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Prediction method for spoiler performance Tam Doo, Peter A. 1977

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A PREDICTION METHOD FOR SPOILER PERFORMANCE by PETER A.jjTAM DOO B.Sc,  University of Manitoba, 1972  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1977 (c) Peter A. Tarn 0 0 0 , 1 9 7 7  In presenting this thesis in partial  fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for  reference and study.  I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of  Mechanical  Engineering  The University of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  2Rth  Orl-nW  1Q77  ABSTRACT  The p r e d i c t i o n  o f the  aerodynamic  c h a r a c t e r i s t i c s of a f i n i t e  r e c t a n g u l a r wing w i t h p a r t span s p o i l e r s i s attempted u s i n g t h e l i f t i n g line  theory o f P r a n d t l .  values of l i f t of  Required  i n p u t s t o the  theory a r e s e c t i o n a l  c o e f f i c i e n t , p i t c h i n g moment c o e f f i c i e n t , zero l i f t  angle  a t t a c k , and aerodynamic c e n t e r a t s e l e c t e d p o i n t s along t h e span. The  v a l u e o f these parameters f o r t h e s p o i l e r e d wing s e c t i o n s i s c a l c u l a t e d by Brown's l i n e a r i s e d t h i n a i r f o i l  theory f o r s p o i l e r s . This theory, i n  common w i t h o t h e r s e c t i o n a l s p o i l e r t h e o r i e s , r e q u i r e s as i n p u t the base p r e s s u r e c o e f f i c i e n t i n the s p o i l e r wake. The base p r e s s u r e must be determined predicted The  by experiment, s i n c e a t the p r e s e n t time i t cannot be  theoretically. e f f e c t o f base v e n t i n g on s p o i l e r e d  examined e x p e r i m e n t a l l y . I t i s found ten are  coefficient  section characteristics i s  t h a t f o r s m a l l base v e n t s o f around  p e r c e n t o f s p o i l e r h e i g h t o r l e s s , t h e vented little  d i f f e r e n t from  t h e unvented.  p r e l i m i n a r y d e s i g n , t h e unvented with l i t t l e  l o s s of accuracy,  section characteristics  Thus f o r the  purposes o f  s e c t i o n c h a r a c t e r i s t i c s may be used  i f the s p o i l e r v e n t i s about t e n p e r c e n t o f  s p o i l e r height or l e s s . The r e s u l t s o f the f i n i t e wing t h e o r y a r e compared w i t h experiment. Good agreement i s found. The method i s s u b j e c t t o t h e l i m i t a t i o n s o f the lifting  l i n e t h e o r y , which l i m i t s i t s a p p l i c a b i l i t y t o unswept wings o f  moderate t o h i g h a s p e c t r a t i o s o p e r a t i n g a t low subsonic method i s a l s o s u b j e c t t o  the a d d i t i o n a l l i m i t a t i o n s  s e c t i o n a l t h e o r i e s employed.  speeds.  imposed  The by the  iii  TABLE OF  CONTENTS  Chapter  Page  1  INTRODUCTION  1  2  THEORY  4  2.1.1  The L i f t i n g Line Theory  4  2.1.2  The Jones Edge Correction Factor  8  2.2  A p p l i c a t i o n to Wings with Spoilers  10  2.3  Brown's Thin A i r f o i l Theory f o r Spoilers  12  2.3.1  The Acceleration P o t e n t i a l  14  2.3.2  Conformal Transformations  2.3.3  Boundary Conditions  17  2.3.4  Flow Model  18  2.3.5  Base Vented Spoilers  22  2.4  Experimental Two Dimensional Base Pressures  2.5  Experimental F i n i t e Span Base Pressures  2.6  Empirical Relationships for Base Pressures  2.6.1  C , Averaged across Span pb  2.6.2  V a r i a t i o n of Sectional Properties with  . .  15  ..  23 25  o ...  Base Pressure 2.6.3  27  31  Use of Experimental Two Dimensional Spoilered Section Parameters  3  27  i n F i n i t e Wing Theory  33  EXPERIMENTS  35  3.1.1  Base Venting Experiments  35  3.2  F i n i t e Wing Experiments  36  3.3  Base Pressure Measurements  37  iv  Chapter  Page 3.4  4  Wind Tunnel W a l l C o r r e c t i o n s  38  RESULTS AND COMPARISONS 4.1  Base V e n t i n g Experiments  4.2.1  Two D i m e n s i o n a l NACA 0015 A i r f o i l  40  Experiments 4.2.2  40  R e c t a n g u l a r wings o f NACA 0015 S e c t i o n F i t t e d w i t h P a r t Span S p o i l e r s  5  41  4.3.1  Two D i m e n s i o n a l Base P r e s s u r e Experiments  4.3.2  Three D i m e n s i o n a l Base P r e s s u r e Experiments  CONCLUSIONS  .... ..  44 44 46  FIGURES  48  REFERENCES  98  APPENDIX  99  V  LIST  OF  TABLES  Table I  Page Values o f C , pb  f o r two d i m e n s i o n a l C l a r k Y and  NACA 0015 A i r f o i l s w i t h Normal Unvented S p o i l e r s II  Base P r e s s u r e C o e f f i c i e n t s f o r S p o i l e r e d  NACA  0015 & 12.9% C l a r k Y A i r f o i l s III  Sectional  24  '.  2  g  C h a r a c t e r i s t i c s f o r a NACA 0 0 1 5 . A i r f o i l  F i t t e d w i t h 9.7% Unvented, Normal S p o i l e r s  i  31  vi LIST  OF  FIGURES  Figure  Page  1  F i n i t e Wing w i t h S p o i l e r and I t s V o r t e x Model  48  2  Airfoil  49  3  Complex T r a n s f o r m P l a n e s  50  4  V a r i a t i o n of m w i t h C , f o r a NACA 0015 A i r f o i l os pb S e c t i o n w i t h Normal Unvented S p o i l e r  51  5  Variation of C & x /c w i t h C , f o r NACA 0015 macs acs pb Airfoil  6  i n the P h y s i c a l P l a n e  S e c t i o n w i t h Normal Unvented S p o i l e r  V a r i a t i o n o f a, los  w i t h C , f o r NACA 0015 A i r f o i l pb  S e c t i o n w i t h Normal Unvented S p o i l e r 7  Modified  Joukowsky A i r f o i l  Section  53  o f 11% T h i c k n e s s  & 2.4% Camber w i t h Base Vented Normal S p o i l e r 8  NACA 0015 A i r f o i l  55  12.9% T h i c k C l a r k Y A i r f o i l  S e c t i o n w i t h 10%  Unvented Normal S p o i l e r  56  10  Base P r e s s u r e Measurement  11  L i f t C o e f f i c i e n t f o r Joukowsky A i r f o i l  System  57 Section  w i t h Base Vented Normal S p o i l e r s 12  L i f t C o e f f i c i e n t f o r Joukowsky A i r f o i l  58 Section  w i t h Base Vented Normal S p o i l e r s 13  P i t c h i n g Moment C o e f f i c i e n t f o r Joukowsky Section  14  Lift  54  S e c t i o n w i t h 9.7% Unvented Normal  Spoiler 9  52  59 Airfoil  w i t h Base Vented Normal S p o i l e r s  60  & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015  Section  61  vii  Figure 15  Page Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015  Airfoil 16  Lift  Lift  w i t h Normal Unvented S p o i l e r  Section  w i t h Normal Unvented S p o i l e r  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n 18  Lift  Lift  20  Lift  21  Half  w i t h Normal Unvented S p o i l e r  w i t h Normal Unvented S p o i l e r  68  w i t h Normal Unvented S p o i l e r  70  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f  Lift  w i t h Normal Unvented S p o i l e r  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n 26  69  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f  NACA 0015 S e c t i o n 25  67  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f  NACA 0015 S e c t i o n 24  66  Half  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 23  65  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f NACA 0015 S e c t i o n  22  64  Half  w i t h Normal Unvented S p o i l e r  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n  Half  w i t h Normal Unvented S p o i l e r  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n  63  w i t h Normal Unvented S p o i l e r ,  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n 19  62  & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015  Airfoil 17  Section  Lift  Half  w i t h Normal Unvented S p o i l e r  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  Wing o f NACA 0015 S e c t i o n  71  72  Half  w i t h Normal Unvented S p o i l e r  73  viii  Figure 27  Page Lift  & P i t c h i n g Moment C o e f f i c i e n t s  f o r Rectangular H a l f  Wing o f NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 28  Lift  & Pitching  Moment C o e f f i c i e n t s  74  f o r Rectangular H a l f  Wing o f NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 29  R o l l i n g Moment C o e f f i c i e n t  f o r R e c t a n g u l a r H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 30  R o l l i n g Moment C o e f f i c i e n t  75  ,  f o r R e c t a n g u l a r H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 31  R o l l i n g Moment C o e f f i c i e n t  E f f e c t i v e Moment Arm o f I n c r e m e n t a l L i f t Unvented S p o i l e r  34  Due t o Normal  on R e c t a n g u l a r Wing o f NACA 0015 S e c t i o n  Spanwise L i f t  on R e c t a n g u l a r Wing o f NACA 0015 S e c t i o n  Spanwise L i f t  Spanwise L i f t  D i s t r i b u t i o n f o r R e c t a n g u l a r Wings w i t h  D i s t r i b u t i o n f o r Rectangular  83 Wings  o f NACA 0015 S e c t i o n 38  39  Spanwise L i f t  81  82  S y m m e t r i c a l l y Deployed S p o i l e r s 37  80  D i s t r i b u t i o n f o r R e c t a n g u l a r Wings w i t h  S y m m e t r i c a l l y Deployed S p o i l e r s 36  79  E f f e c t i v e Moment Arm o f I n c r e m e n t a l L i f t Due t o Normal Unvented S p o i l e r  35  78  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r  33  77  f o r R e c t a n g u l a r H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r 32  76  D i s t r i b u t i o n f o r Rectangular  84 Wings  o f NACA 0015 S e c t i o n  85  C , D i s t r i b u t i o n f o r NACA 0015 S e c t i o n w i t h 9.7% pb Normal Unvented S p o i l e r  86  ix  Figure 40  41  42  Page C  D i s t r i b u t i o n f o r a 12.9% T h i c k C l a r k Y S e c t i o n  w i t h 10% Normal Unvented S p o i l e r  87  V a r i a t i o n o f C , w i t h S p o i l e r P o s i t i o n A l o n g Chord pb f o r R e c t a n g u l a r Wings w i t h S p o i l e r s  88  V a r i a t i o n o f C , w i t h S p o i l e r Span f o r R e c t a n g u l a r Wings pb with Spoilers  43  -  V a r i a t i o n of C , with Aspect Ratio f o r Rectangular pb Wings w i t h S p o i l e r s  44  90  V a r i a t i o n o f C , w i t h S p o i l e r P o s i t i o n Along Chord pb • - • f o r R e c t a n g u l a r Wings w i t h S p o i l e r s  45  92  V a r i a t i o n of C ^ with Aspect Ratio f o r Rectangular Wings w i t h S p o i l e r s  47  V a r i a t i o n of C ^  93  w i t h S p o i l e r Span f o r R e c t a n g u l a r  Wing w i t h Normal Unvented S p o i l e r s 48  50  94  V a r i a t i o n o f C , w i t h S p o i l e r Span f o r R e c t a n g u l a r pb Wings w i t h Normal Unvented S p o i l e r s  49  91  V a r i a t i o n o f C , w i t h S p o i l e r Span f o r R e c t a n g u l a r pb Wings w i t h S p o i l e r s  46  89  95  V a r i a t i o n o f C ^ w i t h S p o i l e r Span f o r R e c t a n g u l a r Wings w i t h Normal Unvented S p o i l e r s  96  V a r i a t i o n o f C_  97  t  / C*  u  and G/G* w i t h s/c  ACKNOWLEDGEMENTS  The author wishes t o thank Dr. G.V. P a r k i n s o n f o r h i s guidance i n the of  p r e p a r a t i o n o f t h i s t h e s i s . H i s many s u g g e s t i o n s d u r i n g t h e c o u r s e t h i s r e s e a r c h were i n v a l u a b l e and much a p p r e c i a t e d . The author a l s o wishes t o thank t h e M e c h a n i c a l E n g i n e e r i n g Machine  Shop f o r b u i l d i n g t h e models and the Computing Center a t t h e U n i v e r s i t y of  B r i t i s h Columbia  f o r the use o f t h e i r  facilities.  T h i s r e s e a r c h was supported i n i t i a l l y by t h e Defense Research Board under Grant Number 9551-13 and by t h e N a t i o n a l Research C o u n c i l Grant Number A586.  under  xi  LIST  a  x  =  IJx  ~dz  a  z  =  3z  ~  =  =  9x Bx  x  c  o  m  P  Z  c  o  m  P °  o  n  e  n  e  OF  SYMBOLS  n  t  °* a c c e l e r a t i o n  n  t  °f a c c e l e r a t i o n  acceleration vector A  Fourier  n  coefficient  2 AR = 2b /S  a s p e c t r a t i o o f wing  b  wing  semispan  b  g  spoiler  span  b  t  coordinate along s p o i l e r  span measured w i t h r e s p e c t t o  inboard s p o i l e r t i p c c c  wing chord r o o t semichord o f e l l i p t i c  e  r  flap  n  disk  chord  spoilered  sectional  drag  unspoilered sectional C, Is  spoilered  C mo  unspoilered pitching  C  m o g  spoilered  C  c  r  sectional  pitching  coefficient  lift  lift  coefficient coefficient  moment c o e f f i c i e n t  moment c o e f f i c i e n t  unspoilered pitching  moment c o e f f i c i e n t  about  about  origin  origin  about t h e aerodynamic  center C  spoilered  pitching  moment c o e f f i c i e n t  about t h e aerodynamic  center C  h a l f wing l i f t  L  ^MAC  coefficient  h a l f wing p i t c h i n g  moment c o e f f i c i e n t  about  h a l f wing p i t c h i n g  moment c o e f f i c i e n t  about t h e aerodynamic  center Cp  g  pressure c o e f f i c i e n t  f o r spoilered  section  origin  xii  CL R  h a l f wing r o l l i n g moment c o e f f i c i e n t  C , pb  base p r e s s u r e c o e f f i c i e n t i n s p o i l e r wake  C , pb  base p r e s s u r e c o e f f i c i e n t i n s p o i l e r wake averaged over incidence  C , pb  base p r e s s u r e c o e f f i c i e n t i n s p o i l e r wake averaged over incidence  and s p o i l e r span  =* C ' pb  = Value o f C . pb  d  h e i g h t o f s p o i l e r base v e n t  E  semiperimeter o f wing/semispan, Jones Edge C o r r e c t i o n  F(Z)=  (J) + i ^  a t t h e 0.7 chord  complex a c c e l e r a t i o n  position  Factor  potential  G  * G  v a l u e o f G a t t h e 0.7 chord  h k  s p o i l e r height chordwise c o o r d i n a t e o f e l l i p t i c  k  g  chordwise c o o r d i n a t e o f  position  edge o f e l l i p t i c  K = -C , pb  c a v i t a t i o n number  L'  sectional l i f t  f o r unspoilered  1/  sectional l i f t  f o r spoilered  Z  cavity  m  unspoilered  Q  m  os  unspoilered  M  Q S  spoilered  M  9.C S  section  section  dC^/da (/rad.) f o r a i r f o i l  s p o i l e r e d dC-/da ±  Q  ac  disk  length  M  M  disk  unspoilered spoilered  ( /rad.) f o r a i r f o i l  section section  s e c t i o n a l p i t c h i n g moment about o r i g i n s e c t i o n a l p i t c h i n g moment about o r i g i n s e c t i o n a l p i t c h i n g moment about aerodynamic c e n t e r s e c t i o n a l p i t c h i n g moment about aerodynamic c e n t e r  h a l f wing p i t c h i n g  moment about o r i g i n  h a l f wing p i t c h i n g moment about aerodynamic chordwise c o o r d i n a t e o f f l a p pressure  i n flow  pressure  i n s p o i l e r wake  center  hinge  field  f r e e stream p r e s s u r e r o l l i n g moment o f h a l f wing chordwise c o o r d i n a t e o f s p o i l e r  base  h a l f wing a r e a chordwise c o o r d i n a t e o f free  spoiler t i p  stream v e l o c i t y  non-dimensional p e r t u r b a t i o n v e l o c i t y i n x d i r e c t i o n non-dimensional p e r t u r b a t i o n v e l o c i t y i n z d i r e c t i o n downwash chordwise  coordinate  chordwise c o o r d i n a t e f o r aerodynamic c e n t e r o f u n s p o i l e r e d section chordwise c o o r d i n a t e f o r aerodynamic c e n t e r o f s p o i l e r e d section chordwise c o o r d i n a t e o f wing aerodynamic c e n t e r spanwise c o o r d i n a t e spanwise p o s i t i o n  a t which t h e induced  angle o f a t t a c k i s  evaluated spanwise c o o r d i n a t e o f i n b o a r d spanwise c o o r d i n a t e o f outboard  spoiler t i p spoiler t i p  c o o r d i n a t e p e r p e n d i c u l a r t o xy p l a n e  z'  complex  Z = x + iz  complex a i r f o i l  a  g e o m e t r i c a n g l e of a t t a c k f o r a i r f o i l  a  a  transform plane plane  = a + a, lo  a b s o l u t e a n g l e of a t t a c k  f o r unspoilered  = a + a..  absolute angle of attack  for spoilered a i r f o i l  a 3.S  e f f e c t i v e angle of attack induced a n g l e o f  lo  a  airfoil  section  section  XO S  cx = a + a . e a 1  a^  section  o g  stall  for a i r f o i l  section  attack  zero l i f t  angle of attack  for unspoilered  zero l i f t  angle of attack  for spoilered  stall  a n g l e f o r wing or a i r f o i l  6  spoiler erection  £, v  complex  ri  flap  8  a n g u l a r v a r i a b l e i n £-plane  u=cos ^(y/b)  span v a r i a b l e f o r f i n i t e  section section  section  angle  transform planes  angle  p o s i t i o n of inner  wing  s p o i l e r t i p i n terms o f span v a r i a b l e y  V<2  p o s i t i o n of o u t e r s p o i l e r t i p i n terms of span v a r i a b l e u  p  a i r density  <|>  acceleration  potential  <J>  velocity potential for e l l i p t i c  ip  acceleration  e  T=cos \ k / k T  e  )  stream  function  chordwise v a r i a b l e f o r e l l i p t i c circulation  disk  disk  1  1.  INTRODUCTION  S p o i l e r s a r e v e r s a t i l e aerodynamic c o n t r o l s u r f a c e s w h i c h a r e used on many modern a i r c r a f t . They may b e s t be d e s c r i b e d a s s m a l l f l a p s t h a t have been moved ahead o f t h e t r a i l i n g surface.  S p o i l e r s may be d e p l o y e d  o r a s y m m e t r i c a l l y t o produce  edge t o t h e upper o r l o w e r wing  s y m m e t r i c a l l y t o c o n t r o l l i f t and d r a g ,  r o l l and yaw.  S i n c e s p o i l e r s may  t o g e t h e r w i t h f u l l span f l a p s , t h e i r p r e s e n c e  be used  i n V/STOL a i r c r a f t  is  becoming i n c r e a s i n g l y common. The b e h a v i o r o f s p o i l e r s on w i n g s i s , however, d i f f i c u l t because f l o w s e p a r a t e s f r o m t h e s p o i l e r edges, and  to predict  a turbulent  wake i s  formed b e h i n d t h e s p o i l e r . The s t a n d a r d p o t e n t i a l f l o w methods o f a i r f o i l t h e o r y cannot t h e r e f o r e be u s e d . the presence  o f t h e wake.  p r o b l e m . He d e v e l o p e d  m o d i f i e d t o account  Woods (1) was among t h e f i r s t  for  to tackle the  a l i n e a r i s e d t h i n a i r f o i l theory f o r s p o i l e r e d  w i n g s e c t i o n s . B a r n e s (2) presence  They must be  l a t e r m o d i f i e d t h e t h e o r y t o account  f o r the  o f t h e boundary l a y e r on t h e a i r f o i l . Here a t t h e U n i v e r s i t y o f  B r i t i s h C o l u m b i a , work on s p o i l e r t h e o r y was begun by J a n d a l i and Brown i n an e f f o r t t o improve on t h e a c c u r a c y o f p r e d i c t i o n . J a n d a l i an a n a l y t i c t h i c k a i r f o i l t h e o r y s p o i l e r s (3). Brown developed  f o r a i r f o i l sections  a t h i n a i r f o i l theory,  developed  w i t h normal  and a  numerical  t h i c k a i r f o i l t h e o r y (4). A l l o f t h e above t h e o r i e s a p p l y f o r s p o i l e r e d a i r f o i l s w i t h wakes w h i c h do n o t r e a t t a c h t o t h e a i r f o i l s u r f a c e . The p r e s e n t work extends  the s p o i l e r theory into three  dimensions.  S i n c e l o w speed a p p l i c a t i o n s a r e o f p r i n c i p a l i n t e r e s t i n Canada, s i n c e t h e s e c t i o n a l t h e o r i e s developed  and  by J a n d a l i and Brown a r e a p p l i c a b l e  2  to  i n c o m p r e s s i b l e f l o w , i t was d e c i d e d t h a t i t would be a p p r o p i a t e t o  extend t h e t h e o r y i n t o t h r e e dimensions  by means o f P r a n d t l ' s l i f t i n g  l i n e t h e o r y ( 5 ) . The l i f t i n g l i n e t h e o r y o v e r p r e d i c t s t h e l i f t f o r s m a l l e r a s p e c t r a t i o s . J o n e s (6) has proposed  a modification to the  l i f t i n g l i n e t h e o r y w h i c h c o r r e c t s t h e o v e r p r e d i c t i o n and t h i s i s i n c o r p orated into the theory.  The l i n e a r i s e d t h i n a i r f o i l t h e o r y o f Brown i s  used t o c a l c u l a t e t h e s p o i l e r e d s e c t i o n parameters  which a r e r e q u i r e d as  i n p u t t o t h e l i f t i n g l i n e t h e o r y . Other t h e o r i e s , such a s t h o s e o f Woods or  J a n d a l i , o r Brown's n u m e r i c a l t h i c k a i r f o i l t h e o r y , may o f c o u r s e be  used. If  t h e s e c t i o n a l parameters  pressure inputs  d e r i v e d from  two d i m e n s i o n a l  base  t o t h e s p o i l e r t h e o r i e s a r e used i n t h e l i f t i n g  line  t h e o r y , e r r o r s w i l l r e s u l t because t h e t h r e e d i m e n s i o n a l base p r e s s u r e s d i f f e r from t h e c o r r e s p o n d i n g two d i m e n s i o n a l v a l u e s .  F l o w around t h e  v e r t i c a l edges o f t h e f i n i t e span s p o i l e r r e d u c e s t h e n e g a t i v e base p r e s s u r e and c r e a t e s spanwise g r a d i e n t s o f base p r e s s u r e w h i c h a r e absent from two d i m e n s i o n a l f l o w s . S i n c e s p o i l e r e d s e c t i o n c h a r a c t e r i s t i c s a r e a f u n c t i o n o f base p r e s s u r e c o e f f i c i e n t s , i t i s c l e a r t h a t t h r e e d i m e n s i o n a l b a s e p r e s s u r e - c o e f f i c i e n t s must be used a s i n p u t t o t h e s e c t i o n a l t h e o r i e s i n o r d e r t o o b t a i n s e c t i o n a l parameters to  appropiate  f i n i t e span f l o w s . A l l o f t h e above mentioned s p o i l e r t h e o r i e s  a r e developed  for  unvented s p o i l e r s . W h i l e t h e p o s s i b i l i t y e x i s t s t h a t some o f t h e t h e o r i e s may be m o d i f i e d t o t a k e i n t o account course of a c t i o n i s n o t attempted  t h e e f f e c t o f base v e n t s , t h i s  i n the present t h e s i s . Instead  the  e f f e c t o f base v e n t i n g i s examined e x p e r i m e n t a l l y . I t i s found t h a t f o r s m a l l base v e n t s o f about t e n p e r c e n t s p o i l e r h e i g h t o r l e s s , t h e v e n t e d  s p o i l e r behavior i s l i t t l e d i f f e r e n t  from the unvented. Thus f o r the  purposes o f p r e l i m i n a r y d e s i g n , t h e unvented s p o i l e r may  characteristics  be used f o r s p o i l e r s w i t h base v e n t s o f about t e n p e r c e n t o r l e s s .  The r e s t r i c t i o n t o base v e n t s o f  about t e n p e r c e n t or l e s s i s  serious l i m i t a t i o n , since i n practice, size.  most base v e n t s a r e about  not a this  2.  2.1.1  THEORY  The L i f t i n g L i n e Theory The  l i n e a r i s e d l i f t i n g l i n e t h e o r y as f o r m u l a t e d by P r a n d t l , i s  applicable  t o unswept w i n g s o f moderate t o l a r g e a s p e c t r a t i o o p e r a t i n g  a t low mach numbers. The wing  i s placed i n  c o o r d i n a t e system as shown i n F i g u r e 1. located  a t the aerodynamic c e n t e r  a r i g h t handed  The o r i g i n o f  orthogonal  t h e system i s  o f the w i n g r o o t s e c t i o n . The  free  s t r e a m v e l o c i t y i s i n the p o s i t i v e x d i r e c t i o n . The wing i s m o d e l l e d a l i f t i n g l i n e o f bound v o r t i c e s l o c a t e d  on the y a x i s ,  c o v e r i n g the  span o f t h e w i n g , and a system o f t r a i l i n g v o r t i c e s i n the p l a n e o f f r e e s t r e a m v e l o c i t y . The  as  the  t r a i l i n g v o r t i c e s i n d u c e a downward v e l o c i t y  o v e r t h e w i n g , c a l l e d t h e downwash w^,  which a l t e r s the d i r e c t i o n of  the o n s e t f l o w , and t h u s r e d u c e s the e f f e c t i v e s e c t i o n a l a n g l e o f a t t a c k by an amount known as t h e i n d u c e d a n g l e o f a t t a c k a^. The  sectional  lift  c o e f f i c i e n t i s t h e r e f o r e g i v e n by  C  l  =  m  o  (  a  a  +  a  i > "  TTT^pU  The  sectional l i f t  L  is still  c  g i v e n by t h e K u t t a - J o u k o w s k y  = pOT  S u b s t i t u t i o n of equation  Law,  (2)  (2) i n t o e q u a t i o n  (1) g i v e s  5  C  1  = ~ = m ( a + a.) Uc o a 1  In t h i s equation,  (3)  o n l y t h e c i r c u l a t i o n i s unknown, s i n c e t h e i n d u c e d a n g l e  o f a t t a c k may be e x p r e s s e d i n terms o f t h e c i r c u l a t i o n by t h e B i o t - S a v a r t Law. Thus  w  i  f  1  dr/dy -b  Glauert  J  o  J  (7) has shown t h a t t h e c i r c u l a t i o n o f a f i n i t e wing may. be  e x p r e s s e d i n terms o f -the F o u r i e r  r = 4bU  £  series  S i n ny  A  (5)  n=l  where U i s t h e f r e e stream v e l o c i t y , b t h e wing semi-span, and  yis  d e f i n e d i n terms o f t h e spanwise c o o r d i n a t e by  (6)  y = b Cos y  S u b s t i t u t i o n of equations equation  (5) and (4) i n t o  f o r t h e unknown c o e f f i c i e n t s A  00  A ,n n=l E  (3) l e a d s t o t h e fundamental  R  QU  S i n ny [  mc o  Siny  + n ] = a Siny a  (7)  6  where c, m  o  and a  a  a r e the s e c t i o n a l c h o r d , l i f t '  angle o f a t t a c k . The parameters c, m  Q  and  curve s l o p e and a b s o l u t e * .  may v a r y a l o n g the span,  depending on whether o r n o t t h e wing has s e c t i o n changes, t a p e r and t w i s t . The  e q u a t i o n must be s a t i s f i e d  f o r a l l p o i n t s between 0 and TT f o r y .  S o l u t i o n f o r t h e unknown A ' s i s a c h i e v e d by c o n s i d e r i n g a f i n i t e n  number o f terms o f the F o u r i e r s e r i e s , say m terms. number o f span v a r i a b l e s y ,  equal to the  By c h o o s i n g the  number o f terms  t r u n c a t e d F o u r i e r s e r i e s , a system o f m e q u a t i o n s  i n the  i n m unknowns i s ob-  t a i n e d , and hence a s o l u t i o n f o r the unknown A ' s . S i n c e o n l y m terms o f n the F o u r i e r s e r i e s a r e c o n s i d e r e d , e q u a t i o n p o i n t s a l o n g t h e span. W i e s e l s b e r g e r  (7) i s s a t i s f i e d  (8) has shown t h a t i t i s n e c e s s a r y  to i n c l u d e t h e wing t i p s i n c a l c u l a t i o n s f o r f l a p p e d wings. equation  f o r only m  Since  (7) i s degenerate a t t h e t i p s where y i s e q u a l t o 0 o r TT,  L ' H o s p i t a l ' s r u l e must be a p p l i e d t o o b t a i n t h e r e s u l t  a  a  (0) = E , n=l  n A 2  n  (8) a  oo  a  (ir) = E , n=l  n A  n  (-l)  n + 1  Once t h e s o l u t i o n f o r t h e unknown F o u r i e r c o e f f i c i e n t s i s found, t h e lift  L, r o l l i n g moment R, and t h e p i t c h i n g moment about t h e o r i g i n ,  may be computed. S i n c e  dL = L'dy = pUT(y) dy  7  dM  0  = ( -L* x  ac  + M  ac  (9) '  ) dy J  dR = L*y dy  L,R, and Mg a r e o b t a i n e d by i n t e g r a t i n g a c r o s s t h e span. The p i t c h i n g moment about t h e o r i g i n M aerodynamic c e n t e r M  M  0  i s r e l a t e d t o t h e p i t c h i n g moment about t h e  Q  by  " AC AC  =  LX  +  <>  M  10  S i n c e t h e r e s u l t s a r e t o be compared w i t h r e f l e c t i o n p l a n e t h e i n t e g r a t i o n s a r e made o v e r t h e semi-span.  experiments,  The r e s u l t s may be e x p r e s s e d  i n c o e f f i c i e n t form by means o f t h e f o r m u l a e  C  =  L  % u s  L  2  P  M C  MAC  c R  -  =  AC  T %pU Sc  < > n  R  %pU Sb 2  where S i s t h e p l a n f o r m a r e a o f t h e h a l f w i n g . When e q u a t i o n s (9) a r e i n t e g r a t e d and s u b s t i t u t e d are obtained:  i n t o equations (11), the following  results  8  L  R  S  A  (2i-1)  (2i-3)(2i+l)  U  J  ;  3 C n MO M  =  \  2  f  «= C mac 2  J  TT/2 x o  In e q u a t i o n  2.1.2  The  (14),. a  ac  m E . i=l n  dy -  2  Cos  a  r  A , „ . S i n ( 2 i - l ) u Sinu dy (2i-l)  (14)  1 N  i s the a n g l e of a t t a c k of the wing r o o t s e c t i o n .  Jones Edge C o r r e c t i o n F a c t o r  Jones (6) has proposed a c o r r e c t i o n f o r the l i f t i n g which i s known to o v e r p r e d i c t l i f t  line  theory,  f o r the s m a l l e r a s p e c t r a t i o s . In wing  s e c t i o n t h e o r y the K u t t a c o n d i t i o n , which determines c i r c u l a t i o n and lift,  depends on the edge v e l o c i t y induced  hence  by the r e l a t i v e normal v e l o c i t y  of  the s e c t i o n . Jones has  of  an e l l i p t i c  of  the c o o r d i n a t e system, and moving w i t h u n i t v e l o c i t y i n z d i r e c t i o n , i s  given  shown t h a t the v e l o c i t y p o t e n t i a l on the s u r f a c e  d i s k , l y i n g i n the xy p l a n e w i t h i t s c e n t e r at the  by  k (|> = '-f e  Sin T  origin  where T i s the c h o r d w i s e v a r i a b l e g i v e n by semi-perimeter  T = cos  -1  ( k / k ) , E i s the' g  o f the d i s k d i v i d e d by the semi-span, k i s the  c o o r d i n a t e , and  k^  the chordwise c o o r d i n a t e o f the edge of the  2  chordwise elliptic  2  d i s k , the v a l u e o f which i s g i v e n by k^ = c ( 1 - y /b g  ) . Here y i s the  spanwise c o o r d i n a t e , b the l e n g t h of the semi-major a x i s (semi-span) and c^ the l e n g t h o f t h e semi-minor a x i s ( r o o t semi-chord), disk.  elliptic  . F o r an i n f i n i t e  for  of the  the i n f i n i t e  and  d i s k , E = 1. finite  Thus the r a t i o s o f the edge v e l o c i t i e s  e l l i p t i c d i s k s i s 1/E.  the Jones Edge C o r r e c t i o n F a c t o r . I t i s an exact wings, but  i s approximate f o r o t h e r p l a n f o r m s .  be seen t h a t the s e c t i o n a l v a l u e s o f l i f t by a f a c t o r of 1/E S e c t i o n 2.2.1  i n three dimensional  must be m o d i f i e d  and  The  factor E i s called  correction for e l l i p t i c  From t h i s c o r r e c t i o n i t can c i r c u l a t i o n must be  f l o w s . The  reduced  v a r i o u s equations  a c c o r d i n g l y . In p a r t i c u l a r e q u a t i o n  in  (7)  must be r e w r i t t e n a s ,  z  oo  n=l  A  n  S i n ny  [  8bE m c o  Siny + n ] = a  Siny  (7a)  10  2.2  A p p l i c a t i o n to Wings w i t h S p o i l e r s Experimental  and t h e o r e t i c a l i n v e s t i g a t i o n s o f two d i m e n s i o n a l  f o i l s with s p o i l e r s  ( 2 ) , have shown t h a t t h e e f f e c t o f the s p o i l e r on the  s e c t i o n a l c h a r a c t e r i s t i c s i s to a l t e r  curve s l o p e m , t h e zero o a n g l e o f a t t a c k ^ » the aerodynamic c e n t e r ^ > and the p i t c h i n g  lift  the l i f t  a  Q  a Q  moment about the aerodynamic c e n t e r , M is m  related  to t h e geometric  and a, be t h e l i f t los •  os  air-  ( The a b s o l u t e a n g l e o f a t t a c k  ac  a n g l e o f a t t a c k by ( a  c u r v e s l o p e and zero l i f t K  s p o i l e r e d s e c t i o n s . Then i n a p p l y i n g e q u a t i o n span s p o i l e r , a  = a - ^ ))» l a  &  e  t  0  angle o f a t t a c k o f t h e &  (7a) t o a wing w i t h a p a r t  , the absolute angle of a t t a c k of the s p o i l e r e d s e c t i o n 3.S  (a  = a - a  cLS  n  ) , and m  XOS  OS  must r e p l a c e a  3.  and m  over  O  s e c t i o n s . This r e s u l t s i n d i s c o n t i n u i t i e s i n the l i f t  the s p o i l e r e d wing curve s l o p e and  a n g l e o f a t t a c k d i s t r i b u t i o n s a c r o s s t h e wing. F o r an i n f i n i t e series,  Fourier  the p o s i t i o n s o f the d i s c o n t i n u i t i e s a r e e x a c t l y f i x e d , s i n c e a l l  v a l u e s o f t h e span v a r i a b l e y a r e covered  by the s e r i e s . F o r  F o u r i e r s e r i e s however, the v a l u e s o f the l i f t a t t a c k must change from one v a l u e t o another  a finite  curve s l o p e and angle o f  over two a d j a c e n t v a l u e s o f .  y. T h i s may be c o n s i d e r e d t o be a g r a d u a l change i n s e c t i o n a l l i f t slope of lie  and a n g l e o f a t t a c k over a f i n i t e range o f y , and i s an  curve  approximation  t h e r e a l s i t u a t i o n . The p o s i t i o n s o f each d i s c o n t i n u i t y may be made t o midway between two a d j a c e n t v a l u e s o f y . Once e q u a t i o n  may be found  (7a) i s s o l v e d , the v a r i o u s aerodynamic  by a p p l y i n g e q u a t i o n s  r e c t a n g u l a r wing o f c o n s t a n t  (12)  to (14) .  coefficients  F o r an u n t w i s t e d  s e c t i o n w i t h p a r t span s p o i l e r s ,  (14) may be f u r t h e r s i m p l i f i e d . L e t  C IUcLCS  and x  equation  be t h e p i t c h i n g  3.CS  moment c o e f f i c i e n t about t h e aerodynamic c e n t e r , and t h e aerodynamic  c e n t e r o f the s p o i l e r e d s e c t i o n s . C , the p i t c h i n g moment mac about the aerodynamic c e n t e r o f t h e u n s p o i l e r e d s i n c e the wing i s o f c o n s t a n t  section. Also, x  ac  s e c t i o n s , i s constant =0  wing over the u n s p o i l e r e d s e c t i o n s . Hence e q u a t i o n  m E i=l  where b  a  y  The  A  ^  e  obtained  f o r a rectangular  (14) reduces to  x S i n ( 2 i - l ) u S i n y dy  (15)  c  1  spoiler  of the i n n e r and o u t e r m  y  i s t h e s p o i l e r span, y  t h e i n n e r and o u t e r  SlAC  - 2 / y  coefficient  and y„ a r e t h e spanwise p o s i t i o n s o f  t i p s . . S i m i l a r l y y ^ and y ^ a r e t h e p o s i t i o n s  s p o i l e r t i p s i n terms o f the span v a r i a b l e y .  from C  by a p p l y i n g e q u a t i o n ( 1 0 ) .  s p o i l e r e d s e c t i o n parameters m  os  , a, , C , and x may be l o s macs acs  c a l c u l a t e d u s i n g any o f t h e p r e v i o u s l y mentioned t h e o r i e s f o r s p o i l e r s . In t h e p r e s e n t work, Brown's l i n e a r i s e d t h i n a i r f o i l (2) i s used. Brown has developed t h e t h e o r y (9) has extended t h e t h e o r y moment c o e f f i c i e n t .  theory  to predict l i f t  for spoilers only.  Bernier  t o i n c l u d e the p r e d i c t i o n o f the p i t c h i n g  2.3  Brown's Thin A i r f o i l Theory f o r Spoilers The a i r f o i l section of chord c, i s positioned i n the p h y s i c a l plane  as  shown i n Figure 2,  with i t s leading edge at  the o r i g i n . The free  stream v e l o c i t y U, i s i n the p o s i t i v e x d i r e c t i o n ,  and the a i r f o i l i s  i n c l i n e d at a small angle a to the free stream. The s p o i l e r , i n c l i n e d at an angle 6 to the chord, i s of height h. I t s base  i s located  at a  distance s from the leading edge. The a i r f o i l may also have a f l a p of chord c^ deflected at an angle T)  to the chord.  The wake behind  s p o i l e r i s modelled as a constant pressure cavity of f i n i t e  the  length. The  pressure i n the cavity may be defined i n terms of the base pressure c o e f f i c i e n t , which i s given by  where P  £  i s the pressure i n the s p o i l e r wake,  or i n terms of the  Cavitation Number, K given by  The a i r f o i l - c a v i t y combination  i s of t o t a l length %.  In the l i n e a r i s e d p h y s i c a l plane as shown i n Figure 3, the wetted a i r f o i l surface and cavity boundary occupies a s l i t on the p o s i t i v e x a x i s . The f i e l d i n the l i n e a r i s e d p h y s i c a l plane i s mapped conformally onto the upper half £-plane external to a unit semicircle centered at  the  o r i g i n by a s e r i e s o f t r a n s f o r m a t i o n s . The wetted a i r f o i l  itself  surface  i s mapped onto t h e u n i t s e m i c i r c l e , w h i l e the c a v i t y b o u n d a r i e s  occupy the r e a l a x i s e x t e r n a l t o the u n i t The f l o w model adopted i s s i m i l a r  semicircle.  to t h a t o f P a r k i n ( 1 0 ) , who has  extended the complex a c c e l e r a t i o n p o t e n t i a l method o f B i o t (11) t o s o l v e the  f o i l - c a v i t y problem.  As t h e t h e o r y i s l i n e a r i s e d ,  the v a r i o u s  geometric parameters o f t h e a i r f o i l may be c o n s i d e r e d s e p a r a t e l y and superimposed  t o g i v e t h e complete s o l u t i o n .  Complex  acceleration  p o t e n t i a l s s a t i s f y i n g t h e boundary c o n d i t i o n s imposed by t h e g e o m e t r i c parameters o f the a i r f o i l  a r e found i n the £-plane  and superposed t o  g i v e t h e complete s o l u t i o n f o r t h e a i r f o i l w i t h s p o i l e r . The s o l u t i o n in  t h e Z-plane i s found by matching c o r r e s p o n d i n g p o i n t s i n t h e Z and  ^ - p l a n e s . The a i r f o i l  c o e f f i c i e n t s may be o b t a i n e d by a p p l y i n g the  B l a s i u s Equations  (16)  The aerodynamic aerodynamic  c e n t e r and t h e p i t c h i n g moment c o e f f i c i e n t about t h e  c e n t e r a r e found u s i n g  C mos  -xacs c  macs  (17) dC dC  mos  1  -xacs c  2.3.1  The  Acceleration  Potential  Newton's second law  suggests the e x i s t e n c e  f o r an  incompressible  fluid  element  o f an a c c e l e r a t i o n p o t e n t i a l <}>, which has  the  property  Vcf) = a = -V  Biot  (9) has  -  shown t h a t the a c c e l e r a t i o n p o t e n t i a l f u n c t i o n i s harmonic,  so t h a t a c o n j u g a t e f u n c t i o n ip, the a c c e l e r a t i o n stream f u n c t i o n e x i s t s as w e l l as the complex a c c e l e r a t i o n p o t e n t i a l  F(Z)  = <j> +  Hp  For a s m a l l p e r t u r b a t i o n and  to the f r e e stream v e l o c i t y U,  Cauchy-Riemann e q u a t i o n s a r e r e l a t e d to <j> and  order d i f f e r e n t i a l  9t  the  ip by the l i n e a r  Euler first  equations,  +  5 u 8x  u  =  D  3 i 8x  (18) 9t  +  U  3x  U  9x  where u, w,  (f> and  flows, equations  a r e n o n - d i m e n sional  b u t n o t U, x, z , o r t . F o r steady  (18) may be i n t e g r a t e d t o g i v e  u =  c> j +  K/2 (19)  w = - \\>  where K i s t h e c a v i t a t i o n number. The c o n s t a n t s o f i n t e g r a t i o n a r e determined  by c h o o s i n g  c a v i t y boundaries,  and by t h e c o n d i t i o n s a t i n f i n i t y . The l i n e a r i s e d  pressure c o e f f i c i e n t  C  2.3.2  Conformal The a i r f o i l  The  s p o i l e r base  t h e c o n s t a n t v a l u e o f (j) t o be e q u a l t o zero on t h e  p  i s g i v e n by  = -2u = -2(f) - K  (20)  Transformations i n t h e l i n e a r i s e d p h y s i c a l Z-plane i s shown i n F i g u r e 3. and t i p a r e a t x = s and x = t r e s p e c t i v e l y .  h i n g e p o i n t i s a t x = n, t h e t r a i l i n g  edge a t x = c,  and  The f l a p  the cavity  t e r m i n a t i o n i s a t x = H. The  first  transformation  maps t h e c a v i t y t e r m i n a t i o n t o i n f i n i t y ,  and t h e p o i n t a t i n f i n i t y  to -1.  The second t r a n s f o r m a t i o n  h  V = a ( z' )  where  a =(  9  r  *  )  maps t h e e n t i r e z' p l a n e onto t h e upper h a l f o f t h e v - p l a n e . The a i r f o i l l i e s between -1 _< v _< b on t h e r e a l a x i s . The upper and lower b o u n d a r i e s of t h e c a v i t y l i e on t h e r e a l a x i s between b <_ v <_ <*>, and -°° _< v <^ -1 r e s p e c t i v e l y . The f i n a l  V = ^  i s a Joukowsky  transformation  ( C +  | ) -  ^  where  b = a (^  t r a n s f o r m a t i o n which maps t h e wetted a i r f o i l  a u n i t s e m i c i r c l e c e n t e r e d a t t h e o r i g i n . The  s u r f a c e onto  c a v i t y b o u n d a r i e s remain  on t h e r e a l a x i s , e x t e r n a l t o t h e u n i t s e m i c i r c l e . By combining t h e transformations, the equation  = fco = ~ l a  Z  2 [  ^(b+pq+c" ) - ha-b)] 1  z  1 + a " [ Mb+l)(C+? ) 2  _1  .  ha-b)}  2  i s o b t a i n e d . Major p o i n t s o f i n t e r e s t i n t h e £-plane a r e a) t h e a i r f o i l  6  nose  o -  C  o  s  b) t h e s p o i l e r base  _  1  <r i r  > »  )  c) the f l a p hinge p o i n t  d) t h e p o i n t a t  infinity  2.3.3  Conditions  Boundary  The boundary c o n d i t i o n s  f o r the problem a r e :  (i)  cf) = 0 on the c a v i t y b o u n d a r i e s ;  (ii)  the K u t t a airfoil  (iii)  c o n d i t i o n , cj) i s c o n t i n u o u s a t the s p o i l e r t i p and the  trailing  the a i r f o i l  edge;  s u r f a c e normal boundary  condition  «--< - S = (iv)  the boundary c o n d i t i o n a t  F(Z)  (v)  = -K/2  the body-cavity  infinity  ;  system  to  be the e q u i v a l e n t  of a s i n g l e closed  body. I n p o t e n t i a l f l o w , t h i s e q u i v a l e n t  body must have zero d r a g .  E q u a t i o n (16) g i v e s  Im  2.3.4  F(Z)dZ = 0  Flow Model Complex a c c e l e r a t i o n p o t e n t i a l f u n c t i o n s a r e found i n t h e £-plane  to  s a t i s f y the boundary  Separate functions  conditions  f o r incidence,  enumerated i n  camber, t h i c k n e s s ,  Section  2.3.3.  s p o i l e r and f l a p , a r e  found and superimposed to g i v e the complete s o l u t i o n f o r the a i r f o i l . complex a c c e l e r a t i o n p o t e n t i a l f u n c t i o n s  i n the v a r i o u s  The  planes are  i n v a r i a n t a t c o r r e s p o n d i n g p o i n t s . The a c c e l e r a t i o n s d i f f e r o n l y by the d e r i v a t i v e s o f the mapping f u n c t i o n s .  dF  The f u n c t i o n s  dF dZ  Thus  dZ d?  given below s a t i s f y the boundary c o n d i t i o n s  The s i g n i f i c a n c e o f t h e v a r i o u s  terms i n each f u n c t i o n  ( i ) to  (iii).  i s explained  in  Reference ( 2 ) .  Incidence  function  + iD o  Camber  function  (22)  19 M  00  -i Z  F (?) c  -J  (23)  Thickness f u n c t i o n  oo (C-e )( -e- °) ieo  N  n=0 ? E  ie  ?  (24)  n  Spoiler function 19]  161 S  IT  L?e  i 6 o  -l  ce"  160  -!  16,  +ln r  (25)  -e  Flap f u n c t i o n i(e -ir)  i(8 -7T) 2  F (0 = ~ f  LCe  i 0 o  + In + - l ' ?e °-l  16: / C-e  _ie  In these equations D , M , and N q  2  fl  -  M n  -  "I/  i  dx  dy -j— dx  t  - I I  dy -3-=- d9 dx  + co ;  TT  TT  N  /  (26) 5  are r e a l constants whose values are  given by : IT  -19  Cos n9 d9 5  ( Cos9  o  - Cos9 ) d0 ; ' '  TT IT J  n  where y  £  and y  the a i r f o i l . B  ( Cos9  dx  - Cos9 ) Cos n6 d9  0  n > 1  ;  a r e f u n c t i o n s r e p r e s e n t i n g t h e camber and t h i c k n e s s o f q  and C  a r e r e a l c o n s t a n t s whose v a l u e s a r e determined by  q  boundary c o n d i t i o n ( i v ) ,  F  la l» c i ( 5  F  ( C  ) + F  t^ )+F (C )+F (C ) - - * , 1  8  1  f  1  where £  i s t h e p o i n t a t i n f i n i t y . The unknown c o n s t a n t s  F  The r e a l and imaginary  ± n  (? )i  taneous e q u a t i o n s ,  parts of equation  (27)  a r e contained i n  (27) g i v e two s i m u l -  which a r e s o l v e d t o g i v e  R l X i t l m E - (a-%M ) ] - ImXiRlE + ^KlmXi B  o  = RlXiImX  - ImXiRlX  2  2  (28) R l E - B R1X - %K 2  c  —  ;  •RlXi  where  Xi  = i  C.e "-! i  C e- '-l  19  =  _  The r e m a i n i n g number K.  ±  = i . < C - f- )  *2  E  + 1  ie  F  c  ( 5  ±  )  _  F  t  (  5  i  )  _  F  g  (  C  i  )  _  F  f  (  q  )  unknowns a r e the c a v i t y l e n g t h I,  and the c a v i t a t i o n  The c a v i t a t i o n number cannot be p r e d i c t e d t h e o r e t i c a l l y a t t h e  21  p r e s e n t time. K and £ a r e r e l a t e d  through boundary c o n d i t i o n ( v ) . Thus  o n l y K i s r e q u i r e d as i n p u t to the t h e o r y . By c h o o s i n g a contour |z| »  Z may be expressed  as a L a u r e n t  such t h a t •  s e r i e s e x p a n s i o n , and t h e  c l o s u r e c o n d i t i o n becomes,  Rl  The is  [ c o e f f i c i e n t z"  ] = 0 .  1  s o l u t i o n o f t h i s e q u a t i o n i n terms o f K and % i s e q u a t i o n g i v e n i n t h e appendix.  An i t e r a t i v e t e c h n i q u e must be used  t h i s e q u a t i o n , because 9 o , 0 i , and The  to solve  62 a r e complex f u n c t i o n s o f  s o l u t i o n t o t h e problem may now be completed  pressure, l i f t ,  (29), which  by d e t e r m i n i n g t h e  and p i t c h i n g moment c o e f f i c i e n t s . The p r e s s u r e  is  o b t a i n e d from e q u a t i o n  to  (26), which a r e t h e a c c e l e r a t i o n p o t e n t i a l s and s u b s t i t u t i n g  equation  (20). By adding t h e r e a l p a r t s o f e q u a t i o n s (22)  (20), t h e p r e s s u r e c o e f f i c i e n t  on t h e a i r f o i l may be r e l a t e d equation  coefficient  into  i n t h e £-plane ±s o b t a i n e d . P o i n t s  t o c o r r e s p o n d i n g p o i n t s on t h e c i r c l e by  (21). Thus  la~ [h  (b+1) C o s 6 - % ( l - b ) ]  2  1 + a " [ ^(b+l)Cos9 2  The  lift  and  t h e L a u r e n t s e r i e s expansion  -  2  h(l-b)}  2  and p i t c h i n g moment c o e f f i c i e n t s a r e o b t a i n e d from e q u a t i o n s (16) o f £. The e q u a t i o n s f o r p r e s s u r e ,  and p i t c h i n g moment c o e f f i c i e n t a r e g i v e n i n t h e appendix.  lift,  22  2.3.5  Base Vented In  Spoilers  some a p p l i c a t i o n s , base v e n t e d s p o i l e r s a r e u s e d . T y p i c a l l y the  base v e n t i s about 10% o f s p o i l e r h e i g h t . A l t h o u g h the p o s s i b i l i t y that Jandali's thick a i r f o i l t h e o r y may  t h e o r y o r Brown's n u m e r i c a l t h i c k  exists  airfoil  be m o d i f i e d t o i n c l u d e t h e e f f e c t s o f base v e n t i n g ,  this  c o u r s e o f a c t i o n i s n o t attempted i n t h e p r e s e n t work. I n s t e a d the e f f e c t s o f base v e n t i n g a r e examined  e x p e r i m e n t a l l y . A two d i m e n s i o n a l  Joukowsky a i r f o i l o f 11% t h i c k n e s s and 2.4% of  camber i s t e s t e d w i t h a s e r i e s  base v e n t e d s p o i l e r s o f h e i g h t e q u a l to 10% o f c h o r d . The base v e n t s  on t h e s p o i l e r s ranged i n s i z e from 10 to 50% o f s p o i l e r  height.  The r e s u l t s o f the base v e n t i n g experiments a r e p r e s e n t e d i n F i g u r e s 11 to 13. These f i g u r e s show t h a t f o r base v e n t s o f about 10% o f s p o i l e r h e i g h t o r l e s s , the v e n t e d s p o i l e r c h a r a c t e r i s t i c s a r e q u i t e c l o s e to the unvented. Thus i t may  be c o n c l u d e d t h a t f o r base v e n t s o f about 10% o f  s p o i l e r h e i g h t o r l e s s , t h e unvented s p o i l e r e d s e c t i o n c h a r a c t e r i s t i c s are  c l o s e enough t o the v e n t e d to be used f o r p r e l i m i n a r y d e s i g n p u r p o s e s .  23  2.4  Experimental Two Dimensional Base Pressures Brown's theory, i n common with the other spoiler theories mentioned  previously, requires as input,  the base pressure c o e f f i c i e n t behind the  spoiler.  the base pressure c o e f f i c i e n t cannot be  At the present time,  predicted t h e o r e t i c a l l y , so experimentally determined values must be used. Figure 39  shows the r e s u l t s of  an experiment  designed to f i n d the base  pressure c o e f f i c i e n t behind a two dimensional a i r f o i l of NACA 0015 f i t t e d with a  normal unvented s p o i l e r of height equal to  The s p o i l e r i s f i t t e d to the 0.77  a i r f o i l at positions ranging from  chord. Figure 40 presents a s i m i l a r r e s u l t f o r a 12.9%  a i r f o i l with a 10%  9.7%  unvented normal s p o i l e r .  the chord ranged between 0.5 and 0.7  section  of chord. 0.48  to  thick Clark Y  The s p o i l e r p o s i t i o n along  chord.  If the experimental values of base pressure c o e f f i c i e n t shown  in  the above mentioned figures are used d i r e c t l y i n the theory, then a nonl i n e a r l i f t curve, inappropiate to a l i n e a r theory i s obtained. To overcome t h i s , Brown l i n e a r i s e d the base pressure d i s t r i b u t i o n . In the present work, an average value of base pressure c o e f f i c i e n t , denoted by  ^  s  used f o r the following reasons: (i)  Since the base pressure c o e f f i c i e n t varies i n a highly non-linear manner with angle of attack for most a i r f o i l s , the use of l i n e a r i s e d values i s no more appropriate than the use of an averaged value. Predictions using both inputs are shown i n Figures  15 and 16.  It  may be seen that good r e s u l t s may be obtained using either input. ( i i ) In taking three dimensional base pressure measurements behind  finite  span s p o i l e r s mounted on f i n i t e span wings, the downwash induced by the t r a i l i n g v o r t i c i t y reduces the e f f e c t i v e angle of attack, so that a  e  = a  a  + a., where a. i s the induced angle of attack. In general a. x i ° i  24  w i l l v a r y a l o n g the span and i s n o t easy t o measure. Thus i t w i l l b e -  hard t o l i n e a r i s e t h e t h r e e d i m e n s i o n a l base p r e s s u r e s because o f t h e difficulty  a s s o c i a t e d w i t h f i n d i n g the e f f e c t i v e angle of a t t a c k a t  which each s p o i l e r e d  section i s operating.  The base p r e s s u r e c o e f f i c i e n t C ^ I g i v e n by 0 _< a _<  a s t a  -Q»  s  averaged  over t h e i n c i d e n c e range  where a i s t h e a n g l e o f a t t a c k o f t h e s p o i l e r e d  s e c t i o n , measured w i t h r e s p e c t t o t h e zero l i f t s e c t i o n . The r a t i o n a l e behind  angle of the unspoilered  t h i s choice o f angles i s that i n p r a c t i c e ,  the s p o i l e r s w i l l o n l y be used when t h e b a s i c wing i s g e n e r a t i n g p o s i t i v e lift  and o p e r a t i n g below s t a l l . V a l u e s o f C , f o r b o t h t h e C l a r k Y and pb NACA 0015 a i r f o i l s e c t i o n s i s g i v e n i n T a b l e I .  AIRFOIL  s/c  AIRFOIL  si c  "  8  p b  NACA 0015  0.48  0.552  CLARK Y  0.50  0.624  with  0.58  0.559  w i t h 10%  0.60  0.612  0.68  0.562  spoilers  0.70  0.610  0.77  0.551  9.7%  spoilers  Table I .  (12.9% t h i c k )  V a l u e s o f C , f o r Two D i m e n s i o n a l pb w i t h Normal Unvented  C l a r k Y and NACA 0015 A i r f o i l s  Spoilers.  T y p i c a l comparisons between t h e o r y and experiment 15 and 16. The p r e d i c t i o n o f l i f t  a r e shown i n F i g u r e s  i s good. The p r e d i c t i o n o f p i t c h i n g  moment i s however l e s s a c c u r a t e , because Brown's t h e o r e t i c a l model p r e d i c t s a s i n g u l a r i t y a t t h e s p o i l e r base. T h i s s i n g u l a r i t y , which i s c h a r a c t e r i s t i c of  linearised  thin a i r f o i l  t h e o r i e s , causes  a p o s i t i v e i n c r e a s e i n the  p i t c h i n g moment - p r e d i c t i o n . A more a c c u r a t e r e s u l t f o r p i t c h i n g moment  24a  would be obtained by using t h i c k a i r f o i l theories f o r s p o i l e r e d a i r f o i l s , such as those of J a n d a l i and Brown ( 3 , 4 ) . In these t h e o r i e s , a stagnation point would replace the s i n g u l a r i t y at the s p o i l e r base. However, even i f t h i c k a i r f o i l t h e o r i e s are used, there would s t i l l be e r r o r s i n the p i t c h ing moment p r e d i c t i o n , because i n r e a l flows a separation bubble would be formed i n the region immediately i n f r o n t of the s p o i l e r , and the stagnation pressure would not be achieved.  25  2.5  E x p e r i m e n t a l F i n i t e Span Base P r e s s u r e s Although  Brown's t h e o r y i s  i n good agreement w i t h  s e c t i o n a l parameters o b t a i n e d from two  experiments,  d i m e n s i o n a l base p r e s s u r e i n p u t s  a r e i n a p p r o p i a t e f o r use w i t h f i n i t e  span s p o i l e r s , which have wakes t h a t  are s i g n i f i c a n t l y d i f f e r e n t  dimensional s p o i l e r s .  from two  Flow around  the v e r t i c a l edges o f a f i n i t e span s p o i l e r c r e a t e s spanwise g r a d i e n t s o f base p r e s s u r e which a r e absent  from two  pressure c o e f f i c i e n t s for f i n i t e than the c o r r e s p o n d i n g two  d i m e n s i o n a l s p o i l e r f l o w s . Base  span s p o i l e r s may  be l a r g e r o r s m a l l e r  d i m e n s i o n a l v a l u e , depending on the l e n g t h t o  h e i g h t r a t i o o f the s p o i l e r , and on p o s i t i o n a l o n g the s p o i l e r S i n c e s p o i l e r e d s e c t i o n parameters  span.  a r e dependent on base p r e s s u r e , i t i s  c l e a r that t h e i r v a l u e s i n three dimensional flows w i l l d i f f e r  from  two  t h a t the  d i m e n s i o n a l c a s e . I n t h i s c o n t e x t , i t s h o u l d a l s o be noted  use of s p o i l e r e d s e c t i o n parameters d e r i v e d from two periments  dimensional  w i l l a l s o be i n a p p r o p r i a t e . Such e x p e r i m e n t a l v a l u e s  m o d i f i e d t o account  F i g u r e s 41 to 43 show t h e r e s u l t s o f t e s t s d e s i g n e d pressures behind  h a l f wings of NACA 0015  a s p e c t r a t i o s r a n g i n g from 3.87  to 7.73  c o n f i g u r a t i o n . S p o i l e r spans o f 20, h e i g h t e q u a l to 9.7%  v a r y i n g from 0.48  must be and  2.6. t o f i n d the base  f i n i t e span unvented s p o i l e r s mounted normal t o the wing  surface.' Rectangular  of  ex-  f o r the d i f f e r e n c e i n base p r e s s u r e between two  three dimensional flows. This i s considered i n Section  the  s e c t i o n and o f e q u i v a l e n t  a r e t e s t e d i n the r e f l e c t i o n  30, 40, and  plane  50% of wing semispan, and  o f chord a r e mounted on the wings a t p o s i t i o n s  to 0.77  c h o r d . The  i n b o a r d t i p s of the s p o i l e r s a r e  always f i x e d a t midspan. The h e i g h t and chordwise a r e the same as f o r the two  l o c a t i o n o f the  d i m e n s i o n a l t e s t s mentioned i n S e c t i o n  spoilers 2.4.  F i g u r e 41 shows the e f f e c t o f v a r y i n g the chordwise  l o c a t i o n o f the  s p o i l e r w i t h t h e a s p e c t r a t i o and s p o i l e r span h e l d c o n s t a n t .  F i g u r e 42  shows t h e v a r i a t i o n o f the base p r e s s u r e c o e f f i c i e n t as a f u n c t i o n  of  s p o i l e r span, w i t h a s p e c t r a t i o and s p o i l e r p o s i t i o n a l o n g the chord h e l d c o n s t a n t . F i g u r e 43 shows t h e v a r i a t i o n o f base p r e s s u r e c o e f f i c i e n t as a f u n c t i o n o f a s p e c t r a t i o , w i t h s p o i l e r l o c a t i o n a l o n g t h e chord and s p o i l e r p e r c e n t o f span h e l d c o n s t a n t . The f i g u r e s g i v e some i d e a about the c o m p l e x i t y o f t h e v a r i a t i o n o f base p r e s s u r e s a l o n g the s p o i l e r when changes a r e made to wing a s p e c t r a t i o ,  span  s p o i l e r span, and s p o i l e r  p o s i t i o n along the chord. For f i n i t e equation  span wings w i t h p a r t span s p o i l e r s ,  (7a) i s s o l v e d f o r a f i n i t e  the l i f t i n g  line  number o f terms, m by c h o o s i n g m  v a l u e s o f the span v a r i a b l e u, and forming a system o f m e q u a t i o n s i n m unknowns. Some o f t h e m p o i n t s w i l l  f a l l on the s p o i l e r e d s e c t i o n s o f  the wing. The base p r e s s u r e c o e f f i c i e n t s a t these p o i n t s a r e used as i n p u t s to  the two d i m e n s i o n a l t h e o r y t o o b t a i n t h e s p o i l e r e d s e c t i o n parameters  a. , m , x , and C which a r e r e q u i r e d as i n p u t s t o e q u a t i o n ( 7 a ) . los os acs macs ^ ^ r  Since  base p r e s s u r e v a r i e s a c r o s s t h e s p o i l e r span, t h e s p o i l e r e d  characteristics w i l l with equations  a l s o v a r y . The s o l u t i o n o f e q u a t i o n  (7a) t o g e t h e r  (12) t o ( 1 5 ) , g i v e the aerodynamic c o e f f i c i e n t s o f the  s p o i l e r e d wing. Such t h e o r e t i c a l p r e d i c t i o n s a r e compared w i t h i n F i g u r e s 17 t o 24. Agreement between t h e o r y and experiment good.  section  experiments  i s seen t o be  27  2.6  E m p i r i c a l R e l a t i o n s h i p s f o r Base P r e s s u r e s The  t h e o r y as developed  t o t h i s p o i n t r e q u i r e s as i n p u t the C ^  d i s t r i b u t i o n a c r o s s the s p o i l e r span. T h i s i n f o r m a t i o n must a t p r e s e n t be o b t a i n e d from wind t u n n e l t e s t s , s i n c e no p r e d i c t i o n methods a r e a v a i l a b l e . T h i s i s a s e r i o u s d e f e c t , s i n c e one o f the t h e o r y ' s is  t o p r o v i d e performance p r e d i c t i o n s f o r a v a r i e t y o f w i n g - s p o i l e r  combinations  without  i n c u r r i n g the c o s t and  time p e n a l t i e s a s s o c i a t e d w i t h  the wind t u n n e l t e s t i n g o f e v e r y c o n f i g u r a t i o n . From t h e p r e d i c t i o n s , the d e s i g n e r may  i f base p r e s s u r e  have to be measured e x p e r i m e n t a l l y f o r each t h e o r y can be a p p l i e d . Furthermore, take l i f t  most  distributions  c o n f i g u r a t i o n b e f o r e the  i t would be a r e l a t i v e l y  simple  task  and moment measurements t o g e t h e r w i t h t h e base p r e s s u r e s .  Experimental values of l i f t  and moment c o e f f i c i e n t s , more a c c u r a t e  the t h e o r e t i c a l p r e d i c t i o n s may t h e o r y s u p e r f l u o u s . The  than  then be c a l c u l a t e d , thus r e n d e r i n g the  t h e o r y would be l i t t l e more than an  academic e x e r c i s e . To be of u t i l i t y , dict  theoretical  s e l e c t the w i n g - s p o i l e r c o m b i n a t i o n  s u i t e d to h i s needs. T h i s advantage i s l o s t  to  purposes  interesting  some method must be d e v i s e d to p r e -  the t h r e e d i m e n s i o n a l base p r e s s u r e s b e h i n d  the s p o i l e r ,  either  from e x p e r i m e n t a l measurements o r from t h e o r y . I n t h e f o l l o w i n g s e c t i o n s , an attempt  to p r e d i c t the t h r e e d i m e n s i o n a l base p r e s s u r e s i s made, based  on e m p i r i c a l measurements. The method does not e l i m i n a t e e x p e r i m e n t a l d e t e r m i n a t i o n s of base p r e s s u r e c o e f f i c i e n t s e n t i r e l y , but r a t h e r reduces s u b s t a n t i a l l y , the amount o f  2.6.1  experimentation.  Averaged A c r o s s Span F i g u r e s 47 to 49 show v a l u e s o f  averaged  a c r o s s the s p o i l e r span,  which w i l l h e n c e f o r t h be denoted  by  d i m e n s i o n a l s p o i l e r span b /h. The s the s p o i l e r a s p e c t r a t i o .  p l o t t e d as a f u n c t i o n of non- • q u a n t i t y b /h may s  be c o n s i d e r e d t o be  The w i n g - s p o i l e r combinations  t h o s e mentioned i n s e c t i o n 2.5.  The  a r e the same  F i g u r e s show t h a t f o r each  as  chordwise  l o c a t i o n o f the s p o i l e r s , the v a l u e o f C , remains n e a r l y c o n s t a n t w i t h  pb r e s p e c t to s p o i l e r a s p e c t r a t i o , o v e r the measured range T h i s may  appear t o be a s u r p r i s i n g r e s u l t , s i n c e i t may  4 <^ b /h < be expected  20. that  as the s p o i l e r a s p e c t r a t i o i s i n c r e a s e d , the f l o w would become i n c r e a s i n g l y two  dimensional,  sectional value  and  t h a t the v a l u e o f C ^ would approach the  Hoerner (12) p r e s e n t s a s i m i l a r  result for flat  p l a t e s normal t o the f l o w , which i s a somewhat s i m i l a r f l o w t o t h a t o f a s p o i l e r mounted on a w i n g . By combining  the r e s u l t s from s e v e r a l s o u r c e s ,  he showed t h a t f o r 1 <^ b/h _< 10, where b/h o f the p l a t e , t h e d r a g c o e f f i c i e n t g r e a t e r than 10,  i s t h e w i d t h to h e i g h t  i s n e a r l y c o n s t a n t . For v a l u e s o f  the d r a g c o e f f i c i e n t r i s e s s l o w l y towards t h e  dimensional v a l u e .  However, the two  approached u n t i l b/h  ratio b/h  two  dimensional drag c o e f f i c i e n t  i s not  i s about 50 o r more. Thus i t would appear t h a t  d i m e n s i o n a l i t y i n f l o w i s not approached u n t i l v e r y h i g h a s p e c t  two  ratios  a r e r e a c h e d . S i n c e the d r a g o f a f l a t p l a t e normal t o the f l o w i s m o s t l y p r e s s u r e d r a g , a c o n s t a n t v a l u e o f d r a g i m p l i e s t h a t the average  pressure  o v e r t h e f r o n t and r e a r o f t h e p l a t e , and hence the base p r e s s u r e , i s also  constant. I t may  immediately  s e c t i o n a l t h e o r y , has between 4 and  20  be seen t h a t t h e use o f C ^  the advantage t h a t i f one  as i n p u t t o the  s p o i l e r of aspect  i s t e s t e d on a f i n i t e wing, then t h e v a l u e o f C ^  o b t a i n e d w i l l be v a l i d  f o r a l l s p o i l e r s o f the same h e i g h t and  ratio so  a n g l e of  29  deflection  6, mounted on wings o f the same s e c t i o n and a t t h e same c h o r d -  wise p o s i t i o n , The  w i t h i n t h e s p o i l e r a s p e c t r a t i o range o f  amount o f e x p e r i m e n t a t i o n  C ^ f°  r  i s thus reduced  each wing s e c t i o n , chordwise  range o f 4 < b / h g  that  g  to a s i n g l e d e t e r m i n a t i o n o f  spoiler position,  s p o i l e r i n c l i n a t i o n . I t should be noted  4 _< b / h < 20.  s p o i l e r h e i g h t , and  the s p o i l e r a s p e c t  ratio  20 i s q u i t e wide, and w i l l l i k e l y cover a l l s p o i l e r  l e n g t h s t h a t may be used. A summary o f the two and t h r e e d i m e n s i o n a l base p r e s s u r e c o e f f i c i e n t measurements i s g i v e n i n T a b l e I I . The v a l u e s o f C ^ shown, a r e averages f o r a l l s p o i l e r l e n g t h s a t the g i v e n chordwise the r a t i o  location.  G i s d e f i n e d as  C_ /C_ _. u  %  AIRFOIL  s/c  NACA 0015 w i t h  0.48  0.552  0.495  0.896  9.7% unvented  0.58  0.559  0.473  0.846  normal  0.68  0.562  0.470  0.836  spoiler  0.77  0.551  0.448  0.825  12.9%  0.50  0.624  0.547  0.876  w i t h 10% normal  0.60  0.612  0.519  0.848  unvented  0.70  0.610  0.512  0.839  Clark Y  spoiler  T a b l e I I . Base P r e s s u r e C o e f f i c i e n t s Clark Y A i r f o i l s  -C , pb  -C , pb  G  f o r S p o i l e r e d NACA 0015 & 12.9%  A l t h o u g h T a b l e I I shows.that t h e v a l u e s o f C the two a i r f o i l s , a p l o t o f  =*  = C  p b  =ft / C  p b  p b  are quite d i f f e r e n t f o r  a g a i n s t s/c ( F i g u r e 5 0 ) , where  = is  t h e v a l u e o f C ^ a t t h e 0.7 chord p o s i t i o n =  v a r i a t i o n of C  p b  /  with s p o i l e r p o s i t i o n  r e v e a l s that the  a l o n g t h e chord i s almost  the same f o r b o t h a i r f o i l s . A l s o a p l o t o f G/G* a g a i n s t s/c ( F i g u r e 50) where G* i s t h e v a l u e o f G a t t h e 0.7 chord p o s i t i o n ,  yields a similar  r e s u l t . The c u r v e s i n F i g u r e 50 show t h a t t h e r e i s a s m a l l d e c r e a s e i n C^  as s p o i l e r p o s i t i o n  a l o n g t h e chord i s moved toward t h e t r a i l i n g  edge. T h i s decrease may be approximated  C  Thus  p b  = C*  b  [ 1.0  by t h e l i n e a r  relation  - 0.445(s/c -0.7) ]  (32)  need o n l y be measured a t s/c=0.7 f o r a g i v e n a i r f o i l  F o r any o t h e r chordwise equation  section.  l o c a t i o n o f t h e s p o i l e r between 0.5 <s/c _< 0.8  (32) may be used  t o f i n d C j ^ . I f t h e two d i m e n s i o n a l base  p r e s s u r e c o e f f i c i e n t i s known a t the 0.7 chord p o s i t i o n ,  then the r e l a t i o n -  ship,  C  p b  _* may be used. C  p b  Equations foils,  = 0.830 C  p b  [1.0  _ i s the value of C  - 0.255 ( s / c - 0.7) ]  p b  a t t h e 0.7 chord  (33)  position.  (32) and (33) a r e v a l i d f o r b o t h t h e C l a r k Y and NACA 0015 a i r -  f i t t e d w i t h 10% s p o i l e r s . Use o f t h e s e e q u a t i o n s i s suggested f o r  o t h e r a i r f o i l s , s p o i l e r h e i g h t s and i n c l i n a t i o n s , as i t i s u n l i k e l y that the v a r i a t i o n s  w i l l be too d i f f e r e n t  from t h e above.  In T a b l e I I I , t h e s e c t i o n a l c h a r a c t e r i s t i c s i n t h r e e dimensionalf l o w s o f t h e NACA 0015 s e c t i o n f i t t e d w i t h 9.7% unvented normal s p o i l e r s , as p r e d i c t e d by Brown's t h e o r y , i s g i v e n . The v a l u e s o f C ^ used as i n p u t a r e o b t a i n e d by u s i n g e q u a t i o n (32) t o g e t h e r w i t h the v a l u e o f C , pb from T a b l e I I . -  s/c  pb  m OS  los  C macs  x  acs  /c  0.48  0.497  4.595  0.2116  0.0786  -0.0378  0.58  0.482  5.033  0.2049  0.1161  -0.0252  0.68  0.467  5.448  0.1997  0.1569  -0.0120  0.77  0.451  5.827  0.1960  0.1977  0.0010  Table I I I .  S e c t i o n a l C h a r a c t e r i s t i c s o f a NACA 0015 A i r f o i l F i t t e d w i t h 9.7% Unvented, Normal S p o i l e r s  2.6.2 In  V a r i a t i o n o f S e c t i o n a l P r o p e r t i e s w i t h Base P r e s s u r e t h e p r e v i o u s s e c t i o n , i t i s shown t h a t t h e u s e o f C , , t h e base pb  p r e s s u r e c o e f f i c i e n t averaged a c r o s s span and i n c i d e n c e , as i n p u t t o t h e t h e o r y , has t h e advantage o f r e d u c i n g by a l a r g e amount, t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f base p r e s s u r u r e s . However t h e u s e o f s u c h an average w i l l o n l y be v a l i d  i f t h e s e c t i o n a l c h a r a c t e r i s t i c s v a r y l i n e a r l y w i t h t h e base  pressure. Although  t h e t h e o r y i s l i n e a r , i t must n o t b e expected  v a r i a t i o n o f s e c t i o n a l p r o p e r t i e s w i t h base p r e s s u r e w i l l In  that the  a l s o be l i n e a r .  t h e s a t i s f a c t i o n o f boundary c o n d i t i o n (v) i n S e c t i o n 2.3.3, a non-  l i n e a r r e l a t i o n s h i p between K, t h e c a v i t a t i o n number, (and hence C ^ ) and  H, t h e c a v i t y l e n g t h i s e s t a b l i s h e d through e q u a t i o n 9i,  (29). The a n g l e s 60,  and 6 2 , which appear i n t h e complex a c c e l e r a t i o n p o t e n t i a l f u n c t i o n s  in  S e c t i o n 2.3.4 a r e r e l a t e d t o JL through  the c o n f o r m a l t r a n s f o r m a t i o n s .  Hence t h e s o l u t i o n t o the problem i s dependent on n o n - l i n e a r manner w i t h t h e base p r e s s u r e  I,  which v a r i e s i n a  coefficient.  F i g u r e s 4 to 6 show t h e v a r i a t i o n o f s e c t i o n a l p r o p e r t i e s w i t h base pressure c o e f f i c i e n t ,  f o r a NACA 0015 a i r f o i l  section f i t t e d with a  normal unvented s p o i l e r o f h e i g h t e q u a l t o 9.7% o f c h o r d . The s p o i l e r i s mounted a t chordwise  l o c a t i o n s r a n g i n g from 0.48 t o 0.77 c h o r d . These  f i g u r e s show t h a t t h e s e n s i t i v i t y o f s e c t i o n c h a r a c t e r i s t i c s t o changes in  base p r e s s u r e c o e f f i c i e n t  i s decreased  as t h e s p o i l e r p o s i t i o n a l o n g  the chord i s moved towards t h e t r a i l i n g edge. A l s o , t h e v a r i a t i o n o f s e c t i o n c h a r a c t e r i s t i c s i s o n l y v e r y s l i g h t l y n o n - l i n e a r . Thus t h e use o f C . pb  i s justified, In  t h e p r e d i c t i o n o f r o l l i n g moment, an a d d i t i o n a l c o m p l i c a t i o n  a r i s e s . S i n c e the r o l l i n g moment about t h e o r i g i n i s g i v e n by,  b p  where y i s t h e spanwise c o o r d i n a t e , t h e s e c t i o n a l l i f t  near  t h e wing, t i p s  w i l l c o n t r i b u t e more t o the r o l l i n g moment because o f t h e w e i g h t i n g  factor  y . F o r example, i f t h e C ^ d i s t r i b u t i o n a c r o s s t h e span i s such t h a t t h e spoilered section l i f t  1 / i n c r e a s e s as y i n c r e a s e s , then t h e p r e d i c t i o n o f  r o l l i n g moment u s i n g C ^ w i l l be low. C o n v e r s e l y , is  i f the C ^ d i s t r i b u t i o n  such t h a t 1 / d e c r e a s e s as y i n c r e a s e s , then t h e r e v e r s e w i l l , b e t r u e .  t h i s e f f e c t i s minimised i f the l i f t d i s t r i b u t i o n across the spoilered section i s symmetrical about the midspan of the s p o i l e r . Fortunately t h i s i s approximately true, as Figures 41 to 46 show.  In these figures the  d i s t r i b u t i o n across the s p o i l e r span i s approximately symmetrical. Since the sectional l i f t varies i n a nearly l i n e a r manner with means  this  that the s e c t i o n a l l i f t d i s t r i b u t i o n across the span i s also  approximately symmetric. A second factor which tends to minimise e f f e c t i s the i n s e n s i t i v i t y of the l i f t  this  to changes i n the base pressure  coefficient,  2.6.3  Use of Experimental Two Dimensional Spoilered Section Parameters in F i n i t e Wing Theory The theory as developed to this point i s capable of predicting the  aerodynamic c o e f f i c i e n t s of a f i n i t e wing with s p o i l e r , using only C , pb  as an empirical input. Experimental spoilered section c h a r a c t e r i s t i c s , i f a v a i l a b l e , may of course, also be used. I t has already been shown, that the use of sectional c h a r a c t e r i s t i c s derived from two dimensional tests i n three dimensional  theory i s inappropriate because of d i f f e r -  ences i n base pressures between two and three dimensional flows. Experimental two dimensional section c h a r a c t e r i s t i c s must be modified to account for the difference i n base pressures before they can be used i n three dimensional theory. In  section 2.6.1, i t i s shown that i f the two dimensional base  pressure averaged over incidence, C , , i s known for a s p o i l e r mounted at pb the 0.7 chord position, then the value of C ^ may be obtained by using equation (33). Theoretical predictions may  then be made using C ^ p  and  C , as i n p u t s . The d i f f e r e n c e pb difference ences may  i n sectional  c h a r a c t e r i s t i c s due  i n base p r e s s u r e c o e f f i c i e n t s may then be deducted  be c a l c u l a t e d .  from the e x p e r i m e n t a l  two  The  to the differ-  dimensional .  s e c t i o n a l parameters t o o b t a i n v a l u e s a p p r o p r i a t e f o r t h r e e d i m e n s i o n a l flows.  35  3.  EXPERIMENTS  The experimental part of t h i s t h e s i s c o n s i s t s of three s e r i e s of experiments.  In the f i r s t , the e f f e c t  of base venting on s p o i l e r e d  s e c t i o n c h a r a c t e r i s t i c s i s examined. In the second the forces and moments generated by f i n i t e wings w i t h part span s p o i l e r s are measured,  and  aerodynamic c o e f f i c i e n t s c a l c u l a t e d . In the t h i r d , the base pressure d i s t r i b u t i o n behind s p o i l e r s mounted on two and three dimensional wings i s measured.  3.1.1  »  Base Venting  Experiments  The purpose of these experiments was to determine the e f f e c t of base v e n t i n g on s e c t i o n c h a r a c t e r i s t i c s . A Joukowsky a i r f o i l of 11% t h i c k n e s s , 2.4% camber and 12.08 mainly of wood,  i n c h chord was used. The a i r f o i l was constructed  w i t h an aluminium center s e c t i o n c o n t a i n i n g 37 pressure  taps of which 24 were on the upper s u r f a c e . Since the Joukowsky p r o f i l e was s t r u c t u r a l l y weak near the cusped t r a i l i n g edge, the upper surface i n t h i s r e g i o n was modified to give an approximately constant thickness of 1/8 i n c h . The modified p r o f i l e i s shown i n Figure 7. The a i r f o i l was w i t h end p l a t e s on which s p o i l e r s could be mounted at the 0.5, 0.6,  fitted 0.7,  0.8 and 0.9 chord p o s i t i o n s , normal to the a i r f o i l s u r f a c e . T h i s a i r f o i l was used by J a n d a l i to v e r i f y h i s s p o i l e r theory, and a f u l l d e s c r i p t i o n of i t i s given i n reference ( 3 ) . A set of 5 s p o i l e r s of height equal to 10% of chord, w i t h base vents of 10,20,30,40 and 50 % of s p o i l e r height were made f o r the a i r f o i l . The t e s t s were conducted i n the small low speed a e r o n a u t i c a l wind tunnel i n the Department of Mechanical Engineering at the U n i v e r s i t y of  36  B r i t i s h Columbia. I t has a t e s t s e c t i o n o f 27 i n c h h e i g h t and 36 i n c h w i d t h . The  t u n n e l has good f l o w u n i f o r m i t y and a t u r b u l e n c e l e v e l o f l e s s than 0.1  p e r c e n t o v e r i t s speed r a n g e . The a i r f o i l  was mounted v e r t i c a l l y and spanned  the t e s t s e c t i o n , w i t h s m a l l c l e a r a n c e s a t t h e r o o f and f l o o r . The a i r f o i l was  a t t a c h e d t o a s i x component p y r a m i d a l b a l a n c e l o c a t e d under the t u n n e l ,  at  t h e q u a r t e r c h o r d p o s i t i o n . F o r c e and moment measurements were taken  w i t h t h e s p o i l e r s a t t a c h e d a t t h e 0.5, 0.6, 0.7 and 0.8 chord over a f u l l  positions,  a n g l e o f a t t a c k range. P r e s s u r e measurements were a l s o taken a t  some a n g l e s o f a t t a c k u s i n g a m u l t i - t u b e manometer. T e s t Reynolds number was  4.4 ( 1 0 ) . 5  3.2  F i n i t e Wing  Experiments  F o r t h e f i n i t e wing experiments, to  h a l f wing models were used  i n order  o b t a i n a good r a n g e o f a s p e c t r a t i o s , w i t h as l a r g e a c h o r d and Reynolds  number as p o s s i b l e .  R e c t a n g u l a r wings o f NACA 0015 s e c t i o n were mounted  v e r t i c a l l y a t t h e q u a r t e r chord p o s i t i o n , i n the same t u n n e l - b a l a n c e system mentioned i n S e c t i o n 3.1.  The wings were machined  from  solid  aluminium i n spanwise s e c t i o n s o f 0.5 and 2.0 i n c h e s . The chord was 5.17 i n c h e s . By combining a p p r o p r i a t e numbers o f each o f t h e two s i z e s o f spanwise s e c t i o n s , h a l f wing models o f e q u i v a l e n t f u l l 4.83,  a s p e c t r a t i o s o f 3.87,  5.80, 6.77, and 7.73 were assembled. H o l e s were d r i l l e d and tapped on  the upper s u r f a c e o f t h e wing, so t h a t s p o i l e r s o f 20, 30, 40 and 50 % o f h a l f span, c o u l d be mounted a t t h e 0.48, 0.58, 0.68 and 0.77 chord  positions.  The unvented s p o i l e r s , o f h e i g h t e q u a l t o 9.7% o f chord were mounted on the wings so t h a t i n a l l c a s e s , t h e i n b o a r d t i p o f t h e s p o i l e r was p o s i t i o n e d at  midspan. F o r c e and moment measurements were made f o r a l l p o s s i b l e  37  c o n f i g u r a t i o n s o v e r a f u l l range o f a n g l e o f a t t a c k . In  a d d i t i o n , two d i m e n s i o n a l t e s t s were made t o o b t a i n t h e s p o i l e r e d  section characteristics. the  A two d i m e n s i o n a l w i n g was made by a s s e m b l i n g  5.17 i n c h chord s e c t i o n s i n t o a w i n g s p a n n i n g t h e t e s t s e c t i o n o f t h e  w i n d t u n n e l v e r t i c a l l y , w i t h s m a l l c l e a r a n c e s a t t h e r o o f and f l o o r .  Full  span 9.7% normal u n v e n t e d s p o i l e r s were mounted on t h e w i n g a t t h e 0.48, 0.58, 0.68 and 0.77 c h o r d p o s i t i o n s . F o r c e and moment measurements were made. The two d i m e n s i o n a l t e s t s were made i n o r d e r t o o b t a i n comparisons between experiment and t h e p r e d i c t i o n s o f t h e Brown t h e o r y . T e s t R e y n o l d s number was 3 (10)"*. The NACA 0015 s e c t i o n  3.3  i s shown i n F i g u r e 8.  Base P r e s s u r e Measurements. Measurements o f t h e base p r e s s u r e s b e h i n d t h e s p o i l e r s f i t t e d t o t h e  two and t h r e e d i m e n s i o n a l wings t e s t e d i n S e c t i o n 3.2 were t a k e n i n t h i s s e r i e s of experiments. This i n f o r m a t i o n  was r e q u i r e d a s i n p u t t o t h e  s e c t i o n a l t h e o r y . A n a d d i t i o n a l s e t o f base p r e s s u r e s was t a k e n u s i n g 12.9% t h i c k C l a r k Y wings o f 5.9 i n c h c h o r d . The w i n g s were made o f wood i n spanwise s e c t i o n s o f 3, 6, and 12 i n c h e s , w h i c h were assembled t o g i v e a two d i m e n s i o n a l model s p a n n i n g t h e t e s t s e c t i o n , and r e c t a n g u l a r h a l f wings w i t h e q u i v a l e n t a s p e c t r a t i o s o f 4.07, 6.10, and 7.12. I n a l l  tests  w i t h t h e C l a r k Y w i n g , 10% u n v e n t e d , normal s p o i l e r s were u s e d . They were t a p e d t o the s u r f a c e o f t h e wings a t t h e 0.5, 0.6 and 0.7 c h o r d p o s i t i o n s . F o r the two d i m e n s i o n a l t e s t s , t h e s p o i l e r s were f u l l span. F o r the  f i n i t e wing t e s t s , t h e s p o i l e r s were o f l e n g t h e q u a l t o 20, 30, 40,  and 50% o f t h e h a l f w i n g span. They were mounted on t h e wings so t h a t t h e i n n e r t i p o f t h e s p o i l e r s were a l w a y s f i x e d a t midspan, a s was t h e c a s e  38  for  the NACA 0015 For the two  midspan, halfway  wings. The  C l a r k Y s e c t i o n i s shown i n F i g u r e  9.  dimensional t e s t s , a s i n g l e pressure tap, l o c a t e d between the s p o i l e r base and  at  the t r a i l i n g edge, was  For the t h r e e d i m e n s i o n a l t e s t s , t e n t a p s , e q u a l l y spaced  used.  i n the spanwise  d i r e c t i o n , s t a r t i n g a t a p o i n t 5% of s p o i l e r span away from the i n n e r spoiler  t i p and moving towards the o u t e r s p o i l e r t i p was  a t i o n o f a l l t e n t a p s i n the chordwise s p o i l e r base and  d i r e c t i o n was  u s e d . The  halfway  loc-  between the  the t r a i l i n g edge. The base p r e s s u r e s were measured u s i n g  the system shown i n F i g u r e 10. Whenever a s e t o f base p r e s s u r e s was measured, the e x t e r n a l s t a r t s w i t c h was  depressed. T h i s a c t i v a t e d  the  PDP-11 computer, which i n s t r u c t e d the s c a n i v a l v e to b e g i n s c a n n i n g t u n n e l dynamic p r e s s u r e t a p s , and  the p r e s s u r e t a p s behind  the  t o be  the  spoiler.  The p r e s s u r e s a t each tap were s e q u e n t i a l l y t r a n s m i t t e d by the s c a n i v a l v e to  the b a r o c e l , which c o n v e r t e d the p r e s s u r e s i n t o v o l t a g e s . The  d i g i t a l c o n v e r t e r then d i g i t a l i s e d  the v o l t a g e s and  analog to  t r a n s m i t t e d them t o  the computer memory f o r s t o r a g e . A f t e r a l l the p r e s s u r e t a p s had  been  scanned, the base p r e s s u r e c o e f f i c i e n t s were computed and p r i n t e d on a typewriter.  3.4  T e s t Reynolds Number was  3  (10)\  Wind Tunnel W a l l C o r r e c t i o n s F o r the Joukowski a i r f o i l ,  employed was  the wind t u n n e l w a l l c o r r e c t i o n  the same as t h a t of J a n d a l i  ( 3 ) , who  used  e s t a b l i s h e d by Pope and Harper ( 1 3 ) . The n o n - d i m e n s i o n a l term was  however, m o d i f i e d t o % ( c / H ) C , , d  found  technique  the c o r r e c t i o n s wake b l o c k a g e  i n s t e a d o f %(c/H)C,,  as  Jandali  a  t h a t measurements f o r a i r f o i l s o f v a r y i n g chord l e n g t h s c o l l a p s e d  b e t t e r u s i n g the m o d i f i e d term. For p r e s s u r e c o e f f i c i e n t s , J a n d a l i  used  the  equation,  (34)  where C  P  and C, a r e the t r u e p r e s s u r e and l i f t 1  a n g l e o f a t t a c k , and C  pu  c o e f f i c i e n t s a t a given  and C, a r e t h e u n c o r r e c t e d lu  p r e s s u r e and  lift  coefficients. The  d a t a f o r t h e C l a r k Y and NACA 0015 a i r f o i l s were n o t c o r r e c t e d  f o r wind t u n n e l w a l l e f f e c t s because o f t h e s m a l l s i z e o f the wings i n r e l a t i o n t o the t u n n e l dimensions (c/H < 0.2, S/C < 0.2). The c o r r e c t i o n s to the v a r i o u s c o e f f i c i e n t s were l e s s t h a n 3% and were t h e r e f o r e i g n o r e d .  4.  4.1  RESULTS  AND  COMPARISONS  Base V e n t i n g Experiments A sample o f t h e r e s u l t s o f experiments u s i n g t h e two d i m e n s i o n a l  Joukowsky a i r f o i l  f i t t e d w i t h 10% normal s p o i l e r s w i t h base v e n t s o f  v a r i o u s s i z e s i s p r e s e n t e d i n F i g u r e s 11 t o 13. I t may be seen from t h e s e f i g u r e s t h a t t h e e f f e c t o f i n c r e a s i n g t h e s i z e o f the base v e n t i s t o increase l i f t . the  F o r t h e l a r g e r base v e n t s , t h i s e f f e c t i s s u b s t a n t i a l a t  lower a n g l e s o f a t t a c k . A t h i g h e r i n c i d e n c e , t h e c u r v e s converge  towards t h e unvented r e s u l t . F o r s m a l l base v e n t s o f around 10% o f s p o i l e r h e i g h t o r l e s s however, t h e i n c r e a s e i n l i f t  over t h e unvented  s p o i l e r i s s m a l l a t a l l a n g l e s o f a t t a c k . The same r e s u l t p i t c h i n g moment about t h e aerodynamic  i s true f o r the  c e n t e r . Thus f o r base v e n t s o f  about 10% s p o i l e r h e i g h t o r l e s s , t h e unvented s p o i l e r c h a r a c t e r i s t i c s a r e s u f f i c i e n t l y c l o s e t o t h e v e n t e d t o be used f o r p r e l i m i n a r y d e s i g n purposes.  4.2.1  Two D i m e n s i o n a l NACA 0015 A i r f o i l  Experiments  F i g u r e 14 shows t h e u n s p o i l e r e d NACA 0015 s e c t i o n The l i f t  c u r v e shows some n o n - l i n e a r i t y a t t h e h i g h e r a n g l e s o f a t t a c k  and t h e l i f t  c u r v e s l o p e i s s i g n i f i c a n t l y lower than t h e t h e o r e t i c a l .  These e f f e c t s a r e due t o t h e low Reynolds number the  characteristics.  [ 3(10)"*] a t which t h e  t e s t s were c o n d u c t e d . F o r the t h e o r e t i c a l f i n i t e wing  predictions  t h e r e f o r e , l i n e a r i s e d e x p e r i m e n t a l v a l u e s o f the s e c t i o n parameters a, , and C lo mac  a r e used,  m, Q  41  Figures 15 and 16 are.comparisons between experimental  and t h e o r e t i c a l  section c h a r a c t e r i s t i c s of the NACA 0015 a i r f o i l , with 9.7% unvented normal spoilers mounted at the 0.48 and 0.68 chord p o s i t i o n s .  Two t h e o r e t i c a l  curves are shown. The broken l i n e s represent the prediction using a base pressure c o e f f i c i e n t l i n e a r i s e d with respect to incidence, and the unbroken l i n e s represent the prediction using C , , the base pressure c o e f f i c i e n t  pb averaged over incidence. The use of these inputs i s discussed i n Section  :  2.4. Both inputs give good predictions for l i f t . The prediction for p i t c h ing moment i s less accurate than for l i f t . The reasons for t h i s are already discussed i n Section 2.4  4.2.2  Rectangular Wings of NACA 0015 Section F i t t e d with Part Span Spoilers  Figures 17 to 20 show experimental  and t h e o r e t i c a l l i f t and pitching  moment comparisons for rectangular h a l f wings of equivalent aspect r a t i o equal to 7.73, f i t t e d with part span s p o i l e r s . The unvented 9.7% normal s p o i l e r s are f i t t e d to the wings at the 0.48 chord p o s i t i o n . Their spans are equal to 20, 30, 40 and 50% of semi-span. The spoilers are mounted so that the inboard t i p of the s p o i l e r s are always at mid-span. Figures 21 to 24 are the corresponding  r o l l i n g moment c o e f f i c i e n t s for the above mentioned  half wings. Since the tests  are made with half wing models, the wing  r o l l i n g moment i s defined as  R  J  5PU Sb 2  rather than the more usual d e f i n i t i o n of  42  R  ?5pU S(2b) 2  as the former d e f i n i t i o n i s more appropriate to h a l f wing t e s t s . In Figures 17 to 24, two t h e o r e t i c a l curves are shown. The broken l i n e s represent the p r e d i c t i o n using  the base pressure c o e f f i c i e n t  averaged over incidence  but varying across the s p o i l e r span. The s o l i d l i n e s represent the prediction using C ^ , the base pressure c o e f f i c i e n t averaged over both incidence and s p o i l e r span. C , pt)  i s obtained from equation (32),  with the value of C , pb  coming from Table 2. The use of these two inputs i s discussed i n Sections 2.5 and 2.6, where the advantages of using C ^  are discussed. The pred-  i c t i o n s given by both inputs are seen to be very close. This confirms that the use of C ^  as predicted by equation (32) i s s u f f i c i e n t l y accurate for-  preliminary design purposes. Figures 25 to 32 are s i m i l a r to Figures 17 to 24, except that the equivalent aspect r a t i o i s 3.87, and the s p o i l e r s are mounted at the 0.68 chord p o s i t i o n . Only one t h e o r e t i c a l prediction i s shown, that using C . pb as the input to the sectional theory. the prediction of l i f t and r o l l  In a l l of the above mentioned cases,  f o r the f i n i t e rectangular wings i s seen  to be good. The prediction of p i t c h i n g moment i s l e s s accurate. This i s to be expected, since the sectional theory's p r e d i c t i o n of pitching moment i s l e s s accurate than f o r l i f t . Any inaccuracies i n the prediction of sectional c h a r a c t e r i s t i c s , w i l l of course be c a r r i e d over into the three dimensional theory. Figure 33 compares the predicted and measured v a r i a t i o n with respect to r e l a t i v e s p o i l e r span b /h, of the e f f e c t i v e moment arm of the incremental g  43  lift  caused by s p o i l e r e r e c t i o n .  combinations  The  figure refers  spoiler  which a r e the same as f o r F i g u r e s 17 to 24. The d a t a i s  p r e s e n t e d i n the form A C / A C , p l o t t e d D  K.  to F i g u r e 33, except  T  a g a i n s t b /b. F i g u r e 34 i s s i m i l a r  . L  S  t h a t the w i n g - s p o i l e r combinations  f o r F i g u r e s 25 to 32. The variation  to wing  dashed l i n e s i n the two  t h a t would o c c u r i f the i n c r e m e n t a l l i f t  are the same as  figures  r e p r e s e n t the  a c t e d a t the midspan  of the s p o i l e r . The e x p e r i m e n t a l v a l u e s a r e averages  over the i n c i d e n c e  range of -4° to j u s t below s t a l l . The agreement between t h e o r y and e x p e r i ment i s seen to be good. An i n s i g h t  i n t o the r e a s o n f o r the inward  moment arm of the i n c r e m e n t a l l i f t may 35 and  s h i f t o f the  effective  be o b t a i n e d by examining  36, which show the spanwise d i s t r i b u t i o n of the  Figures  non-dimensionalised  c i r c u l a t i o n r/4bU. The c u r v e s show c l e a r l y t h a t the e f f e c t o f the is  not c o n f i n e d to the s p o i l e r e d  p o r t i o n s o f the wing. There i s a l s o  l o s s o f c i r c u l a t i o n and hence l i f t , wings. The l o s s o f l i f t larger  over the u n s p o i l e r e d s e c t i o n s o f the  over the u n s p o i l e r e d s e c t i o n s of the wings i s  towards the wing r o o t .  F i g u r e 37 shows the spanwise d i s t r i b u t i o n of the c i r c u l a t i o n r/4bU f o r r e c t a n g u l a r wings of NACA 0015  9.7%  7.73,  dimensionless section  mounted a t the 0.48  s y m m e t r i c a l l y deployed  spoilers  are  spoilers  t i p . The lower c u r v e i s f o r  ( s p o i l e r s up on b o t h h a l f w i n g s ) .  curve i s f o r asymmetric deployment  normal,  spoilers  chord p o s i t i o n , w i t h the i n b o a r d t i p s of the  p o s i t i o n e d midway between the wing r o o t and  '  and of a s p e c t  f i t t e d w i t h s y m m e t r i c a l l y and a s y m m e t r i c a l l y deployed  unvented s p o i l e r s o f span e q u a l to 40% of semi-span. The  middle  a  i n b o a r d of the s p o i l e r . Hence the c e n t e r of i n c r e m e n t a l l i f t i s  s h i f t e d inwards,  ratio  spoiler  The  ( s p o i l e r up on one h a l f wing  44  but retracted on the o t h e r ) . The upper curve i s f o r the unspoilered wing. Figure 38 i s similar to Figure 37 except that the aspect r a t i o i s 3.87, and the s p o i l e r s are mounted at the 0.68 chord p o s i t i o n . show that the curves f o r symmetric and asymmetric s p o i l e r  The two figures deployment are  i n close agreement. This implies that while the half wing tests s t r i c t l y correspond to cases of symmetric s p o i l e r deployment f o r complete wings, they may be used to model cases of asymmetric deployment as w e l l .  4.3.1  Two Dimensional Base Pressure  Experiments  Figure 39 shows the v a r i a t i o n of base pressure with incidence for a r  two dimensional a i r f o i l of NACA 0015 section f i t t e d with a 9.7% unvented normal s p o i l e r . The p o s i t i o n of the s p o i l e r i s varied from 0.48 to 0.77 chord. Figure 40 i s s i m i l a r to Figure 39, except that the section i s a 12.9% thick Clark Y, the s p o i l e r s are 10%, and the p o s i t i o n of the s p o i l e r i s varied between 0.5 and 0.7 chord. A comparison of the curves for the NACA section shows that a rearward s h i f t i n s p o i l e r p o s i t i o n along the chord r e s u l t s i n an e a r l i e r peak i n the base pressure d i s t r i b u t i o n . However, the average value of the base pressure c o e f f i c i e n t C , , as  pb defined i n Section 2.4, does not appear to vary by a large amount. The same trends are apparent for the 12.9% thick Clark Y section. Although the s p o i l e r s for both a i r f o i l s are s i m i l a r i n geometry and height, the base pressure d i s t r i b u t i o n s are quite d i f f e r e n t . C ^  a  r  4.3.2  e  n  o  t  t o  ° f  a  r  However, the values of  apart.  Three Dimensional Base Pressure  Experiments  Figures 41 to 43 show the spanwise d i s t r i b u t i o n of C ^ , the base pressure d i s t r i b u t i o n averaged over incidence, f o r part span spoilers  mounted on r e c t a n g u l a r wings o f NACA 0015 s e c t i o n . The w i n g - s p o i l e r c o n f i g u r a t i o n s a r e the same a s i n S e c t i o n 3.3.  I n the f i g u r e s b  i s the  spanwise c o o r d i n a t e , measured w i t h r e s p e c t t o t h e i n n e r s p o i l e r t i p , and moving outwards, toward the o u t e r  s p o i l e r t i p . F i g u r e 41 shows the e f f e c t  of v a r y i n g the chordwise p o s i t i o n o f the s p o i l e r w i t h a s p e c t  r a t i o and  s p o i l e r span h e l d c o n s t a n t . F i g u r e 42 shows t h e v a r i a t i o n o f C ^ as a f u n c t i o n o f s p o i l e r span, w i t h a s p e c t the chord  tion  r a t i o and s p o i l e r p o s i t i o n a l o n g  h e l d c o n s t a n t . F i g u r e 43 shows t h e v a r i a t i o n o f C , pb  of aspect r a t i o , with s p o i l e r percent  as a f u n c -  o f span and chordwise  spoiler  p o s i t i o n h e l d c o n s t a n t . F i g u r e s 44 t o 46 a r e p l o t s s i m i l a r t o F i g u r e s 42 to 44 except t h a t t h e s e c t i o n i s a 12.9% C l a r k Y. F i g u r e s 41 t o 46 r e v e a l t h e complex manner i n which C , v a r i e s pb across C  p b  t h e s p o i l e r span, as t h e v a r i o u s parameters a r e changed. P l o t s o f  a g a i n s t non d i m e n s i o n a l  s p o i l e r span b / h r e v e a l a c o n s i d e r a b l e g  s i m p l i f i c a t i o n . F i g u r e s 47 49 show p l o t s o f 5 ,  a g a i n s t b /h. These  f i g u r e s / t h a t a t each chordwise l o c a t i o n t h e v a l u e o f C ^ constant The  f o r each a i r f o i l ,  regardless of aspect  implications of t h i s are discussed  remains n e a r l y  r a t i o and s p o i l e r  i n S e c t i o n 2.6.1  length.  CONCLUSIONS  The use o f t h e m o d i f i e d l i n e a r l i f t i n g good p r e d i c t i o n s o f l i f t  l i n e t h e o r y i s shown t o g i v e  and r o l l i n g moment f o r f i n i t e wings f i t t e d  with  p a r t span s p o i l e r s . The p r e d i c t i o n o f p i t c h i n g moment i s n o t as good as for  lift.  airfoil  T h i s i s due t o t h e f a c t  t h a t t h e Brown t h e o r y f o r f o r s p o i l e r e d  s e c t i o n s g i v e s p r e d i c t i o n s o f p i t c h i n g moment which a r e l e s s  a c c u r a t e than f o r  lift.  A r e q u i r e d i n p u t t o t h e t h e o r y i s t h e base p r e s s u r e behind  coefficient  t h e f i n i t e span s p o i l e r s . A t the p r e s e n t time, t h e base p r e s s u r e  c o e f f i c i e n t cannot  be p r e d i c t e d t h e o r e t i c a l l y . Experiments  conducted i n  support o f t h i s t h e s i s show t h a t t h e base p r e s s u r e c o e f f i c i e n t v a r i e s i n a complex manner w i t h a i r f o i l and s p o i l e r geometry. However i t i s found t h a t t h e base p r e s s u r e c o e f f i c i e n t averaged  on any g i v e n a i r f o i l  over span and i n c i d e n c e , C , pb  s e c t i o n , i s independent  o f s p o i l e r l e n g t h , and i t s  use as an i n p u t t o t h e t h e o r y g i v e s good r e s u l t s . A method o f p r e d i c t i n g C^,  which g r e a t l y reduces  t h e amount o f e x p e r i m e n t a l measurements i s  presented. The n e c e s s i t y f o r a base p r e s s u r e i n p u t based  on e x p e r i m e n t a l  measurements remains a weakness o f the t h e o r y . Measurements must s t i l l made f o r each s p o i l e r h e i g h t and i n c l i n a t i o n and a i r f o i l  section. In  a d d i t i o n , i f the wing i s f l a p p e d , the f l a p a n g l e and s l o t s i z e affect  be  the base p r e s s u r e . S i n c e t h e e x p e r i m e n t a l p a r t o f t h i s  will thesis  d e a l s o n l y w i t h 10% s p o i l e r s mounted on u n f l a p p e d wings, no attempt can be made to develop  e m p i r i c a l l y based  formulas  f o r the p r e d i c t i o n of  base p r e s s u r e , as changes a r e made t o s p o i l e r h e i g h t and i n c l i n a t i o n ,  flap angle and s l o t s i z e . Further experiments w i l l have to be made before this can be attempted. Experiments on base vented spoilers show that base vents of about 10% of s p o i l e r height or less give sectional c h a r a c t e r i s t i c s which are l i t t l e d i f f e r e n t from those of the corresponding unvented s p o i l e r s . Thus for the purposes of preliminary design, the unvented s p o i l e r c h a r a c t e r i s t i c s may used, provided that the s p o i l e r s have base vents of about 10% or l e s s .  be  48  Z  F i g u r e 2.  PLANE  A i r f o i l i n the P h y s i c a l Plane  Z  PLANE st  n  Z '  ;  c  I  P L A N E  1  -1  v P L A N E ia b  C P e  F i g u r e 3. Complex T r a n s f o r m  L  /Ae *' 1  Planes  A  N  E  10  9 m  0S 8  S/C = 0-77 S/C = 0-6 8 S/C =0-58 S/C=0-48  h/c=0097 8 = 90° 3  2 \-  -0-30 Figure 4 &  -0-35  -0-40  Variation of m  -0-45  -0-50  -0-55  -0-60  w i t h C , f o r NACA 0015 A i r f o i l os pb w i t h Normal Unvented S p o i l e r  -0-65  Section  52  h/c = 0-097 8=90°  001  s/c=077  n  \j  -001 x  s/c=0-68  acs c  h/c = 0 0 9 7 8=90°  -002 '.  ^__s/c=0-58  -003 ~^~-^s/c=0-48 -004  -005 -0-30  i -0-35  i -0-40  i -0-45  i  -0-50  i -0-55 _  i -0-60  i -0-65  Cpb Figure 5  V a r i a t i o n of C & x Ic w i t h C , f o r NACA 0015 A i r f o i l macs acs pb S e c t i o n w i t h Normal Unvented S p o i l e r  Figure 7  M o d i f i e d Joukowsky A i r f o i l S e c t i o n o f 11% T h i c k n e s s & 2.4% Camber w i t h Base Vented Normal Spoiler  Figure 8  NACA 0015 A i r f o i l S e c t i o n , w i t h 9.7% Unvented Normal S p o i l e r  Figure 9  12.9% T h i c k C l a r k Y  A i r f o i l S e c t i o n w i t h 10% Unvented Normal S p o i l e r  J  3  TO WING PRESSURE TAPS  SCANIVALVE  BAROCEL  OUTPUT I (VOLTS)  PDP-II COMPUTER WITH A/D CONVERTER  F i g u r e 10  Base P r e s s u r e Measurement System  EXT. START SW.  TO DYNAMIC PRESSURE TAPS  58  Figure 12  L i f t C o e f f i c i e n t f o r Joukowsky A i r f o i l Section with Base Vented Normal Spoilers  -8  -4  0  4  8  12  16  20 a  Figure  13  P i t c h i n g Moment C o e f f i c i e n t f o r Joukowsky A i r f o i l S e c t i o n Base Vented Normal  Spoilers  with  J-.l-O F i g u r e 14.  Lift  & Pitching  Moment C o e f f i c i e n t s  f o r a NACA 0015 S e c t i o n  l i n e a r i s a t i o n ; o e x p e r i m e n t a l C. ; A e x p e r i m e n t a l C  62  h/c = 0-097 s/c = 0-48  0-3J--I-2 F i g u r e 15  Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015 A i r f o i l  w i t h Normal Unvented S p o i l e r .  theory  Section  (C , i n p u t ) ; theory pb ( l i n e a r i s e d C ^ i n p u t ) , o C^ experimental;A C experimental m a c s  h/c = 0-097 s/c=0-68  -0-3 J—1-2 F i g u r e 16  Lift  & P i t c h i n g Moment C o e f f i e i e n t s  w i t h Normal Unvented S p o i l e r .  f o r NACA 0015 A i r f o i l  theory  (C ^ i n p u t ) ;  ( l i n e a r i s e d C , i n p u t ) : o C. e x p e r i m e n t a l ; A C pb 1 macs r  r  Section theory  experimental  64 0-3TI2  48  A  A  A  1 6  a  2 0  AR=773 s/c = 0-48 b /b=0-2 s  h/c = 0 - 0 9 7  -0-2- -0-8 L  F i g u r e 17  Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . input); —  theory  (C ^ i n p u t ) ;o C  L  H a l f Wing o f theory ( C  p b  experiment;A C^^, experiment  65  AR=7-73 s/c =0-48 b /b = 0-3 h/c = 0097 s  -0-2^--0-8 F i g u r e 18  Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . input),  (C  p b  input);o C  L  H a l f Wing o f t h e o r y (C  experiment; A C ^ ^ experiment  Pb  66  16  a  20  AR=7 73 s/c = 0-48 b/b= 0-4 h/c = 0 097 s  0-2 - - 0-8 L  Figure 19  L i f t & Pitching Moment C o e f f i c i e n t s for Rectangular  Half Wing of  NACA 0015 Section with Normal Unvented S p o i l e r .  theory (C  input),  theory (C  b  input) ;o C  L  experiment;A  pb  experiment  67  0-3  T  1-2  AR = 7-73 s/c=0-48 b /b=0-5 h/c=0097 s  0-2F i g u r e 20  Lift  0-8  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . input;  theory  ( C ^ i n p u t ) ;o  H a l f Wing o f theory  experiment;A C j ^ , experiment  68  F i g u r e 21  R o l l i n g Moment C o e f f i c i e n t f o r R e c t a n g u l a r H a l f Wing o f NACA 0015 Section —  w i t h Normal Unvented S p o i l e r .  theory  (C , i n p u t ) ; o  experiment  theory  (Cp  b  input);  69  0-6  16  20 a  AR = 3-87 s/c = 0-68 b /b=0-2 h /c = 0097 s  0-6  -0-2-1--0-8 F i g u r e 25  Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . o C^ experiment; A C^^,  experiment  H a l f Wing theory;  73  F i g u r e 26  Lift  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . o C  experiment;  A C  experiment  theory;  F i g u r e 27 L i f t  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r  of NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . o C  experiment; A C  experiment  H a l f Wing theory;  75  0-3T  AR=3-87 s/c = 0-68 b /b = 0-5 h/c=0097 s  0-6  - 0-2 -- - 0-8 1  F i g u r e 28 ' L i f t  & P i t c h i n g Moment C o e f f i c i e n t s f o r R e c t a n g u l a r H a l f  of NACA 0015 S e c t i o n o C  experiment; A C  w i t h Normal Unvented experiment  Spoiler.  Wing  Theory;  76  0-6  O  0-2  AR=3-87 s/c = 0-68 b/b=0-2 h/c = 0097 s  0-3  -L-0-4 F i g u r e 29  R o l l i n g Moment C o e f f i c i e n t  f o r Rectangular  H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . o C experiment; R  theory;  -r-0-6  4--0-2 0  AR=3-87 s/c = 0-6 8 b /b = 0-3 h/c = 0097 s  + -0-3  -L-0-4 Figure 30  R o l l i n g Moment C o e f f i c i e n t for Rectangular  Half Wing of  NACA 0015 Section with Normal Unvented Spoiler. o C  R  experiment  theory;  F i g u r e 31  R o l l i n g Moment C o e f f i c i e n t  f o r Rectangular  H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . o experiment  theory;  F i g u r e 32  R o l l i n g Moment C o e f f i c i e n t  f o r Rectangular  H a l f Wing o f  NACA 0015 S e c t i o n w i t h Normal Unvented S p o i l e r . theory; o experiment  0-4  AR = 7-73 s/c=0-48 h/cs 0-097  0-3  0-2  0-2  0 Figure 33  0-3  0  4  b  s  / b  E f f e c t i v e Moment Arm o f I n c r e m e n t a l L i f t Due t o Normal Unvented Spoiler  on R e c t a n g u l a r Wing o f NACA 0015 S e c t i o n .  o experiment;  A C  T  acting  a t midspan o f s p o i l e r  Theory;  0  5  81  10  0-9  0-8 R  AC  0 -L 7 0-6 h  0-5  0-4  AR=3-87 s/c=0-68 h/c = 0 097  0-3 h  0-2 0-1 \-  0-2  0 F i g u r e 34  0-3  0-4  b /b s  E f f e c t i v e Moment Arm o f I n c r e m e n t a l L i f t Due t o Normal Unvented Spoilers o  on R e c t a n g u l a r Wings o f NACA 0015 S e c t i o n .  experiment;  AC^ a c t i n g  a t midspan o f s p o i l e r  theory;  0-5  0 ;ure 35  I  1  01  1  0-2  1  0-3  1  0-4  I  0-5  I  0-6  I  0-7  I  0-8  t  y/b  Spanwise L i f t D i s t r i b u t i o n f o r R e c t a n g u l a r Wings w i t h S y m m e t r i c a l l y Deployed  0-9  Spoilers  I  10  028  Figure  37  Spanwise L i f t D i s t r i b u t i o n f o r R e c t a n g u l a r Wings o f NACA 0015  Section  0-28  0-24  r  4b U 0-20  016  NACA 0 015 AR = 3-87 s/c = 0 - 6 8 b /b = 0 - 4 h/c = 0 0 9 7 8=90° a= I RAD d/h=00 s  012  008  ®  UNSPOILERED  ®  ASYMMETRICALLY  ©  SYMMETRICALLY  SPOILERED SPOILERED  004  0-2 Figure 38  0-3  0-4  0-5  0-6  0-7  Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wing of NACA 0015 Section  0-8 y/b 0 - 9 oo  s/c  0-48 0-58 0-60 h  068 0-77  NACA 0015 AR= 5-80 b /b = 0-3 h/c= 0097 d/h=00 8=90°  symbol  A O •  s  V  0-45 h  0-40 \-  01  0-2  0-3  0-4  0-5  0-6  0-7  °- b /b °8  t  F i g u r e 41  Variation of C ^ with Spoiler Position  9  10  s  A l o n g Chord f o r R e c t a n g u l a r Wings w i t h  Spoilers  AR  NACA 0015  symbol  3- 87 4- 83 5- 80 6-77 7- 7 3  0-2 F i g u r e 43  Variation of C  O •  b /b=0-3 s  h/c= 0-097 8 =90°  o  0-7  0-8  w i t h A s p e c t R a t i o f o r R e c t a n g u l a r Wing w i t h  Spoilers  0-3 p b  s/c=0-68  0-4  0-5  0-6  0-9  IC  -0-70  symbol  s/ c  CLARK Y  AR=6I0  0-5  -0-65  12-9%  O •  0-6 0-7  b /b=0-3 s  h/c=0097 8=90°  -0-60  -0-55  -O50  -0-45  -0-40 0-2  0-3  0-4  0-5  0-6  07  08 .  t V a r i a t i o n o f C ^ w i t h S p o i l e r P o s i t i o n A l o n g Chord f o r R e c t a n g u l a r Wings w i t h b  u r e 44  .,09 /  b  s Spoilers  -0-65  AR  12-9%  symbol  s/c = 0-7  407 610 712  -0-60  CLARK Y  O •  b /b=0-3 s  h/c=0-097 8=90°  -0-55  0-50  -0-45  0-40 0 F i g u r e 46  01  0-2  Variation of C  p b  0-3  0-4  -L  0-5  w i t h Aspect Ratio f o r Rectangular  0-6  0-7  _L 08  0-9  Wings w i t h S p o i l e r s  U3  -0-7  AR  -0 6  symbol  NACA 0 0 1 5  3- 87  A  s/c = 0-48  d/h=00  4- 83  O  h/c = 0 0 9 7  8=90°  5- 8 0  •  6- 77  O  -0-5  -04  8  10  12  14  16  b /h  18  20  s  NACA 0 0 1 5 -0-7  Cpb  s/c = 0-58  d/h = 0 0  h/c=0097  8=90°  -0-6  -0-5  o -04  J  _l  0  2  4  6  o  I  I  I  L  8  10  12  14  \7 16  18  20  b /h s  F i g u r e 47  Variation of C  fe  w i t h S p o i l e r Span f o r R e c t a n g u l a r Wing w i t h Normal Unvented S p o i l e r 4>  -0-7  AR  Cpb  3-87  A  s/c = 0-68  d/h = 0 0  4-83  O  h/c = 0 097  8=90°  5- 80  •  -0-6  NACA 0015  symbol  6-77  o  7-73  -0-5  o  O' -0-4  0  i  1  i  i  2  4  6  8  V ° Q  •  o  0  v  i  i  1  I  i  10  12  14  16  18  b /h  1  20  s  NACA 0015 -0-6  A  -0-5  O  A •  -0-4  -0-3  O  10  d/h=00  h/c = 0 097  8=90°  •  o  O  8  s/c = 0-77  12  14  0  16  b /h  18  s  F i g u r e 48  Variation  o f C , w i t h S p o i l e r Span f o r R e c t a n g u l a r Wings w i t h Normal Unvented S p o i l e r s pb ,  20  -0-7  AR  Cpb  -0-6  12 9 %  symbol  4 07  A  s/c = 0-5  610  •  h/c=0l  712  O  m  0  • r-.  *  96  CLARK Y  0  o  •  -0-5 X 8  10  12  b /h  l 4  1 6  18  s  -0-4 12-9%  -o-7r  CLARK Y  s/c =0-6 h/c = 0 l  Cpb  -0-6  o  Q  -0-5  _L  0-4 8  10  12  1  4  b /h  1  6  18  s  12 9 % CLARK Y S/C = 0-7 h/c=0l  -0 7 Spb  o  -0-6  -0-5  -0-4 8 F i g u r e 49  10  12  l  4  b,/h  1  6  V a r i a t i o n of C ^ w i t h S p o i l e r Span f o r R e c t a n g u l a r Wings w i t h Normal Unvented  Spoilers  18  •i r Cpb =* Cpb  AO  0-9 A 0-8 I  0 0 1 5 , h / c = 0 097, d/h = 0 ,  NACA  O CLARK Y, h/c=0l, d/h = 0 ,  8=90°  0-7 0-4  0-5  06  =90°  07  0-8  0-9  0-8  09  J  S/C  10  1-2 6 G*  A O 10  A  09  08  J  _L  0-4  05  0-6  0-7  10 S/C  F i g u r e 50 V a r i a t i o n o f C , / C , and G/G* w i t h s/c  pb  pb  REFERENCES  Woods, L . C , "The Theory o f Subsonic P l a n e Flow", Cambridge P r e s s , 1961. Barnes, C.S., "A Developed Theory o f S p o i l e r s on A i r f o i l s , R & M, CP 887, J u l y 1965. J a n d a l i , T., "A P o t e n t i a l Flow Theory f o r A i r f o i l T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1971.  University  A.R.C.  S p o i l e r s " , Ph.D.  Brown, G.P., "Steady and Non-Steady P o t e n t i a l Flow Methods f o r A i r f o i l s w i t h S p o i l e r s " , Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1971. P r a n d t l , L., " A p p l i c a t i o n s o f Modern Hydrodynamics NACA Report No. 116, 1921.  to Aeronautics",  Jones, R.T., " C o r r e c t i o n o f the L i f t i n g L i n e Theory f o r t h e E f f e c t of t h e Chord", NACA T e c h n i c a l Note No. 817, 1941. G l a u e r t , H., "Elements o f A i r f o i l U n i v e r s i t y P r e s s , 1927.  and A i r s c r e w  Theory",  Cambridge  W i e s e l s b e r g e r , C , " T h e o r e t i c a l I n v e s t i g a t i o n s o f the E f f e c t o f the A i l e r o n s on t h e Wing o f an A e r o p l a n e " , NACA TM 510, 1928. B e r n i e r , R., "Steady & T r a n s i e n t Aerodynamics o f S p o i l e r s on A i r f o i l s " , M.A.Sc. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1977. P a r k i n , B.R., " L i n e a r i s e d Theory o f C a v i t y Flows i n Two Dimensions", RAND R e p o r t , P-1745, 1959. B i o t , M.A., "Some S i m p l i f i e d Methods i n A i r f o i l Theory", J o u r n a l o f the A e r o n a u t i c a l S c i e n c e s , 9, 1942, p l 8 5 . Hoerner, S.F., " F l u i d Dynamic Drag", P u b l i s h e d  by A u t h o r , 1965.  Pope, A. & Harper, J . J . , "Low Speed Wind T u n n e l T e s t i n g " , W i l e y , 1966  APPENDIX  RlXi K  =  2R1  Jla  Ii(l+ a  /  i9 n  e  -18, -, \ -1)  6iSin6 +  2  — .iS Oo n  ,  (a e  ( a - %M -  2  a -e o  a  2  , , x\2 -1)  - i 6  -e  ±1  2  ±6i  a  ImX RlXi 2  n=l  a  - i 6 a e -1 o - i 9 o -. \  i9„. .  a  e  n+1  -19,  a e -1 o o _ "n - 1 , i9n o o " (  - i 6 i a -e  nM  19,  a  I E Nn n=0  -  2  Q  E  2  1-n  oo  o  RlX ImXi  , a -e i— o  TT  o  ImXiReX2  2  n TT  i 9  ImXiRlE  R1X [ImE-(a-%M )]-ImX RlE  °°  n  ) ] -  Sin6  02-TT  + TT  TT  -  RlXiImX  o  i9o (a  [ ImE  ^ )  1  )  0  (  a  o  e  _  1  )  -19  19,  ImX2 Rl RlX ImXi-RlXiImX 2  ImX r RlXiImX2  where  2  (a e u. o  1  (1 + i _ - ImXiRlX  2  a  ) Q  1  9  o  - l )  2  (a  o  e~  i 6  °-l)  2  -  (29)  100  h  2 1-b l  1+b  o  a, = 1  jiia  1+b  +  ia  2  ps  =  -2  C  +  o  ln  TT  0i • + •  M  n  S i n n9  1  Im  1B  a (1 +  Cos  K  C  0  o  +  (30)  (0 -7T)n  0iSin6  2  + —  TT  o  TT  +  , i 6 ,N 2 ( a e -1)  (a  o  o  e-  ±  e  o  la.  - l )  , a - ei e <- o  TT  0 0  a  o  -e  -i6i  TT  1  a  o  -e  ie  2  •,  l-n +  iJla.E I  n  n=0  SinG  -ie, 0  L  o  n0 s  -  1  10, la.  4B  fi  SinJ2(0+6 )  Sin  Q  a  ^ + Cos6  s±nh\0-0 2  In  n  ls= IT | o* l V -  C  Sin8 •z Cos9o~  2  E N ° Cos0o  +  1+b  TT  2n  1  2ia  Sin6  Sin%(0+0i)  0 0  2E  +  ]  " J  1+b J  1-b 1+b  ' s i n ^ | e-e.i |  2Sin6  2ia 1  f J^b  L I 1+b  1  1+b  n(e -Tr) C  [  a  N 1 1  o n-l  a  e  a (a  o e  e i e  ie  i8o  o  -e  +  -ie  i£a E n=l  nM  1  2  a  n+1  -ie,  0  ,  -1  °-l)(a  a  o  e  e-  -ie _  i  0  9  o  -l)  1  (31)  101  2  4TT  Im  mo s  iC  a  £ a„ o 2  <  o o  i£ a  E n=l  2  o V  a  M  i£ a„ E N n=0  n  1  + i B I a_ o 2  - 1  )  ?  - 1  r-l)a o o L  (a  •ft,  r  l o  1  -  - 1  2  a  a  n n+1 a. o  a  - 1  <=±i> a  o 2  o 2 a  n-1 n a o  a o -5 o a o -C o  2 l  a  _  (n-1)(n-2) n+1 2! a'  a a o 2 0  a*(n-l) a _ ( a -£ ) a ( a -£ ) 2 o o o o o n  a  2  o  a  o  n-1, 2 (a -£ ) o o o x  - 1 o  ( a  " o ?  )  a  2  a  9  (  z.  a  „ - ? ^ )  o  o  / >- \ 2 , - \ 2 n-1 a ( a -£ ) (a ) a 2 o o o o o 0  a  ?  ) a  " (  a  o  „ " ?  o  a  r  t  o  )  n-2 . - . 2 ( a - £ ) ( a - £ ) o o o o o w  + £ a,  - 1 ( o- o  a  2  2  a  <  a  l^o o V  a  -1 1  )  a  2  (a crC o - 1 )  (  a  l^o o o?  -1 1  )  a  2  2 f l  102  + ^  £ a 2 2  9  TT  (  +  2(a -C ) a  2  2(a -£ ) a  2  2  o  1  a  o ^ l  5  ( a  >  2  TT  o~V  < a  o- 2 5  1  o  )  ( a  Z  o- 2> ?  (32) (a -C ) a  2  o  2  where  a  2  ia 1+b  f  I  a  2 4  Q  2  Q  2  2  and a ^ a r e t h e same as i n e q u a t i o n  r  2ia 1+b « . "2ia .1+b  3 1-b 4 1+b , . -.2 1-b] 1+bJ  ia 1+b  f l l a  L1+b  +  (29) and  l^bl  1+bJ 12  

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