12)  p w K dy  = -s  (13)  For the case o f  the  twisted aerofoil  the angle of  a may be e x p r e s s e d a s a = a - e c o s 6 w h e r e a i s  the angle  attack of  19 attack of  the c e n t r e  (root)  of  the a e r o f o i l  t w i s t from the centre to the t i p . into in  the c o e f f i c i e n t equation  two p a r t s ,  method o f  2.7  Elliptic  will  i s discussed i n Appendix  be a minimum f o r  the  the angle  a given  lift.  The m a j o r  The E f f e c t o f  to the l i f t  force.  untwisted  has c o n s t a n t v a l u e a l o n g  the span then  The  the wing the  then the  induced  the  ellipse  circulation  load  by t w i s t i n g a  distribution rectangular  the t i p .  With  the  elliptic  span.  Twist  rectangular  aerofoil  induced v e l o c i t y tends  By t w i s t i n g t h e a e r o f o i l  If  determined  to e .  the  The e l l i p t i c  along  its  t h e s p a n we o b t a i n  is divided  induced drag  is  into  to  (which  elliptic  proportional  the  so t h a t  right it  loading.  segments  across  t o t h e sum o f  i n d u c e d v e l o c i t i e s o f each segment s q u a r e d .  not  towards  by j u s t  the t i p  a number o f  is  increase  length  amount t o d e c r e a s e t h e i n d u c e d v e l o c i t y t o w a r d s  individual  of  of a t t a c k decreases towards  loaded e l l i p t i c a l l y ) the tip.  substituted  axis of  i n d u c e d v e l o c i t y has a c o n s t a n t v a l u e a l o n g  For a r i g i d  the  then  load d i s t r i b u t i o n  c a n be a c h i e v e d by u s i n g an e l l i p t i c w i n g o r  loading  a may be  B.  be t h e s p a n a n d t h e m i n o r a x i s t h e m a g n i t u d e  w i n g so t h a t  angular  Loading  which i s proportional  2.8  for  t o a a n d one p r o p o r t i o n a l  When a w i n g has an e l l i p t i c drag w i l l  value  the  and each c o e f f i c i e n t i s  one p r o p o r t i o n a l  solution  This  and. e i s  the  Clearly  the  20 minimum p o s s i b l e i n d u c e d d r a g w i l l of  each segment  i s t h e same o r when t h e  value across the span. much s o t h a t towards will  the  tip  then  that  all  it  of  twists just  the r i g h t  of attack.  the l e a d i n g  t w i s t forward  towards  2.9  stiffeners  Effect of  edge.  aerofoil  and i f  component  i n both  far  this  leading  c o u l d be c l e a r l y  aerofoil twisted a  elliptic  is  however,  be s o  elliptic  to place  enough f o r w a r d  loading  stiffeners the  centre  the angle  t e s t s w e r e made on  and t r a i l i n g edges  at  t o be a h e a d o f  t w i s t to reduce  project  designed  so t h a t  models  the  effect  shown.  t h e normal tip.  i s yawed f o r w a r d  the span towards  spanwise flows Thus  on t h e  the v o r t e x  induced drag f o r c e are  the flow  the c e n t r e ,  high  rolling  reduced.  then  pressure  this  the  of  Yawing  directed along  c e n t r e to the so the  aerofoil  has t h e e f f e c t o f moving  i t moves  In  may,  amount t o p r o d u c e  This  When t h e a e r o f o i l  oppose  too  decreases  and t h e  attack f o r which  One way t o do t h i s  the t i p .  with stiffeners of  constant  is twisted  velocity  rectangular  c e n t r e of pressure then the s e c t i o n w i l l attack  the a e r o f o i l  velocity  can o c c u r .  angles  towards  if  be d e t r i m e n t a l  i s o n l y one a n g l e o f  induced  i n d u c e d v e l o c i t y has  the induced  For a r i g i d  A f l e x i b l e rectangular that  that  the e f f e c t w i l l  efficient.  g i v e n amount t h e r e loading  T h i s means  the net r e s u l t i s  be l e s s  be o b t a i n e d when t h e  has a tends  to  s i d e from  the  around the w i n g t i p  and  21  CHAPTER  APPARATUS AND  3.1  Water  Columbia's  the  in  yawing  ft/sec  INSTRUMENTATION  Flume  All  .3 f t / s e c  t e s t s were p e r f o r m e d  75 f o o t w a t e r f l u m e the 23.5 test;  a 6 in. weir at  i n the  University of  x 30 i n c h s e c t i o n .  For a l l  the end o f  tests  open.  The y a w i n g  and a f t e r  some m o d i f i c a t i o n s  these  t e s t s w a t e r v e l o c i t y was 3 . 5  c o n d i t i o n s were o b t a i n e d the narrow  a n d by h a v i n g  f t / s e c and depth  24 i n . s e c t i o n .  Water  two s e t s o f  inlet  Further  turbulence.  by a s c r e e n p l a c e d 15 f t  upstream of  was i t s e l f l o c a t e d 28 f t  from the e n t r a n c e .  the flume  and  Figure  18 i n ,  t h e end o f  through a s t i l l i n g tank which c o n t a i n s d e f l e c t o r to reduce  after  the  by  For  these  the flume  enters  the  baffles  smoothing  t h e s t r a i n gauge  the other  had b e e n made t o t h e f l u m e .  by a 12 i n . w e i r a t  12 i n . x  2.4  both v a l v e s of  t e s t was p e r f o r m e d  and  was  an  obtained  P l a t e 1 shows an of  and  flume  balance,  6 shows a s c h e m a t i c l a y o u t  to  except  These c o n d i t i o n s were o b t a i n e d  the flume  tests  equipment.  the  up  w a t e r v e l o c i t y and depth were h e l d c o n s t a n t a t  6 c f s pump f u l l y  view of  British  i n w h i c h v e l o c i t y c a n be v a r i e d  a n d 19 i n . r e s p e c t i v e l y .  by u s i n g  3  the  which overall  22  Plate  1  Overall  View o f  Flume  FIGURE 6  I From  Strain  Ur^  23  Rigid  Mounting  Gauges —Strain  Gauge  Balance  Water S u r f a c e  Flow Model  BALANCING UNIT  VOLT METER  CHART RECORDER  P.O W E R SUPPLY  SCHEMATIC  L A Y O U T OF EXPERIMENTAL  EQUIPMENT  24 3.2  S t r a i n Gauge B a l a n c e '  Force measurements  w e r e t a k e n on a s t r a i n g a u g e  s y s t e m w h i c h was d e s i g n e d  for  i n . x 1 .625  s h a f t machined to 0.75  of  its  in,round  length.  steel  place.  was f a s t e n e d  passing through s l o t s  top of  t o be  adjust  measure  the  the model.  upper  gauge  equipment. of  l o w e r end o f  upper  plate of  t h e model  this  on t h e  plate.  of  position of  to  The f o r c e  the balance  balance.  across  the depth  support  by through the  of  a model  in  the  o f ± 15 d e g r e e s  and  l i f t and drag f o r c e s  t h e model  components  is  i n d i c a t e d by  relative  are measured  1.5  r e c o r d i n g equipment  through  P l a t e 2 shows t h e  and F i g u r e  graduon  strain  electronic  t o r e c o r d a maximum l i f t lbs.  water  on  to a marking  r e c o r d e d on a p p r o p r i a t e  The s y s t e m was d e s i g n e d  of  the  lower p l a t e measured  and t h e o u t p u t i s  and t h e  cantilever  c o u l d be r o t a t e d  permitted  designed  l o c a t i o n and m a g n i t u d e  set-up  the  varied.  10 l b s a n d a maximum d r a g o f  the  the  A s l i d i n g mounting  The a n g u l a r  pairs  The l o w e r e n d o f  i t s angle of a t t a c k over a range  a t i o n markings the  to the  so t h a t  The b a l a n c e s y s t e m i s flume,  most  in.apart,  The s h a f t was s e c u r e d t o a r i g i d m o u n t i n g  the flume.  immersion  sections, 9  along  t o a c c e p t a b o l t a n d a p l a t e w h i c h c o u l d be l o c k e d  The model  30 d e g r e e s .  in.round  on e a c h f a c e a n d t h e s e w e r e c o v e r e d by a  a n d t h e s y s t e m was c o n n e c t e d t o bolts  T h i s c o n s i s t e d o f a 26  s h i e l d to e l i m i n a t e water s p l a s h .  s h a f t was t h r e a d e d in  project.  Two m a c h i n e d r e c t a n g u l a r  c o n t a i n e d s t r a i n gauges plexiglass  this  balance  7 shows t h e  force balance  dimensions  Plate  2  B a l a n c e and R e c o r d i n g  Equipment  FIGURE 7  V  DiQ.  l /e" Did. 5  Q.  IVe  C e n t r e L i n e - S t r a i n Gauges  T  3/  11  '*  i  r  Centre Line - S t r a i n Gauges  1  1  li I  ::  5  /,  i i •  6  ©  o  •  VIEW IN DRAG  VIEW  DIRECTION  IN LIFT  / 2  THE  STRAIN GAUGE  ; • J  BALANCE  ©  © .  DIRECTION  27 3.3  Models  T h e r e w e r e b a s i c a l l y two t y p e s o f m o d e l s u s e d i n t h e t h e s e were the a e r o f o i l  models and t h e w h a l e f l u k e m o d e l s .  The  aero-  r e c t a n g u l a r and s y m m e t r i c and had a t y p i c a l  aero-  foil  models were a l l  foil  s h a p e h a v i n g t h e maximum t h i c k n e s s a b o u t 40% o f  from the l e a d i n g edge. t h e same s i l a s t i c  The t h r e e e p o x y a e r o f o i l  allowed bars flexibility  a l l o w e d a c e r t a i n amount o f  Silastic  15 i n . when a 1 l b  s h r i n k a g e c o e f f i c i e n t , and i s q u i t e  t h e model  and c a s t i n g i s g i v e n to vary  welding rods. edge, 3.  0.3  the c e n t e r of  The f i r s t model  in.and 0.9  in.from  The s e c o n d e p o x y model  a t the t r a i l i n g edge,  0.4  degree,  had t w o b a r s  in.and 0.8  1 1/2  the c a n t i n g  degrees,  test.  has a l o w information  The b a r s ,  0.0938  placed i n the  T h i s model  placed  in.mild  is  twist along angles of  and 5 d e g r e e s ,  in  steel leading  shown i n  Plate placed  edge. i n . and  c o u l d be t w i s t e d t o i t s y i e l d  different  2 degrees  i n . from the  m o d e l was 18 i n . x 5 . 5  i t would r e t a i n a l o c k e d i n  model was t e s t e d w i t h f i v e  since this  The  tip.  had no b a r s a n d t h e t h i r d h a d 2 b a r s  i t s maximum t h i c k n e s s a n d i t  so t h a t  t w i s t , were  of  in.over  the  f l e x i b l e when s e t (more  the edge.  The a l u m i n u m a e r o f o i l at  in.  the epoxy s e t .  l o a d was a p p l i e d a t  i n Appendix C).  from  also  the t i p d e f l e c t e d 1  r u b b e r was c h o s e n f o r t h e m o l d m a t e r i a l  on m o l d i n g  i n . and 0.188  f l e x i b i l i t y , and  l i q u i d before  t h e s e m o d e l s was s u c h t h a t  an e f f e c t i v e s p a n o f  length  I t was d e c i d e d t o c a s t t h e s e m o d e l s  t o be p l a c e d w i t h i n t h e of  the chord  models were c a s t  r u b b e r m o l d a n d w e r e 17 i n . x 4 . 7 5  a t t h e i r maximum t h i c k n e s s . epoxy s i n c e t h i s  project;  its  length.  twist:  0.188  in.  point This  no t w i s t , 1  and was l a t e r u s e d  in  I  Plate  3  Epoxy A e r o f o i l  Model  29 The w h a l e f l u k e m o d e l s w e r e 5 / 8 of the  killer  whale Skana's s t a r b o a r d t a i l  made on t h e w h a l e i n t h e p o o l is  trained  of  scale  symmetric  fluke.  Measurements  to hold her f l u k e s out of  the w a t e r ,  An a c c u r a t e p l a n f o r m was o b t a i n e d  onto a l a r g e p i e c e o f the a d r o i t fluke  use o f a s e t o f  by t r a c i n g t h e  that for  purposes  of  a t t h e i r maximum t h i c k n e s s .  silastic  rubber  this  is  shown i n P l a t e 4 .  of  The a l u m i n u m model was l a t e r m o d i f i e d  that  3.4  the f l a n g e  t h e end s e c t i o n s , a n d m a r k i n g  i t off  i t c o u l d be y a w e d b a c k a n d f o r t h  dimensions  of  work:  is a list  to  The wooden  in a scale of  by  plate  rounding  degrees,  v e r t i c a l l y i n the flume.  so The  8.  Equipment  Following experimental  in.  c o n n e c t i o n on t h e l o w e r  t h e t h r e e m o d e l s a r e shown i n F i g u r e  Electronic  x 15  t h e m o d e l s had r e c t a n g u l a r end s e c t i o n s  holes  off  in  t h a t w o u l d be u s e d t o c a s t a n e p o x y model  All  to f i t  by  The  and t h i s was u s e d  d r i l l e d with four the balance.  shape  A h a n d made wooden  w i t h an aluminum c o r e w h i c h c o u l d h o l d a l o c k e d i n t w i s t . model  flukes  project  T h e s e m o d e l s w e r e 10 i n .  model was c o n s t r u c t e d f r o m t h e f i e l d m e a s u r e m e n t s make a m o l d o f  on t h e  i n s e c t i o n b u t t h i s was n e g l e c t e d  symmetric models would s u f f i c e . in.  whale  s o t h a t i t was p o s s i b l e  l a r g e c a l i p e r s and a t a p e m e a s u r e .  t h e m o d e l s b e c a u s e i t was f e l t  and 1.7  The  c a r d b o a r d and t h i c k n e s s p r o f i l e s were o b t a i n e d  i s s l i g h t l y unsymmetric  overall  were  the Vancouver P u b l i c Aquarium.  t o s t a n d on t h e p o o l - s i d e p l a t f o r m a n d t a k e m e a s u r e m e n t s directly.  reductions  of e l e c t r o n i c apparatus  used i n  the  Voltmeters:  H e w l e t t P a c k a r d , 419 A  D.C.  Dana D i g i t a l  Model  Power S u p p l y :  Harrison  Voltmeter,  620 4B D . C .  Power  Null  Voltmeter  5330 Supply  Chart Recorder:  Also a five Engineering  Honeywell  channel  Electronik  19  Recorder  B a l a n c i n g U n i t d e s i g n e d and b u i l t  Department.  i n the  Ci  Plate 4  Whale  Fluke  Model  r\3  DIMENSIONS OF THE THREE  MODELS  33  CHAPTER  EXPERIMENTAL  4.1  Calibration  PROCEDURE  Methods  The s t r a i n gauge e x t e n s i o n was b o l t e d be a p p l i e d  4  by w i r e s  b a l a n c e was s e t up on a b e n c h , a n d  to the v e r t i c a l in  the h o r i z o n t a l  c a l i b r a t i o n was c o m p l i c a t e d by t h e drag  shaft,  so t h a t  the c r o s s - s e c t i o n i n the  s o t h a t when p u r e d r a g was a p p l i e d  reading.  These  easily  be e l i m i n a t e d  calibration devised, strain  gauge for  c o r r e c t i o n s were  the  curves were f e d  range into  of  the  t h a t would i n t a k e the four  gauge  pairs,  and o u t p u t t h e  strain  gauge  section.  on t h e b e n c h f o r  4.2  Test  lift  gauges  so they  readings  showed a  program  from the  four their  from the  P l a t e 5 shows t h e s t r a i n g a u g e  lift  The  f o r c e s , and  measured  the  could  involved.  and a s i m p l e  and d r a g  effort  lift  linear,  drag f o r c e s  voltage  The  lower  b a l a n c e mounted  calibration.  Procedure  Lift models,  the  computer,  r e s p e c t i v e d i s t a n c e to the c e n t e r o f  could  the c r o s s - s e c t i o n in  direction,  lift  known l o a d s  l i f t and drag d i r e c t i o n s .  fact that  d i r e c t i o n was much s m a l l e r t h a n  an  and d r a g measurements  t h e a l u m i n u m model  f l u k e models.  at various  The t o p end o f  w e r e made on t h e angles  e a c h model  of  three  epoxy  t w i s t and t h e  was a t t a c h e d t o t h e  two w h a l e lower  34  Plate 5  S t r a i n Gauge Balance  35 plate of  t h e b a l a n c e , a n d t h e b a l a n c e was r a i s e d t o a h e i g h t  t h a t t h e l o w e r p l a t e was a b o u t depth of  i m m e r s i o n was a b o u t  o f a t t a c k , no l i f t  3 i n c h e s above the w a t e r s u r f a c e .  15 i n c h e s f o r a l l  Due t o t h e s y m m e t r y  of  t h e model  f o r c e was p r o d u c e d ,  t h e model  B e f o r e e a c h t e s t on t h e a e r o f o i l  because o f  a n d t h e p o s i t i o n o f minimum  occurred during  readings  the t e s t .  zero angle of  effect of full  of water.  were c h e c k e d t o e n s u r e At each a n g l e o f  to a l l o w the turbulence e f f e c t s  aerofoil  depending  i n the l i f t  t h a t no d r i f t  had  voltages  run f o r  The  showed r a t h e r  range o f  15 d e g r e e s .  up large  The a l u m i n u m  so a s t o d e c r e a s e t h e a n g l e o f  the t i p  o f a t t a c k , a n d t h i s was t e s t e d a t  positive  and n e g a t i v e a n g l e s , so t h a t t h e a d v a n t a g e o u s  position  c o u l d be c o m p a r e d w i t h t h e d i s a d v a n t a g e o u s  The a l u m i n u m a e r o f o i l  epoxy  be t e s t e d  m o d e l was t w i s t e d u n i f o r m l y p o s i t i v e angles  2  The m o r e s t i f f a l u m i n u m a n d w h a l e  f l u k e m o d e l s w e r e t e s t e d up t o t h e f u l l  for  two  i n c r e a s e d by e i t h e r 1 ,  f l e x i b l e , and c o u l d o n l y  direction.  the  out.  t o an a n g l e o f a t t a c k o f 8 d e g r e e s , a t w h i c h t h e y deflections  later  after  on t h e r a n g e a n d t h e s e n s i t i v i t y .  s e c t i o n s were r a t h e r  cir-  Immediately  a t t a c k the four  to even  The a n g l e o f a t t a c k was g e n e r a l l y or 3 degrees  attack.  the whale f l u k e s e c t i o n s ,  w e r e r e c o r d e d i n s u c c e s s i o n , e a c h one b e i n g a l l o w e d t o minutes  lift  s e c t i o n s , the four balance bridge  b r i d g e was b a l a n c e d w i t h t h e f l u m e each t e s t the zero  angle  p o s i t i o n w i t h no w a t e r i n t h e f l u m e ,  the l a r g e buoyancy  The  sections.  s e c t i o n at zero  was u s e d t o e s t a b l i s h t h e b a l a n c e p o s i t i o n f o r  c u i t s were s e t t o t h e n u l l  such  model was a l s o u s e d f o r  attack  at  both  positive  negative  position.  t h e y a w i n g t e s t where  t h e a n g l e o f a t t a c k was h e l d c o n s t a n t a t - 8 d e g r e e s  and t h e  model  36 yawed back and f o r t h angles  4.3  from 3 to  13  from the v e r t i c a l  p o s i t i o n through a range  degrees.  Errors  During the c a l i b r a t i o n procedure the s i g n a l  from the  was s t a b l e a n d p r o d u c e d a s t r a i g h t l i n e on t h e c h a r t . made i n  the  range o f  flume,  because o f  The e r r o r  the v a r i a t i o n s  The e r r o r  in  the v o l t a g e r e a d i n g s  d r a g g a u g e s a n d ± 35 mv f o r attack).  the  gauges  The a v e r a g e  error  the  l i f t and d r a g  error  (averaged  i n the drag lb.  r e a d i n g was ± 40 mv w h i c h c o r r e s p o n d s  lb.  from  gauges.  f r o m t h e f l u m e w e r e ± 20 mv f o r  the l i f t  The t o t a l  both  which corresponds to a drag f o r c e o f ± 0.02  ± 0.20  by e y e a n d t h i s was  in the c a l i b r a t i o n f a c t o r s obtained  r e a d i n g s was a b o u t ± 5 mv f o r  of angles of  The a c t u a l  method.  spread of  the l i f t  Measurements  v a l u e s a n d t h e c h a r t i s a s shown i n P l a t e 6 .  deemed a r e a s o n a b l e  recorder  t h e low f r e q u e n c y o s c i l l a t i o n s , showed a  r e a d i n g was o b t a i n e d by a v e r a g i n g  ±  of  i n the  lift  over the  range  r e a d i n g was ± 25 mv The t o t a l  to a l i f t  to drag  the  error  force  r a t i o was  in  of  about  0.8. The e r r o r  markings  be o f  the angle o f a t t a c k from the  on t h e b a l a n c e was ± 0 . 2 5  s e t t i n g the degrees.  in reading  zero  degrees.  The e r r o r  graduated  incurred  a n g l e o f a t t a c k p o s i t i o n was e s t i m a t e d t o be ±  The e r r o r  the order of  i n the angle of ± 0.25  degrees.  from 0.25  t w i s t m e a s u r e m e n t s was j u d g e d  to  Plate 6  Sample Output from Chart Recorder  38  CHAPTER 5  DISCUSSION OF RESULTS  In discussed. In  this  this  s e c t i o n the r e s u l t s fronvthe  The r e s u l t s a r e p r e s e n t e d  s e c t i o n the r e s u l t s are  as a p l o t  of  lift  to drag  various  in tabular  tests will  form i n Appendix  presented i n graphical  form,  r a t i o versus angle of a t t a c k . of  on t h e g r a p h s a n d t h e s e w i l l  now be d i s c u s s e d i n d e t a i l .  the o u t s e t t h a t  of  the t o t a l  d r a g and not  of  the t o t a l  drag.  In to determine Figure  the i n f l u e n c e o f  the l i f t  generally  illustrated  to drag r a t i o  It is  induced drag which i s only  should  in  a  for all  t e s t s f l e x i b l e a e r o f o i l s were  t w i s t on t h e i r l i f t  l e a d i n g edge s t i f f e n e r s  angles of  to drag  attack --  greater  l e a d i n g edge s t i f f e n e r s  has t h e h i g h e r  than 2 degrees.  lift  The model  to drag r a t i o  tested  ratios. aerofoils.  lift  to  trailing  The model w i t h no s t i f f e n e r s ,has a g r e a t e r  around 1 or 2 degrees.  stiffeners  has a g r e a t e r  a t t a c k t h a n t h e model w i t h t h e  t h a n t h e model w i t h  terms  portion  9 shows t h e r e s u l t s f r o m t h e t e s t s on t h e t h r e e e p o x y  stiffeners. ratio  the  the f i r s t s e r i e s of  The model w i t h t h e ratio  the v a r i o u s models a r e  lift  drag edge  to  drag  at small angles  with leading for a l l  of  edge  angles  I n t h i s t e s t i t was d i s a p p o i n t i n g  of  attack  that  the  model w i t h l e a d i n g e d g e s t i f f e n e r s d i d n o t  show g r e a t e r  over the  A t low a n g l e s o f a t t a c k  full  range o f  angles of a t t a c k .  A.  The  advantages and d i s a d v a n t a g e s  be made c l e a r a t  be  efficiency the  PLOT  OF  LIFT FOR  TO  DRAG  THE  RATIO  THREE  VERSUS  EPOXY  ANGLE  AEROFOILS.  OF  ATTACK  40 lift  f o r c e s produced are small  stiffeners  and so t h e t w i s t i n g e f f e c t o f  i s s m a l l , a n d one w o u l d e x p e c t t h e l e a d i n g e d g e c a s e t o  a t l e a s t match t h e p e r f o r m a n c e  of  t h e model w i t h no s t i f f e n e r s .  may h a v e been due t o a s h a p e d e f e c t i n o n e i o r b o t h o f a s t h e e p o x y had a t e n d e n c y t o w a r p a f t e r mold.  It  was v e r y  edge s t i f f e n e r s of angles  the  peak l i f t  i t was removed  from  e f f i c i e n t over a wide  and l e s s s e n s i t i v e t o changes i n a n g l e o f the second s e r i e s of  the  leading  to drag r a t i o over a wide  o f a t t a c k , i t was t h u s q u i t e  In  t h e two m o d e l s ,  i n t e r e s t i n g t o n o t e t h a t t h e model w i t h  held i t s  This  range range  attack.  t e s t s a r i g i d aluminum  aerofoil  was u s e d w h i c h c o u l d be t w i s t e d a n d made t o h o l d a g i v e n t w i s t each t e s t . aerofoil  Figure  10 shows t h e r e s u l t s f r o m t e s t i n g t h e  i n w h i c h t h e model was t w i s t e d t o ' r e d u c e  at the t i p .  The  model  had a peak l i f t  i t was t w i s t e d , b u t w i t h a n a n g l e o f r a t i o was 1 2 . 4  and a t 1.5  degrees  was r e d u c e d t o a b o u t 8 . 0 . at higher  angles  u n t w i s t e d model  t w i s t o f 1 degree  the higher  t h e n t h e model  t h e model w i t h 5 d e g r e e s  of  a t e t h a t the curve f o r 1.5  the  to drag  ratios  angles of  twist.  w i t h 1 degree o f  before  maximum  The  degrees  of  is  of the  then unfortun-  t w i s t d i d not cross the 1  9 degrees.  t w i s t c a s e show i t  t w i s t was  It  it  occurred  11 d e g r e e s ,  t w i s t was m o s t e f f i c i e n t .  t w i s t curve at about  the 2 degrees of  10.2  i t was 9 . 8 w h i l e a t 5 d e g r e e s  m o s t e f f i c i e n t up t o a n a n g l e o f a t t a c k o f a b o u t  of  attack  p r o v e d t o be t h e m o s t e f f i c i e n t up t o an a n g l e  attack of 4 degrees,  degree  aluminum  the angle o f  to |drag r a t i o o f  T h e s e peak l i f t  of attack for  for  However t h e r e s u l t s  from  t o be t h e m o s t e f f i c i e n t f r o m  10  41  FIGURE  0  2  10  4 6 8 A n g l e of A t t a c k ( d e g r e e s )  10  12  P L O T OF LIFT TO DRAG RATIO V E R S U S A N G L E OF A T T A C K FOR T H E A L U M I N U M A E R O F O I L AT VARIOUS A N G L E S OF T W I S T TO R E D U C E A N G L E OF A T T A C K .  42 to  12 d e g r e e s .  the higher  T h i s means t h a t a s t h e a n g l e o f a t t a c k i s  angles of  t w i s t are  t h e most e f f i c i e n t .  f r o m w h a t was s a i d e a r l i e r a b o u t  This  increased follows  the t w i s t i n g e f f e c t being small  for  t h e f l e x i b l e a e r o f o i l w i t h l e a d i n g e d g e s t i f f e n e r s when t h e a n g l e s attack are s m a l l .  As t h e a n g l e o f  a t t a c k i n c r e a s e s the  and so t h e s e c t i o n t w i s t s t o g r e a t e r Figure drag r a t i o  drawn t h r o u g h t h e peaks o f represent  at a l l  each a n g l e of a t t a c k . the i n d i v i d u a l  the curve f o r  that twists just  angles of a t t a c k .  t h e peak o f  the r i g h t Another  a properly  the curve f l a t t e n s out f o r  that  t h e peak o f  as t h e peaks o b t a i n e d  i n the aluminum a e r o f o i l  the w i n g t i p ,  all  t h e no t w i s t  case. In  tested  of  the models  to drag r a t i o .  Figure  10 i s  that  twist.  the  the  same  aerofoil  the opposite  sense  a t t a c k a t the t i p .  All  i n c r e a s i n g the l i f t  at  have a l o w e r l i f t  the next s e r i e s of  to i n v e s t i g a t e the  e f f e c t of  loading  tests.  was t w i s t e d i n  t h e s e t e s t s show t h e d e t r i m e n t a l  flexible  s h o u l d be a b o u t  than decrease the angle of  it  the curve f o r  11 shows t h e r e s u l t s f r o m t h e a l u m i n u m  t e s t s i n w h i c h t h e model  to increase rather of  since i t  should  attack,  increasing angles of  is  of  designed  to  curve  t w i s t curves  t o 'note f r o m F i g u r e  u n t w i s t e d e p o x y model  Figure  too h i g h ,  The  amount t o o b t a i n e l l i p t i c  point  The r e s u l t s f r o m t h e s e t e s t s v e r i f y  series  the best l i f t  t h e b e s t t h a t c a n be done a t a n y g i v e n a n g l e o f  s h o u l d be i d e n t i c a l t o aerofoil  each of  increases  angles.  10 t h u s r e p r e s e n t s an e n v e l o p e o f  t h a t c a n be o b t a i n e d f o r  lift  of  to drag r a t i o  than  t e s t s the whale f l u k e models  were  influence of  t w i s t and s h a p e on t h e  12 shows t h e r e s u l t s f r o m t h e w h a l e  lift  fluke  43  FIGURE  II  2  8  ^^^^^  6 c  4  T w i s t e d to i n c r e a s e a of a t t a c k c]t T i p  0  2  PLOT  OF  4 Angle  of  Attack  LIFT  TO  DRAG  OF A T T A C K VARIOUS  6  FOR  THE  ANGLES  OF  ANGLE  1 -  le  ig  8  10  (degrees)  RATIO  VERSUS  ANGLE  ALUMINUM  AEROFOIL  TWIST  INCREASE  OF  12  TO  ATTACK.  AT  P L O T OF L I F T TO DRAG RATIO VERSUS ANGLE OF ATTACK FOR THE WHALE F L U K E MODELS .  N  series  of  r a t i o of  tests. 8.5  a n d t h e e p o x y t w i s t e d model  The t w i s t e d model efficient  The u n t w i s t e d wooden model  ( w i t h an a n g l e o f  to  drag  had a maximum r a t i o o f  9.5.  t w i s t of about  angles of  0 t o 10 d e g r e e s .  the u n t w i s t e d aluminum a e r o f o i l  On t h e o t h e r  the angle of a t t a c k i n c r e a s e d a t  fell  lower than the u n t w i s t e d case f o r The e p o x y a e r o f o i l the e l l i p t i c l o a d i n g  either  p r o d u c e s optimum  with  theory  lift  as  before,  higher  angles* of  a t t a c k but  than  interest-  the t w i s t e d a e r o f o i l s .  the t i p , all  is quite  was t w i s t e d s o  the l i f t  angles of  to drag  ratios  attack.  l e a d i n g edge s t i f f e n e r s put forward e a r l i e r .  seems  t o d r a g r a t i o s a n d c a n be o b t a i n e d  The e l l i p t i c w i n g w i l l  to  Elliptic  an e l l i p t i c wing o r a r e c t a n g u l a r wing t w i s t e d to r e d u c e  angle of a t t a c k a t the t i p . all  and t h i s  h a n d when t h e e p o x y model  that  loading  Presumably,  lift  The p e a k s o f b o t h c u r v e s a r e s l i g h t l y f l a t t e r  i n g a s a f l a t t e n i n g o u t was a l s o n o t i c e d f o r  verify  higher  t w i s t w o u l d h a v e b e e n more e f f i c i e n t a t t h e  angles of a t t a c k . t h e peak f o r  the range  lift  1 d e g r e e ) was more  over a l a r g e range o f angles of a t t a c k , having  to drag r a t i o s over higher  had a p e a k  by the  be e f f i c i e n t  at  the r e c t a n g u l a r wing w i t h a f i x e d angle  of  t w i s t c a n o n l y m a t c h i t s p e r f o r m a n c e a t one s p e c i f i c a n g l e o f  attack  depending  in  Figure  on t h e a n g l e o f  twist.  10 a s e a c h i n d i v i d u a l  range of  T h i s shows up q u i t e  angle of  angles of attack f o r which i t  clearly  t w i s t h a s i t s own p a r t i c u l a r i s more e f f i c i e n t t h a n  the  untwisted case. The f l e x i b l e r e c t a n g u l a r w i n g d e s i g n e d t o t w i s t a t the r i g h t ing at a l l  angle f o r  each a n g l e of a t t a c k w i l l  angles of a t t a c k .  from the envelope  just  produce e l l i p t i c l o a d -  The f l e x i b l e a e r o f o i l c a n be d e s i g n e d i c u r v e o b t a i n e d from a s e r i e s o f t e s t s such as t h o s e !  I  45  46 of  F i g u r e 10.  If  a l a r g e number o f  incrementing the angle of  An a c c u r a t e e n v e l o p e  peak r a t i o s f o r  f o r which the range of angles  a t any p a r t i c u l a r a n g l e o f  the  twist required  for  a t t a c k c o u l d be  off. The e p o x y a e r o f o i l  peak l i f t  e l l i p t i c loading.  d e s i g n e d c o r r e c t l y and so i t the envelope Figure  w i t h l e a d i n g edge s t i f f e n e r s h e l d  to drag r a t i o over a wide range  influence of  Unfortunately  this aerofoil  d i d not match the performance  curve through the peaks of  the i n d i v i d u a l  the  was  not  given  by  curves  of  10.  t h e u n t w i s t e d model t w i s t e d model place over  shape o f  the e n t i r e range,  perhaps  the whale f l u k e .  It  of  t w i s t and the  i s c l e a r t h a t the unusual  i t s performance.  unusual  shape  The t e s t s on t h e  This  twist The w h a l e  t w i s t e d c a s e t o be more e f f i c i e n t e v e n  a n g l e s o f a t t a c k and we s u r m i s e t h a t  shape e f f e c t .  of  aluminum  was m o s t e f f i c i e n t a n d t h i s was c o n s i s t e n t w i t h t h e t h e o r y . shows t h e  taking  t h e g r e a t e r e f f i c i e n c y was b r o u g h t  showed t h a t a t s m a l l a n g l e s o f a t t a c k t h e c a s e o f no  however,  than  The  was n o t f l e x i b l e s o e l l i p t i c l o a d i n g c o u l d n o t be  s e c t i o n improves  aerofoil  was more e f f i c i e n t  over a l a r g e range of angles of a t t a c k .  a b o u t by a c o m b i n a t i o n o f t h e s m a l l a n g l e  fluke,  its  a n d t h i s m u s t be due t o  The t w i s t e d w h a l e f l u k e e p o x y model  small  of  be  c u r v e c o u l d t h e n be d r a w n t h r o u g h  each t w i s t and from t h i s c u r v e the  the f l e x i b l e a e r o f o i l  this  then a  i n w h i c h t h a t p a r t i c u l a r t w i s t was m o s t e f f i c i e n t w o u l d  small.  read  performed,  t w i s t a s m a l l amount e a c h t i m e ,  s e r i e s o f c u r v e s w o u l d be o b t a i n e d attack  t h e s e t e s t s were  t h i s m u s t be due t o  seems t o c o n t r a d i c t t h e b a s i c t h e o r y  l o a d i n g b e c a u s e we s h o u l d n o t e x p e c t t o do b e t t e r  at  the  of e l l i p t i c  than the  untwisted  47 case a t small angles of series  of  definite  attack.  t e s t s are required  This  to i n v e s t i g a t e  c o n c l u s i o n s s h o u l d n o t be drawn The m o s t i n t e r e s t i n g  that  result  the tapered whale f l u k e b e n e f i t s  thought that tapering of  the  in  the optimum  elliptic  be u s e f u l  tapered whale f l u k e ,  of  is  elliptic  loading at a l l  toward changes  angles  Originally  it  tapering  These r e s u l t s  the  by  combinations is  shows a n i n c r e a s e i n  of a t t a c k .  is was  c o u l d be b r o u g h t a b o u t  by t w i s t i n g , a n d  however,  two d i m e n s i o n a l  small.  efficiency  are  better  theory  t e s t e d here  a wide range o f  angles  of  flexible aerofoil  system i s very  forgiving,  angle  of  a t t a c k produce  There  is  no s h a r p  range  of  angles  many n a t u r a l  the f l e x i b l e a e r o f o i l  which  of  attack is  sensitivity  held  flukes  for  propulsion. of  i t s peak  lift  t o do much b e t t e r it  or  is less little stall  s y s t e m s a n d we m i g h t  birds  lack of  The e p o x y a e r o f o i l  than  i n the  in  lift  the  The f l a p p i n g of  This  is  this  action rotates  a t t a c k and t h e h i g h e r  kind  to drag  in  full with  particularly t h e i r wings  the wing  lift  of  ratio.  i n agreement is  quite  changes  c u r v e and t h e  This  surmise that  lift  leading  properly  and s m a l l  change  produces  r a t i o over  this.  a n d swimming a n i m a l s w h i c h must f l a p  angles  with  to drag  demanding,  o f a t t a c k c a n be u t i l i z e d .  for  its  a t t a c k and we w o u l d e x p e c t a  very  drop o f f  useful  q u i t e a range  of  i n angle of a t t a c k .  edge s t i f f e n e r s  designed  from these t e s t s  loading. The m a i n a d v a n t a g e  elliptic  obtained  loading  i n d i c a t e d by t h e p r e d i c t i o n s o f  and  here.  when t h e amount o f  when t w i s t e d t o r e d u c e t h e a n g l e  extensive  t h e shape e f f e c t ,  from t w i s t .  the form o f an e l l i p s e o r  two s h o u l d o n l y  The h i g h l y  than  i n d i c a t e s t h a t a more  to drag  or  through ratio  48 t h a t c a n be m a i n t a i n e d d u r i n g are a l s o s u b j e c t to natural turbulent  flapping  changes  c h a r a c t e r i s t i c s of  the b e t t e r .  Gliding  i n a n g l e o f a t t a c k due t o  G l a u e r t a n d some t y p i c a l r e s u l t s g a v e t h e  the  i n d u c e d d r a g as 0.0245 l b s as c o m p a r e d ' t o  2.47 for  l b s and a measured  total  drag of  the u n t w i s t e d aluminum a e r o f o i l  degrees.  For 1 degree of  l b s w h i l e the measured 0.20  lbs.  lower  a measured  0.29!lbs,  the  theory  lbs  and  lift  a t an a n g l e o f a t t a c k o f attack of  the flow in  drops  only  being 6  6  degrees  l b s and t h e c o m p u t e d i n d u c e d d r a g was  l i f t was 2 . 0 8  The c o m p u t e d v a l u e s f o r  of  of  these r e s u l t s  the flume.  0.021  l b s and t h e m e a s u r e d d r a g was l i f t a n d i n d u c e d d r a g a r e much  t h a n t h e m e a s u r e d v a l u e s a n d t h i s m u s t be due t o t h e  nature lift  l i f t as 1.16  t w i s t a t an a n g l e o f  t h e c o m p u t e d l i f t was 1 . 0 8  the  wind.  Some c a l c u l a t i o n s w e r e c o m p l e t e d a c c o r d i n g t o of  birds  It  is  turbulent  s i g n i f i c a n t t h a t the  12% w h i l e t h e m e a s u r e d d r a g d r o p s  31%.  This  measured 31% d r o p  i i n d r a g c a n n o t be a d e q u a t e l y theory.  e x p l a i n e d by t h e u s u a l  Presumably the three  s i g n i f i c a n t r o l e and f u r t h e r  dimensional study of  two  dimensional  spanwise flows play a  such t h r e e  dimensional  effects  w o u l d be w o r t h w h i l e . Figure that there  i s very  13 shows t h e r e s u l t s f r o m t h e y a w i n g t e s t s a n d shows little  difference  i s yawed f o r w a r d o r b a c k w a r d . lift it  t o d r a g r a t i o w i t h yaw b u t  i s probable  on t h e r e s u l t s .  that wingtip  i n p e r f o r m a n c e when t h e  T h e s e t e s t s show l i t t l e  aerofoil  variation  of  the r e s u l t s are not c o n c l u s i v e because  s h a p e may h a v e a n o v e r - r i d i n g  influence  10  PLOT OF L I F T TO DRAG RATIO VERSUS ANGLE OF YAW FOR THE ALUMINUM  AEROFOIL .  50 SUMMARY OF  It  has been c o n f i r m e d  by a f l e x i b l e a e r o f o i l permit j u s t  the r i g h t  c o r r e c t amount o f rigid aerofoils each t e s t .  amount o f  i n which the of  should  a t t a c k towards  drag  tapered wings  and t h e s e b e n e f i t s  beneficial  of  than  benefit  makes t h e t o t a l low s p e e d s ,  has been are  a small  would  This tests  amount  than  from washout  the  tip);  on  for gives  fact, In  Furthermore, shows;  the theory  reverse  this  study  commencement  twist washout  in the cases  in  As an  induced  additional  l e s s s e n s i t i v e t o changes stall,  a result  of  birds  and  in which  a n d c o u l d be p a r t i c u l a r l y of motion  dimen-  t w i s t i n g i s more  predicts.  or  to  s i m p l e two  the r e d u c t i o n s  i s no s h a r p d r o p o f f  system l e s s demanding  with  twist (twist  shown t o be b e n e f i c i a l  the t w i s t e d wing i s  say d u r i n g  in  wings  i n c o n t r a d i c t i o n to the  the simple theory  angle of a t t a c k , there  at  attack.  controlled twist, tests  e l l i p t i c loading.  e l l i p t i c loading.  b e i n g much g r e a t e r  important  incremented  of e l l i p t i c loading,  not b e n e f i t  s h o u l d make them a p p r o x i m a t e  theory  of  obtained  which  by m a k i n g a s e r i e s o f  these r i g i d ,  simple theory  reduce the angle of  sional  leading edge,  t w i s t a t each angle  twist is  c a n be  flexible aerofoil characteristics.  considerable taper  tested,  i n the  t w i s t c a n be o b t a i n e d  By t h e  t w i s t i n g of  that e l l i p t i c loading  with stiffeners  The e n v e l o p e  the d e s i r a b l e  CONCLUSIONS  useful  fish.  L I S T OF REFERENCES  Abbott,  Barry,  I . H . a n d von D o e n h o f f , A . E . Theory o f Wing S e c t i o n s I n c l u d i n g a Summary o f A i r f o i l D a t a . New Y o r k , McGraw H i l l , 1949. A.B. Engineering S o n s , 1964.  Measurements.  New Y o r k , W i l e y  and  Glauert,  H. The E l e m e n t s o f A e r o f o i l a n d A i r s c r e w T h e o r y . Cambridge, Cambridge U n i v e r s i t y P r e s s , 1926.  Howard,  F. a n d G u n s t o n , W. Y o r k , Random H o u s e ,  Prandtl,  L. and T i e t j e n s , O . G . Aeromechanics. Trans!. McGraw H i l l , 1 9 3 4 .  Prandtl,  L. and T i e t j e n s , O.G. A p p l i e d H y d r o - and A e r o m e c h a n i c s . T r a n s ! . J . P . Den H a r t o g . New Y o r k , D o v e r P u b l i c a t i o n s , 1957.  The C o n q u e s t 1972.  A n d e r s e n , H.T. The B i o l o g y o f Academic P r e s s , 1969. Norris,  K.S. Whales, University of  of  the A i r .  New  F u n d a m e n t a l s o f H y d r o - and L. Rosenhead. New Y o r k ,  M a r i n e Mammals.  D o l p h i n s and P o r p o i s e s . C a l i f o r n i a P r e s s , 1966.  New Y o r k ,  Berkely,  APPENDIX A TABLE OF RESULTS  EPOXY AEROFOIL ANGLE  LIFT  1 2 3 4 5 6 7 8  T.25 1.23 1.89 2.01 2.31 2.65 2.45 2.83  LIFT  DIST.  S T I F F E N E R S IN DRAG 0.10 0.12 0.16 0.21 0.24 0.32 0.41 0.46  12.6 16.7 11.2 12.5 11.7 11.2 13.4 12.3  EPOXY AEROFOIL 1 2 3 4 5 6  1.41 1.56 1.96 2.35 2.25 2.60  EPOXY AEROFOIL 1 2 3 4 5 6 7 8  0.83 1.01 1.53 1.69 2.14 2.25 2.56 2.83  0.09 0.12 0.17 0.23 0.27 0.37  11.7 12.6 11.7 11.3 12.8 12.0  14.4 15.1 11.7 13.1 11.9 12.4 11.2 11.4  -  LEADING EDGE  DRAG D I S T . 10.1 10.7 10.4 10.6 11.3 11.5 10.6 11.8  L/D  RATIO 13.0 10.5 11.9 9.8 9.5 8.2 6.0 6.2  NO S T I F F E N E R S 10.4 10.6 11.1 11.3 12.6 11.8  16.5 12.7 11.7 10.0 8.2 7.0  S T I F F E N E R S IN T R A I L I N G EDGE 0.08 0.11 0.16 0.22 0.27 0.35 0.41 0.53  8.2 9.1 9.7 10.6 11.1 11.3 12.6 11.9  10.0 9.0 9.7 7.9 7.8 6.4 6.3 5.4  53  ALUMINUM AEROFOIL -  ANGLE  -3 -6 -9  LIFT  L I F T DIST.  DRAG  DRAG D I S T .  1.43 2.47 3.28  14.1 12.7 12.5  0.14 0.29 0.54  12.6 13.7 13.3  ALUMINUM AEROFOIL -  -3 -6 -9 -11 +3 +6 +9 +11 +13  -3 -6 -9 -11 -13 3 6 9 11 13  0 DEGREES OF TWIST  1.44 2.50 3.37 3.42 0.38 1.15 2.08 2.62 3.21  2.05 2.84 3.43 3.65 3.98 0.02 0.57 1.73 2.38 3.13  15.2 12.7 12.4 14.5 2.6 10.8 11.5 11.4 12.0  L/D  RATIO  10.2 8.5 6.0  1 DEGREE OF TWIST  0.16 0.32 0.61 0.82 0.05 0.09 0.20 0.33 0.49  ALUMINUM AEROFOIL -  1.5  12.5 13.1 13.1 13.5 12.9 35.8 20.8 12.7 12.3 11.7  0.20 0.43 0.67 0.84 1.07 0.05 0.08 0.17 0.31 0.45  11.6 14.0 13.5 13.3 9.3 11.6 14.5 13.8 14.1  DEGREES OF  13.0 12.4 14.1 14.6 14.6 11.0 12.8 13.0 12.2 13.6  8.7 7.8 5.5 4.2 7.4 12.3 10.2 8.0 6.5  TWIST  10.2 6.6 5.1 4.3 3.7 0.4 7.6 9.9 7.6 7.0  54  ALUMINUM AEROFOIL  ANGLE  LIFT  -3 -6 -9 -11 3 6 9 11 13  1.60 2.62 3.19 3.85 0.06 0.90 1.82 2.43 2.84  LIFT  DIST.  13.9 12.7 13.6 12.8 6.8 12.6 12.4 12.5 13.0  ALUMINUM AEROFOIL  -3 -6 -9 -11 -13 3 6 9 11 13 15  1.93 2.86 3.39 3.78 4.07 0.42 0.28 1.20 2.13 2.61 3.34  15.4 14.4 14.5 13.7 13.3 13.4 15.8 13.3 11.1 11.6 11.3  -  2 DEGREES OF TWIST  DRAG  0.18 0.36 0.58 0.90 0.06 0.07 0.20 0.28 0.45  -  DRAG  DIST.  12.6 13.5 14.8 12.9 8.6 15.4 12.6 15.3 14.4  L/D  RATIO  8.8 7.3 5.5 4.3 1.1 12.5 8.9 8.7 6.3  5 DEGREES OF TWIST  0.26 0.51 0.75 0.98 1.17 0.07 0.08 0.15 0.20 0.33 0.53  14.2 14.7 16.0 15.1 15.5 9.4 8.1 9.8 12.6 12.6 11.9  7.4 5.6 4.5 3.8 3.5 5.8 3.6 8.2 10.4 7.9 6.3  WHALE FLUKE -  ANGLE  LIFT  1 2 3 4 5 6 7 8 10 12 15  0.49 0.72 0.88 0.94 1 .38 1.66 1 .74 2.09 2.67 3.14 3.53  LIFT  DIST.  11.0 11 . 8 13.2 16.0 13.1 12.4 13.9 13.1 12.2 11.9 12.4  WHALE FLUKE -  -2 -4 -6 -8 -10 -12 2 4 6 8 10 12  0.57 1.26 1.69 1.71 2.51 2.52 0.17 0.95 1.32 1.70 2.10 2.49  17.3 12.3 12.9 15.4 12.7 14.5 23.0 10.3 12.7 13.7 14.0 14.1  0 DEGREES OF TWIST  DRAG  DRAG D I S T .  0.22 0.25 0.21 0.20 0.22 0.22 0.24 0.28 0.30 0.37 0.47  9.2 8.1 9.7 10.2 9.7 10.1 10.5 8.9 10.6 9.9 10.5  1 DEGREE OF  0.18 0.19 0.19 0.25 0.29 0.33 0.24 0.26 0.34 0.43 0.52 0.63  L/D  RATIO  2.2 2.9 4.2 4.7 6.4 7.5 7.4 7.6 8.9 8.5 7.6  TWIST  10.2 9.7 10.7 8.9 9.7 10.3 10.5 11.9 11.4 11.5 12.3 12.8  3.2 6.8 9.1 6.9 8.8 7.5 0.7 3.6 3.8 4.0 4.1 3.9  56  ALUMINUM AEROFOIL  ANGLE*  LIFT  0 3 5 8 10 13  3.11 2.77 3.44 3.49 3.34 3.62  LIFT  DIST.  14.3 15.7 13.2 13.0 13.4 12.3  ALUMINUM AEROFOIL  3 5 8 10 13  2.85 3.44 3.30 3.30 3.31  -  14.4 12.7 13.2 13.1 13.1  -  CANTED  FORWARD  DRAG  DRAG D I S T .  0.41 0.38 0.41 0.42 0.40 0.40  13.0 16.9 14.9 15.2 16.5 17.9  L/D  RATIO  7.6 7.2 8.3 8.2 8.3 9.1  CANTED BACKWARD  0.37 0.39 0.39 0.38 0.36  13.4 11.7 10.7 10.2 9.4  7.7 8.8 8.5 8.6 9.2  N o t e t h a t h e r e t h e a n g l e q u o t e d i s t h e a n g l e o f yaw a n d n o t t h e of attack. The a n g l e o f a t t a c k i s c o n s t a n t a t - 8 d e g r e e s .  angle  57  APPENDIX B METHOD OF  For purposes taken as h a l f  a monoplane  symmetric about  a . o  solution  aerofoil  i t s midpoint  The a s p e c t r a t i o w i l l used f o r  of  SOLUTION  the aluminum  so t h a t  having  a c o  a  _  4 s  =  2 x  2 A  . 287 x  and t h e v a l u e  3.14  _  ^f^-  being  span o f of  be  15 i n c h e s .  3.14 w i l l  be t a k e n  be  as:  287  5.46  3 14 ya  half  the c o n s t a n t terms w i l l  o  will  i s t r e a t e d as  an e f f e c t i v e  be t a k e n as 5 . 4 6  The v a l u e s o f  it  aerofoil  .03 x 6  03  2TT S  These v a l u e s t w i s t of  1°.  coefficients, coefficients for  these  =  =  ye  2  pv  2  correspond  =  .005  2 x 3.14  t o an a n g l e  x (y|)  of  As a f i r s t a p p r o x i m a t i o n A-| a n d A g , w i l l do n o t a p p e a r  values).  2  attack of  (2.4)  6°  2  =  110  w i t h an a n g l e  the values of  be c o m p u t e d  because the  1.94  the f i r s t  ( n o t e t h a t even  s i n e s e r i e s i s not  of  two  numbered symmetric  The f u n d a m e n t a l  £ A  s i n n8 (ny + s i n 8)  n  and t h i s w i l l  First  equation  be d i v i d e d  into  is:  u s i n 6 (a-e c o s 6 )  two p a r t s ,  each being  solved  separately.  part:  Z A  A  1  n  s i n n0  (ny + s i n 0)  s i n 6 (y + s i n  6) + A  =  e  . 7 0 7 A., ( . 5 7 5  A^.575  Solving  Second  =  to solve  + .707)  + 1)  -  +  =  3  ya s i n 6  s i n 36  45,  .707 A g O . 7 2 5  3  =  simultaneously  A  1  =  0 . 7 0 4 ya  =  0.0211  A  3  =  0 . 0 4 0 ya  =  0.0012  +  .707)  =  s i n n0 (ny + s i n  0)  =  .707  ya  for  A-j a n d A^  part:  E A  ya s i n 6  =  90  A ( l .725 + 1 )  t h e s e two e q u a t i o n s  (3y + s i n 6)  - y e cos 6 s i n 6  ya  59 A-j s i n e  (y + s i n e )  1  + 1.72  1.575 A  1  -  A  ]  = -.264  for  3  s i n 36 ( 3 y + s i n e )  A  2.725 A  = -  =  3  -  =  3  y e =  0.152  .5 y  A  1  =  .0211  A  3  =  .0012  Drag  e cose  sine  e  -0.00132  y e =  -0.00076  t h e c o n s t a n t s may be u s e d a s  =  ^  (.0211)  =  ^  [(.0211)  =  0.0245  For the t w i s t e d a e r o f o i l  = -y  0  the u n t w i s t e d a e r o f o i l  Lift  Induced  3  .907 A  A  So t h a t  + A  =  2  1.16  +  lb  3(.0012) ] 2  lb  the c o n s t a n t s  become:  A  1  =  .0211  -  .00132  =  .0198  A  0  =  .0012 -  .00076  =  .0005  follows:  60 Lift  Induced Drag  =  55 ( . 0 1 9 8 )  =  55 [ ( . 0 1 9 8 )  =  2  1.08 l b  + 3 (.0005) ] 2  =  .021  lb  61  APPENDIX C MOLDING AND CASTING  The m a t e r i a l B RTV M o l d m a k i n g of  Paris or  u s e d f o r m o l d m a k i n g was Dow C o r n i n g  Rubber,  t h i s was c h o s e n o v e r  F i b r e g l a s s because o f  of workability.  its  such t h i n g s as P l a s t e r  l o w l i n e a r s h r i n k a g e and e a s e  The m o l d f o r m s w e r e made f r o m p l e x i g l a s s , a n d  g e n e r a l l y a l l o w e d about  0.25  in-of  rubber  at  t h e maximum t h i c k n e s s .  t h e f i r s t a t t e m p t on t h e a e r o f o i l  model  and a 5 t o 1 r a t i o  a n d c a t a l y s t (Dow C o r n i n g  #1 ) u s e d .  between r u b b e r  The i n s i d e s u r f a c e o f  t h e l o w e r c l o s e d e n d a n d by wedges a t t h e u p p e r low rubber  t h e w o r k i n g t i m e t h a t n o t enough to penetrate  the rubber surfaces  tended to s t i c k  of In  the  quite  open e n d .  This  at  attempt  to c a t a l y s t r a t i o so a c c e l e r a t e d the  low v i s c o s i t y  to the mold r e l e a s e  that  coated  form.  the second attempt  t h e f o r m was s e t h o r i z o n t a l l y a n d a n  e n d p l a t e a t t a c h e d w i t h a s l o t so t h a t through  The  the form  i n . n e c k s , a l s o i t was f o u n d  readily  the  236 D i s p e r s i o n ) .  t i m e was a l l o w e d f o r  the narrow 0.25  Catalyst  t h e form and t h e s u r f a c e o f  model was h e l d i n p l a c e i n s i d e t h e f o r m by s c r e w s t h r o u g h  was u n s u c c e s s f u l b e c a u s e t h e  In  t h e f o r m was s e t v e r t i c a l l y  model w e r e c o a t e d w i t h m o l d r e l e a s e (Dow C o r n i n g  rubber  Silastic  the o r i g i n a l  and a t t a c h e d t o t h e o t h e r end o f  t h e f o r m was h a l f  full  of  the rubber.  model  c o u l d be  slipped  t h e f o r m by t h e s c r e w s o n c e The u p p e r  plate of  the  form  had been removed so t h a t f i l l i n g c o u l d t a k e p l a c e o v e r a l a r g e  area,  t h i s was r e p l a c e d when t h e model was i n p l a c e and t h e m o l d h a d  been  filled  to c a p a c i t y .  was a t h i n  film of  This attempt rubber  over  produced a v o i d but  t h e model  a n d a r e a s o n a b l e m o l d was p r o d u c e d . was u s e d on t h e p l e x i g l a s s f a c e s o f  In  since  there  t h i s c o u l d be f i l l e d  separately,  t h i s attempt  release  t h e f o r m , and i t  no m o l d appeared  was more s a t i s f a c t o r y t h a n when m o l d r e l e a s e h a d been u s e d . this  a t t e m p t a 20 t o 1 r a t i o o f  t h e w o r k i n g t i m e was more t h a n  rubber  In  this  successful  The r u b b e r  that  by means o f a s l i t  f r o m wooden m o d e l s  it  is  This  The o r i g i n a l  cut in  vertical  the rubber  important  plate  proved  quite  models were  in  removed  the long d i r e c t i o n .  removed q u i t e  readily.  t o o b t a i n a good  on t h e wood s u r f a c e , e i t h e r by p a i n t o r s h e l l a c , b e f o r e is  in  was a t t a c h e d t o a t o p  c o u l d t h e n be p e e l e d b a c k a n d t h e model  When m o l d i n g  the  t h e model was i n s e r t e d .  a n d no v o i d s w e r e o b s e r v e d .  from the molds  Also  doubled.  c a s e t h e wooden model  and t h e form f i l l e d b e f o r e  this  t o c a t a l y s t had been used so  The m o l d f r o m t h e w h a l e f l u k e was m a d e . i n position.  that  seal  the mold  release  applied. The m a t e r i a l  because  it  different  involved  It  flexibility  the m i x i n g o f a hardener  was f o u n d  for this  m o l d was e n c a s e d i n  degrees  project.  the m i x t u r e .  During  the  that  final  desired the it  rubber  the  t h e f o r m was t i l t e d a b o u t  to e l i m i n a t e the formation  model.  agent so  f l e x i b i l i t i e s of  p l e x i g l a s s form to g i v e  the pouring  The e p o x y model  the o r i g i n a l  and b o n d i n g  For the c a s t i n g o p e r a t i o n  its original  rigidity.  t h i s was c h o s e n  t h a t a 1 to 1 r a t i o produced the  from the v e r t i c a l  t o remove  c a s t i n g was 5V E p o x y ,  mixtures would produce d i f f e r e n t  product.  necessary  used f o r  was r e m o v e d  by o p e n i n g  of bubbles  in  the s l i t  used  30  

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