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A theoretical investigation of a low-correction windtunnel wall configuration for airfoil testing Malek, Ahmed Fouad 1979

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A  T H E O R E T I C A L  W I N D T U N N E L  I N V E S T I G A T I O N  W A L L  O F  C O N F I G U R A T I O N  A  L O W - C O R R E C T I O N  F O R  A I R F O I L  T E S T I N G  by  AHMED FOUAD MALEK B.Sc.(Eng.),  Alexandria  University,  1970  B.Sc.(Math.),  Alexandria  University,  1974  -A T H E S I S  SUBMITTED I N P A R T I A L F U L F I L M E N T  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D  SCIENCE  in  THE FACULTY OF GRADUATE (Department  We a c c e p t to  of  Mechanical  this  the  thesis  required  THE U N I V E R S I T Y  ^  Engineering)  as  conforming  standard  OF B R I T I S H  August  STUDIES  COLUMBIA  1979  Ahmed F o u a d M a l e k ,  1979  OF  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r  an advanced d e g r e e a t the 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 , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s .  for extensive  study.  copying of t h i s thesis  be g r a n t e d by the Head o f my Department o r It i s understood that copying or p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written  permission.  Department The U n i v e r s i t y o f 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  DE-6  BP  75-51  1E  A B S T R A C T  This corrections ally  spaces  wall  the This  free  air  symmetrical slats  are  wall  configuration  method,  airfoil  velocities  calculated  contour. in  This the  with  moment  reduce  wall  employing elements  zero  symmetric-  of  the  incidence,  increasing  uses  the  the  linearly  are  is  determined  the  negligible  or.  flow  is to  very use.  and t h e  by n u m e r i c a l  in  around order  analysis.  the to  to  small  surface  Conditions  simple  distributions  in  close  method  directly  can be d e v e l o p e d  plenum chambers  Kutta  This it  to  conditions  potential  Load"  slats. and  flow  leads  calculated  pressure  method  which  evidence  can be  provides  "Full  and w a l l  by p h y s i c a l  and p i t c h i n g  the  at  to  by  solid  nonuniform,  The t h e o r y  surface  The  airfoils  corrections.  test  approach  testing,  walls.  environment  supported  flow  slotted  the  anew  airfoil  test  on t h e  of  with  rear.  vortex-element  l i f t  deals  high-lift  are  between  towards  wall  in  transversely  slotted  the  thesis  satisfied well The  aerodynamic integration airfoil  include  the  -iiTABLE OF CONTENTS Page Abstract  i  Table o f Contents  i i  L i s t of Figures  i i i  Symbols  v  Acknowledgement  vi  I - ..INTRODUCTION  1  II -  POTENTIAL FLOW ANALYSIS  4  II-l  Introduction  4  II-2  Surface S i n g u l a r i t y Theory  5  II-3  The Kutta C o n d i t i o n  8  I I I - METHOD OF SOLUTION AND NUMERICAL ANALYSIS  10  IV -  THE FLOW IN THE WINDTUNNEL PLENUM  15  IV-1  Introduction  15  IV-2  The Induced T a n g e n t i a l and Normal V e l o c i t i e s  IV-3  Due t o V o r t i c i t y D i s t r i b u t i o n  15  Free Streamline  17  Tracking  V -  RESULTS AND DISCUSSION  19  VI -  CONCLUSIONS  23  References  25  APPENDIX 1 - E v a l u a t i o n o f the I n t e g r a l i n Equation (5) APPENDIX 2 - C a l c u l a t i o n of the V e l o c i t y Component Induced at a P o i n t i n the F i e l d o f Vortex D i s t r i b u t i o n  26 30  Figures  33  -iii-  LIST OF FIGURES Page Figure 1  Comparison o f A i r f o i l  Pressure  Coefficients:  Theory, Ref. (3) Figure 2  Vortex  Figure 3  N o t a t i o n Used t o C a l c u l a t e I n f l u e n c e C o e f f i c i e n t s  35  Figure 4  Streamline  36  Figure 5  Location of T r a i l i n g Control Point  37  Figure 6  L o c a t i o n o f Elements on A i r f o i l  38  Figure 7  Comparison o f Higher Order Methods, Ref. (6)  39  Figure 8  Comparison o f A i r f o i l V e l o c i t y D i s t r i b u t i o n s , Ref. (6) An A i r f o i l I n s i d e Tunnel Test S e c t i o n with Double  40  S l o t t e d Wall  41  Figure 9  Representation  33 o f Two Component A i r f o i l  Contours Around an A i r f o i l  Surface  34  F i g u r e 10 N o t a t i o n Used t o C a l c u l a t e Induced V e l o c i t i e s  42  F i g u r e 11 Geometry o f Higher Order Terms F i g u r e 12 Comparison o f Pressure C o e f f i c i e n t s f o r NACA-0015 A i r f o i l i n Free A i r and Between S o l i d Walls Test Section  43 44  F i g u r e 13 Comparison o f Pressure C o e f f i c i e n t s f o r NACA-23012 A i r f o i l i n Free A i r and Between S o l i d Walls Test Section  45  F i g u r e 14 Comparison o f Pressure C o e f f i c i e n t s f o r CLARK-Y 14% A i r f o i l i n Free A i r and Between S o l i d Walls Test S e c t i o n  46  F i g u r e 15 Comparison o f Pressure C o e f f i c i e n t s f o r NACA-0015 A i r f o i l i n Free A i r and Between S i n g l e Uniformly S l o t t e d Wall Test S e c t i o n  47  F i g u r e 16 Comparison o f Pressure C o e f f i c i e n t s f o r NACA-0015 A i r f o i l i n Free A i r and Between Double Uniformly S l o t t e d Wall Test S e c t i o n  48  -ivPage F i g u r e 17 Comparison o f t h e E f f e c t o f Test A i r f o i l the R e l a t i v e E r r o r i n L i f t C o e f f i c i e n t s  S i z e on  F i g u r e 18 Comparison o f t h e E f f e c t o f Test A i r f o i l S i z e on the R e l a t i v e E r r o r i n P i t c h i n g Moment C o e f f i c i e n t s  49 50  F i g u r e 19 Comparison o f P r e s s u r e C o e f f i c i e n t s f o r NACA-0015 A i r f o i l i n Free A i r and Between Double Non-uniformly S l o t t e d Walls Test S e c t i o n w i t h the Flow i n the Plenum 51 F i g u r e 20 Comparison o f P r e s s u r e C o e f f i c i e n t s f o r NACA-23012 A i r f o i l i n Free A i r and Between Double Non-uniformly S l o t t e d Walls T e s t S e c t i o n w i t h the Flow i n the Plenum 52 F i g u r e 21 Comparison o f Pressure C o e f f i c i e n t s f o r CLARK-Y 14% A i r f o i l i n Free A i r and Between Double Nonu n i f o r m l y S l o t t e d Walls Test S e c t i o n w i t h the Flow i n the Plenum  53  F i g u r e 22 Comparison o f t h e E f f e c t o f Two D i f f e r e n t Angles o f A t t a c k on the R e l a t i v e E r r o r i n L i f t C o e f f i c i e n t s f o r NACA-0015 A i r f o i l 54 F i g u r e 23 Geometry o f the I d e a l i z e d Flow i n the Plenum Chamber f o r NACA-0015 A i r f o i l  55  -v-  SYMBOLS Symbol  Definition Uniform  flow  velocity.  Stream f u n c t i o n .  Y  Vortex d e n s i t y or induced s u r f a c e v e l o c i t y .  K.  I n f l u e n c e c o e f f i c i e n t o f the element " j " on the c o n t r o l point " i " .  N  Number o f v o r t e x  M  Number o f l i f t i n g  11  elements. bodies o r components.  Relative error i n l i f t E_ = (C  T  - C L  T  F  )/C F  * 100  T  L  R e l a t i v e e r r o r i n p i t c h i n g moment c o e f f i c i e n t .  M  E  M  "  ( C  Pressure Lift Mc/4  MC/4  " MC/4 '/ L C  T  C  F  F  *  1  0  0  coefficient.  coefficient.  Quarter chord p i t c h i n g moment c o e f f i c i e n t . Free a i r data.  < >P ( )  T  coefficient  T  Windtunnel data.  UOAR  Uniform  Open Area R a t i o .  AOAR  Average Open Area R a t i o .  c/C  S l a t chord:  C/h  Airfoil  airfoil  chord:  chord r a t i o .  tunnel t e s t  section height r a t i o .  -vi-  ACKNOWLEDGEMENT T h i s r e s e a r c h was c a r r i e d out under the s u p e r v i s i o n o f Dr. G.V. Parkinson, whose e x p e r t advice and guidance i s g r a t e f u l l y acknowledged. A l l the computing was done a t the U.B.C. Computing T h i s r e s e a r c h was supported E n g i n e e r i n g a t U.B.C.  Center.  by the Department o f Mechanical  - 1 CHAPTER I - INTRODUCTION T r a d i t i o n a l l y , t h e subsonic measurements o f the aerodynamic c h a r a c t e r i s t i c s o f a i r f o i l s i n windtunnels  with s o l i d walls r e -  q u i r e d but small c o r r e c t i o n s by the standard methods, provided t h a t the t e s t a i r f o i l s were s m a l l r e l a t i v e t o the t e s t c r o s s - s e c t i o n s and developed  r e l a t i v e l y small l i f t  section  coefficients.  Now, however, w i t h the frequent t e s t i n g o f very h i g h l i f t s e c t i o n s , u s i n g l a r g e models which produce r e a l i s t i c a l l y Reynolds'  high  numbers, the standard w a l l - c o r r e c t i o n theory i s no l o n g e r  a c c u r a t e u n l e s s windtunnels are used.  airfoil  w i t h very l a r g e t e s t c r o s s - s e c t i o n s  Since the c o s t o f c o n s t r u c t i o n o f windtunnels o f  s u f f i c i e n t s i z e i s not economical, develop a windtunnel  the a l t e r n a t i v e i s t h e r e f o r e t o  w a l l c o r r e c t i o n theory f o r subsonic  which can answer t o the c a l l  testing  f o r r e a l i s t i c measurements and  accurate c o r r e c t i o n s . C o n t r i b u t i o n s t o such a theory a r e presented i n t h i s It  i s w e l l known t h a t most c o r r e c t i o n s t o data i n windtunnels  open j e t s a r e o p p o s i t e i n s i g n to those i n windtunnels are s o l i d , Ref. (1). These opposing of  employing  with  whose w a l l s  e f f e c t s suggest the s t r a t e g y  p a r t l y s o l i d , p a r t l y open w a l l s i n p u r s u i t o f can-  c e l l i n g the c o r r e c t i v e e f f e c t s o f the two types o f w a l l . two  thesis.  such designs f o r t h i s purpose have been c o n s i d e r e d .  w a l l s w i t h narrow l o n g i t u d i n a l s l a t s .  Recently, One has  The other has w a l l s  p a t t e r n e d w i t h small h o l e s . Using the l i n e a r theory t o i n v e s t i g a t e those two types o f . w a l l c o n f i g u r a t i o n s , Parkinson and Lim (2) and o t h e r s have found t h a t t h e r e i s a l a c k o f agreement between the experimental  results  and t h o s e w h i c h slotted is  not  must  wall.  are predicted Also  simply  they  by t h e  have  an e m p i r i c a l  be d e t e r m i n e d  impossible  2  found  situation  for  that  function  empirically the  for  theory  of  for  the  the  the  "porosity  open area  each a i r f o i l  practical  longitudinal  use o f  parameter"  ratio  under  porous  but  test, wall  it  an config-  uration. Parkinson the  and L i m  longitudinal  experimentally flows  are  not  slats  of  (2)  and p o r o u s  separated  accounted  nonlinearities  separations  seriously  another  type  slotted  wall,  theoretical ation  to  the  the  lack  theories  the  slats  theories,  them.  degrade  will  of  In  main  to  success  primarily  in  Such  as t h e y  those  the  of  occurrence  and h o l e s .  addition, flow  of  the  add  flow  vincinity  with  uniform  that  Fig.  such.a wall  separation  on t h e  vorticity  the windtunnel  Also  the  has  of  thesis, method,  of  his  type  of  tunnel  the  wall.  wall  test  will  configur-  section. l i f t  values,  moment d a t a  provides  His  produce  free-air  experimental  investigated  transversely  slotted  section  pitching  configuration  transversely wall  this  of  error  and t h e  transverse  present  both walls  using  such a t e s t  (1).  shows t h a t  the  and a l o w e r  the performance  distribution  accuracy,  a surface  gaps,  shows t h a t  predicts  source method,  one w h i c h has an u p p e r  a small percentage  pressure  In  a surface  windtunnel,  improve  data w i t h i n  lower  (3), u s i n g  analysis  The a n a l y s i s  with  in  in  the  walls. Williams  the  wall  flows  for  undesirable  the  have a t t r i b u t e d  are  while  of  ..  investigation  a flow  free  of  slats. a surface is  used t o  slotted  configuration  in  singularity  analysis,  investigate  a  as a n o t h e r order  to  using  windtunnel  approach to  recreate  the  modify free-  - 3 air  s t r e a m l i n e p a t t e r n s about the t e s t a i r f o i l , which would then  experience the corresponding f r e e - a i r l o a d i n g . The approach here uses symmetrically t r a n s v e r s e l y - s l o t t e d upper and lower w a l l s , with symmetrical at  zero i n c i d e n c e .  The  airfoil-shaped solid  flow i n c l i n a t i o n s near the w a l l w i l l  small f o r a l l p r a c t i c a l cases envisaged.  slats be  Hence a l l the w a l l  s l a t s , w i l l operate w i t h i n t h e i r u n s t a l l e d i n c i d e n c e range,  so  t h a t flows near the w a l l w i l l be f r e e of separated wakes. A uniform s p a c i n g of the w a l l s l a t s shows t h a t the upper s u r f a c e of the a i r f o i l  (see F i g . ( 1 6 ) )  s e c t i o n i n the presence  of the  s l o t t e d w a l l tends t o experience a s l i g h t l y lower n e g a t i v e pressure d i s t r i b u t i o n than t h a t of the f r e e - a i r c o n d i t i o n s near the l e a d i n g edge and  s l i g h t l y higher f u r t h e r a f t .  for  but l e a d t o a p p r e c i a b l e e r r o r s i n p i t c h i n g moment.  lift  The e f f e c t s tend t o c a n c e l  s o l u t i o n t o t h i s problem i s to use graded, of  A  narrow gaps upstream  the t e s t a i r f o i l and wider ones downstream, r a t h e r than  uniform  spacing between the w a l l s l a t s , as shown i n F i g . (9a). A l s o the s u r f a c e s i n g u l a r i t y a n a l y s i s has been used here to t r a c k the f r e e s t r e a m l i n e s which enter or leave the t e s t  section  from or t o the upper and the lower plenum, r e s p e c t i v e l y , i n order to  r e p r e s e n t the flows i n t h e r e .  - 4 CHAPTER I I II-l  -  -  POTENTIAL FLOW A N A L Y S I S  Introduction An e f f i c i e n t ,  distribution formal  on t h e  reliable surface  transformation  analyze  sections  of  the  theorem which  the  potential  states  around a c i r c l e .  is  such theorem  of  analysis.  side  the  sources  The component  surface  transformation surface  This  method uses  of  the  to  they  have  form of  the  upper  about the  sources  his  and l o w e r  and,  of  the  cases.  and s i n k s  One  conditions. with  widely (5).  on t h e  surface to  employed which  particular  the  Kutta  magnitudes  at  the  control  Condition, points  = -U ) . t 1 u  n  multi-  conformal  has  of  e d g e e l e m e n t s J (U t  out-  distribution  application  trailing  methods  field  The t e c h n i q u e  some d r a w b a c k s .  there  around  Hess and S m i t h  among o t h e r s ,  model.  as  The m o s t  Williams  velocity  potential  singularities,  a vorticity  however,  the  flow  combined w i t h  windtunnel  transform  easily  than  on  to  same b o u n d a r y can d e a l  can  based  field  of  Con-  (4)  singularity  a set  accurate  (3),  are  simple  potential  of  from t h e i r  equal  the  a distribution  section  into  surface  that  used h a s ,  the  to  methods less  required.  possible  potential  is  investigate  problem arises  no  is  velocity  Theodorsen  not  method  circulation.  this  satisfy  are  on s i n g l e - c o m p o n e n t  singularity  airfoil  generate  and a r e  methods  used  that  of  contour  the  replace  with  which  sections  always  one l o o k s  singularity  sections  is  transforming  contour  vortices,  it  any c l o s e d  These methods  calculating, the  These methods  Such methods  sections,  airfoil or  for  airfoil  shape.  that  around  for  s u c h as t h a t  arbitrary  field  multi-component  of  methods  field no  method  This  in  of can  _ 5_ be c a l l e d the NO LOAD Kutta C o n d i t i o n s i n c e i t e l i m i n a t e s any lift  from the a i r f o i l near the t r a i l i n g edge which i s i n c o n f l i c t  with the aim o f examining s e c t i o n s w i t h l a r g e r e a r l o a d i n g . overcome t h i s problem a b e t t e r method must be  developed.  Recently a d i f f e r e n t s u r f a c e s i n g u l a r i t y method was by Kennedy  developed  (9). T h i s method uses a d i s t r i b u t i o n of v o r t i c e s on  the s u r f a c e o f the a i r f o i l determined  To  directly,  section.  The v o r t e x d e n s i t y , which i s  i s equal t o the s u r f a c e v e l o c i t y .  As i n the  a i r f o i l a n a l y s i s the boundary c o n d i t i o n which i s a p p l i e d here i s t h a t the s o l i d s u r f a c e s o f the a i r f o i l the stream  s e c t i o n are s t r e a m l i n e s and  f u n c t i o n s are r e q u i r e d t o be constant.  II-2 - Surface S i n g u l a r i t y Theory In two dimensional, i n c o m p r e s s i b l e , i r r o t a t i o n a l flow the stream  f u n c t i o n must s a t i s f y Laplace's ifi 2 dx  +  iJL - n 2 dy  For the flow over a i r f o i l at  equation, (i)  s e c t i o n s t h e r e can be no normal v e l o c i t i e s  the s o l i d s u r f a c e s , and thus each s u r f a c e i s a s t r e a m l i n e of  the flow.  Since the stream  f u n c t i o n s if)  (K=l,2,....,M) on the  is. s u r f a c e s o f M components on a multi-component s e c t i o n are c o n s t a n t s , the boundary c o n d i t i o n f o r equation if> = \b The  v  , on the s u r f a c e  stream  (2)  f u n c t i o n f o r a uniform stream  p o s i t i v e X a x i s a t an angle a if)  which s a t i s f i e s equation  i n c i d e n t t o the  i s g i v e n by  = y cos a - x s i n a  equations  (1) can be w r i t t e n as,  (3)  (1). T h i s equation, and a l l subsequent  are i n dimensionless  form.  The d i s t a n c e s length Uoo  stream  The p o i n t stream  -  are dimensionless  C , the v e l o c i t i e s  and t h e  6  with  functions  vortex  of  with  respect  with  to  the  respect  strength  r,  respect  to  free the  located  to  the  system  product  at  chord  (X ,  UooC.  y  q  velocity  ) has  the  function ^  where  "  =  r  2T  £  = I(x-x l  equation  (4)  linearity  o  2  '  )  (  +  (y-y  side  S.  S,  (1)  having  uted  the  over =  )  except  them as  satisfy  in  £ n  s  at  S,  (  r  p  is s  Because  point (2),  (1)  that  in  lies  the  )  S'  and c o n t i n u o u s l y  s  the or  on  point  any the  region  a general  d  of  vortices  at  given  ' '>  r=0.  of  equation  Y(S')  surface  f'V( ')  at  Fig.  stream f u n c t i o n  airfoil  4  1/2  any c o l l e c t i o n  a density  2T  2 '  (1),  of  will  Then t h e  vorticity  o  satisfies  distribution  surface,  ^P  )  r  equation  airfoil of  (  also  of  continuous  n  out-  P due  to  distrib-  by  '  (5)  S Applying f l o w due t o vorticity, %  the  a uniform one  =  y  S  °  O  S  surface  elements.  located  at  made t o  distributed whole  (x^,  stream plus  X  S  S  surface  i  n  is  On e a c h o f  y^),  at  a  the  ~ ITF divided these  which the  Each element j  on i t s S,  ~  a  apply.  surface  condition,  equation  (2),  the  above d i s t r i b u t i o n  combined of  the  obtains  The a i r f o i l  is  boundary  is  surface. then  ^ ^ ( S ' ) S ' up i n there  boundary  some m a n n e r is  a control  condition,  has v o r t i c i t y  The i n t e g r a l  replaced  £n r ( S , S ' )  by  in  of  into  N  (6) N  small  point, C ^ , equation  density  equation  a summation o f  d S'  (6),  (6),  Y (Sj) over  integrals  the  over  the  point, iK k r  N surface  •N E . ,  +  one  C^,  ^— 2ir  /  y (S !) ]  j The r e s u l t s surface at  the  interior  to  in  the  the  the  velocity.  In  for  At  section  form of to  the  this  points each  at  over  the  1  of  equation  (6)  (6)  at  the  which  vortex  across  sheet.  solving  the  the  control  the  location  to of  elements  straight  element  midpoints  in  so t h a t  the  sheet is  these  discontinbeing  equal  equal  to  the  one t h e r e f o r e  control  The s i m p l e s t  and  (7) the  solves  surfaces.  • the  j .  a  velocity  m a k e some a s s u m p t i o n  each element are  sin  l  method are  zero  (S^) (7)  airfoil  necessary  be  a vortex  Thus Y  x.  tangential  results  equation  on t h e  the  has t o  also  a -  analysis  shows t h a t  surfaces  velocity  is  d S ! = y.cos ] ^i  3  an a i r f o i l  solid  it  geometry,  assume t h a t  ( C . , S !)  velocities  point  y(Sj)  r  streamlines,  of  directly  the  the  tangential  density  surface  Applying  Kennedy  of  become  In  required  velocities.  surfaces uity  elements.  -  obtains,  / S  3  7  lines  Y (Sj)  is  points  about  and  the  approximation  with  is  control  a constant  over  element. Using the  each c o n t r o l  above  point  approximation  yields  the  and a p p l y i n g  system of  equation  (7)  at  equations,  N \ where  K^j  control uated  at  airfoil  . E, j=l  +  E  is  K  i j  the  point  i,  Y  j  (  i  1,  =  right  is  K.  the i  notation  c a n be  '  R^  component  efficients  i  coefficient  point  the  R  influence  control  Using  =  written,  Fig.  N)  of  hand s i d e  a n d ip i s K  of  ...  the  (3),  (8)  the  element  j  on  the  of  equation  (7)  eval-  stream  function  for  the  the  BASIC i n f l u e n c e  co-  - 8 -  K..  = j- { (b+A)  J  in  (A)  -  + 2a tan  (b-A) £n • (A  1  1  ~) - 4A  (-~^ a +b  -A  (9) The  d e t a i l s of the c a l c u l a t i o n of t h i s equation  are  provided  i n Appendix 1 . The  K^j and  are p u r e l y f u n c t i o n s of the geometry of the  s u r f a c e elements and the angle of a t t a c k . (8) i s a s e t of N equations  system of  equation  f o r the N unknown y. and M unknown IJJ, ,  where t h e r e are M components. for  The  The M a d d i t i o n a l equations  required  a s o l u t i o n to t h i s problem are termed the Kutta C o n d i t i o n s  there i s one  f o r each component i n the t e s t a i r f o i l  each a i r f o i l  s l a t i n the w a l l , f o r the cases to be  and  s e c t i o n , and considered  later. II-3  - The  Kutta  Condition  Kutta and Joukowski were concerned with a i r f o i l whose geometries are c a l c u l a t e d by a conformal technique  sections  transformation  which maps the flow over a c i r c u l a r c y l i n d e r i n t o the  flow over an a i r f o i l s e c t i o n s have two  s e c t i o n w i t h a cusped t r a i l i n g  s t a g n a t i o n p o i n t s , one  edge and the other near the t r a i l i n g the t r a i l i n g  edge w i l l be,  A l s o the v e l o c i t y at  infinite.  They both proposed t h a t the c i r c u l a t i o n around the c y l i n d e r be adjusted  so t h a t one  edge.  In t h i s case the i n f i n i t e  s t a g n a t i o n p o i n t , o c c u r r i n g together  i n t o the  airfoil  v e l o c i t y and  at the t r a i l i n g  and y i e l d a f i n i t e , non-zero v e l o c i t y t h e r e . Milne-Thompson  circular  of the s t a g n a t i o n p o i n t s i n t h a t  flow be l o c a t e d at the p o i n t which w i l l map trailing  These  l o c a t e d near the l e a d i n g  edge.  i n general,  edge.  the  edge, c a n c e l  I t has been shown by  (7) t h a t a consequence of t h i s assumption i s t h a t  - 9 the s t a g n a t i o n t o i t and Tietjens  streamline  l e a v e s the cusped t r a i l i n g  edge tangent  photographs o f flow v i s u a l i z a t i o n s t u d i e s o f P r a n d t l (8) show t h i s e f f e c t c l e a r l y , see F i g .  and  (4).  T h i s c o n d i t i o n can be modelled by p r o v i d i n g an a d d i t i o n a l c o n t r o l p o i n t j u s t o f f the t r a i l i n g was  edge.  used s u c c e s s f u l l y by Bhateley and  of the t r a i l i n g point placed  Bradley  (9).  The  a small f r a c t i o n of chord downstream of the (5).  through the other  bisector  to these t r a i l i n g  a control trailing  I t i s then assumed that  the  c o n t r o l p o i n t s o f t h a t component  a l s o pass through t h i s c o n t r o l p o i n t .  can be w r i t t e n  Condition  edge i s extended i n t o the f r e e stream and  edge, as i t i s shown i n F i g . streamlines  Such a Kutta  control points, C  t  Equation , and  (8) then a p p l i e s  the Kutta  Condition  as, N  +  m  Z  K  j=l  tD  tp  m  i  ,:i  Y  V  = D  R  tD m tp  There are M such t r a i l i n g component, and  '  (m = 1 ^ 2" " ' M) '  ( m  1  c o n t r o l p o i n t s , one  hence M Kutta C o n d i t i o n  <> 10  M  f o r each  equations.  Thus the problem of p o t e n t i a l flow over an a i r f o i l has t h e r e f o r e been reduced to t h a t of s o l v i n g p r e s c r i b e d by equations  (8) and  vortex d e n s i t i e s , y • , and  (N+M)  (10), simultaneously  M stream f u n c t i o n s  ik.  section  equations, to get N  - 10 CHAPTER I I I  -  The which this the  METHOD OF SOLUTION AND NUMERICAL  first  describe is  to  let  surface  step  in  the  airfoil  the  supplied  elements.  insufficient  To o v e r c o m e  from i t s  leading  edge a t  of  (5),  co-ordinates  xx  the  = |  £  where  "  h  2  =  has  to  points  at  the  the  the  airfoil  may b e  that  doing  end p o i n t s  or  are  they  airfoil  is  there  may  as  be  irregularly  divided  edge a t  located,  of  up,  x=l.  The  shown i n  Fig.  by,  in  trailing  general,  control  ,N)  largest  determined  interpolation  by  a cubic  spline  method,  since  can be e a s i l y  function it  gives  (11)  element, edge  is  as  the  on t h e  has been  found  points  are  shown i n  extended,  as  leading  most  through  the  given  the  trailing The  are  data.  the  it  The use  as t h e  points  U.B.C.  (6), t h e n  shown i n  Fig.  mid-points the  of  reliable and  computer  interpolation.  taken Fig.  airfoil  of  occur.  end p o i n t s  Here  point  because  edge and  generally  to  end  distribution  be  computed. for  the  solution  element  given  smooth curves  SAINT has b e e n u s e d  The c o n t r o l  the  that  This  gradients  y£ of  and e f f i c i e n t l y  order  accurate  near  velocity  in  edge.  a more  points  co-ordinates  trailing  method o f  that  (£ = 0,1,2,  corresponding  surface  elements  I T  concentrates  subroutine  the  TT£  provides,  edge w h e r e  be t h e  trailing  Here N must be an even number be l o c a t e d  define  disadvantage  the  its  elements  ,  £  the  problems  given  (l-cos<J> )  to  co-ordinates  x=0  surface  is  One o b v i o u s  available  these  end p o i n t s  solution  surface.  This  co-ordinates  spaced.  at  the  ANALYSIS  of  bisector  (5), and t h e  each of  the  control  - 11 p o i n t i s l o c a t e d on t h i s e x t e n s i o n a d i s t a n c e O.Olt from the t r a i l i n g edge.  T h i s d i s t a n c e was  found to g i v e the most r e l i a b l e  r e s u l t s f o r a wide range of a i r f o i l s e c t i o n s . The a i r f o i l  system of axes x-y should be r o t a t e d c l o c k w i s e  an angle a, the angle of a t t a c k , then the c o - o r d i n a t e s o f the element end p o i n t s w i t h r e s p e c t to the wind system of axes w i l l be g i v e n  X-Y  by,  X. = x. cosa + y.  sina  Y. = y. cosa - x. l 1 l  sina  Having determined  the c o - o r d i n a t e s o f the element end p o i n t s  1  1  1  1  2  (12)  and c o n t r o l p o i n t s one can proceed  to c a l c u l a t e the BASIC i n f l u e n c e  c o e f f i c i e n t K ^ j , which i s g i v e n by equation  (9), and R^,  a r e s u l t of the r o t a t i o n of the a x i s should be g i v e n  which as  by, (13)  R. = Y. l l  As the v o r t e x d e n s i t i e s are i d e n t i c a l to the s u r f a c e v e l o c i t i e s counter-clockwise  about the a i r f o i l  s e c t i o n , the c o - o r d i n a t e s of  element end p o i n t s and c o n t r o l p o i n t s should be taken i n t h a t order around the p o l y g o n a l  contour.  For a single-component a i r f o i l equations  (8) and i=l  K  i n f r e e a i r , the system of  (10) can be w r i t t e n i n the matrix form as,  1,1  K  1,N  1  Y-  R,  (14)  i=N i=N+l  K  N,l  *  •  '  K  N,N  Kutta C o n d i t i o n  1  YN  R  tp  The above N vortex  system o f  densities  When t h i s sections scaled  the  to  applied. correct the  It  y. a n d t h e  technique  point  the  each  This  edge o f  through which the defined,  is  extended  is  the  given  individual  then necessary  location.  leading  of  to  component  one can c a l c u l a t e  then  solved  for  the  is the  to  multi-component  by  equation  component  (11)  unknown  specifying is  the  translated  rotated.  airfoil  is  before  move e a c h c o m p o n e n t  done by  component  is  s t r e a m f u n c t i o n tf>, .  distribution,  chord is  equations  first  being  to  its  amounts  and t h e  by  angle  With the . geometry  thus  ;  K..  a n d R.  l j  1  which  from equations  (9)  and  •  (13). The m u l t i - c o m p o n e n t function  for  Condition. ponent  gives  each component A two-component rise  to  i=l  K  l , l  i=N  K  K  i=N+l  case g i v e s  rise  to  a different  and each component airfoil  a system o f  with  has  N elements  equations 1,2N  1  0  Y  l  N,1  K  N,2N  1  0  Y  N  N+1,1  K  N+1,2N°  1  Y  *  '  l=2N+2  2N,1 ' * ' 2N,2N ° K u t t a c o n d i t i o n , component K  Kutta  K  c o n d i t i o n , component  i 1 2  Y  Kutta  on each com-  R  written: l  *N  N+1 =  •  i=2N  own  w h i c h can be  K  *  its  stream  2N  *  1  *  2  (15) *N+1  R  2N  tp  2  The U . B . C . solve  the  computer  system of  the dimensionless  13  subroutine  equations  surface  -  (14)  FSLE h a s b e e n u s e d h e r e  and  velocities  stream functions  of  distribution,  l i f t  and t h e  moment  c a n be c a l c u l a t e d  The  the  control  at  dimensionless  coefficient  (15).  solutions  each component..  from the  points The  leading-edge  velocities  as  to  are and  the  pressure  pitching  follows  2 = 1- Y•  C  l  N C  l  . \  =  C  1=1  Ay)  (16)  1  counter' clockwise  around  the  poly-  contours.  of  straight  each element, Kennedy  which  is  however,  higher  order  terms  vortex  density results  so  velocities  referred  to  studied  due t o  while  the  velocities.  two  terms  which  the  into  magnitude  but  of  when t h e y  are  combined.  BASIC m e t h o d w i t h  the  inclusion  the  the  opposite  the  1,  influence  It  is line  of  linear  linear  signs,  straight  effects  results  thus  simplifying  vortex  as t h e  curvature  from appendix  introduce  curvature  here  the  surface  including  Also  makes t h e  and c o n s t a n t  has  show t h a t  far  elements  on each element,  the  surface  described  line  (6)  These  Ax + y .  1  are performed  The t e c h n i q u e tions  (x.  P  summations  gonal  X  n  % and  A  l  N £ C -i=l i  = -  C  P.  density  BASIC  of  including  shown i n  element vortex  therefore elements  varying  Fig.  density  (7).  raises  decreases  that  the  distribution  and  coefficients their  the  curvature  one can n o t i c e  velocity  on  method.  and a l i n e a r l y are  assump-  are  effects  of  tend  recommended t h a t and c o n s t a n t  the to  same cancel  only  vortex  the  density  be  method w i t h  in  elements  In sections, elements slotted  with  deflected  while for  16 s l a t s  extent  of  chord,  C.  the the  shown t h a t  Condition  gives  a section  with  is  a fairly  a cusped t r a i l i n g  edge  for  describe  given  the  section,  the  results sharp  This  is  for  peak a  which 40  and t h e r e  is  excellent  20 d e g r e e s , for  the  with  solid  with  parts for wall  airfoil,  airfoil  of  walls  is  slotted the  each w a l l ,  and  walls  with  has been f o u r  by  30 e l e m e n t s  represented  wall  by 4 0  NACA-23012 w i t h  has been r e p r e s e n t e d  main a i r f o i l  solid  windtunnel  single-component  14% h a v e b e e n r e p r e s e n t e d  two-component  the windtunnel  BASIC  solution.  investigation  (8 s l a t s  the  accurate  (8).  a n d CLARK-Y  The w i n d t u n n e l  64 e l e m e n t s the  current  has  Fig.  exact  (40 e l e m e n t s  elements  of  (6),  in  each w h i l e . t h e  elements  of  the  NACA-0015  flap  flap).  with  used t o  the  Kutta  distribution  airfoil were  agreement  Load"  An e x a m p l e  velocity  Joukowski  investigation  "Full  airfoils.  the  -  used.  Kennedy's  most  14  and  is  by  times  =  70 for  the  128  represented  8 elements  c/C  25%  .15). the  test  for The  by each  total  airfoil  - 15 CHAPTER T V -  IV-1  -  THE FLOW I N THE WINDTUNNEL  Introduction  The geometry occurs  extension  of  the  of  flow  the  of  which  occurs  As  it  is  the  to  positive  its  upper  in  order  current  tangential field,  to  the  plenums in  the  with  the  the  flow  physical  that  there  test-section  negative  flow  side.  In  test  this  current  free  while  airfoil  and  the  test  CD opposite  flows  by c o n s i d e r i n g  stream-  the  side,  section  represented  two  upstream of  pressure the  are  in  the  these  two  analysis.  the  geometry  of  the  theory  should  be d e v e l o p e d  to  provide  velocities  are  actually  plenum  compares  with  the  which  the  define  they  make  that  to  and normal  since  (9a)  the  are  to  test-section.  upstream of  pressure  is  like  (9)  analysis  Fig.  its  more  Figure  current  test-section  streamlines  analysis  test-section,  walls.  shown i n  and l o w e r  In the  the  and o p p o s i t e  enters  current  the  AB a n d CD, AB l e a v e s  airfoil  free  in  slotted  representation actually  the  representation  experimentally  surrounding  lines  PLENUM  necessary  of  the  terms  flow  for  free  at  streamlines the  induced  any p o i n t  tracking  any  in  the  free  streamline.  IV-2  -  The  Induced  The  stream  Tangential  function  and Normal  \1). . a t  Velocities  a control  xj  Due t o  point  c  P.  Vorticity  whose  co-  1  th ordinates as  in  over  Fig. that  with  respect  (10a), d u e t o element  is  to  the  j  vorticity  given  by,  straight having  element  a constant  are  P^={X\,Y\),  density y  - 16 A *ij  2?  =  / -A  1  £ n  r  (  i ' ^  P  d  ^  (17)  Then the v e l o c i t y components can be c a l c u l a t e d equation  (17).  Using the n o t a t i o n o f F i g . (10) the v e l o c i t y  components induced 'j',  from  at the element V i ' due t o a v o r t e x element  are Yu. . = -1 { T a n  x!-A (-J-T-) - Tan  -I -1  x'.+A (—L—) }  1  *1  1  v  ij  =  .4?  £ n  {  {  The d i r e c t i o n s  y'y  ( j~ ) )/ (yx  +  (18a)  A  2  + A) >}  2 +  (i8b)  2  o f u.. and v.. a t P. a r e p a r a l l e l and  ID  normal to the d i r e c t i o n o f element respect to cartesian  i  i l 1  j',  respectively.  With  'wind axes' X and Y (X i s the wind d i r e c t i o n ) ,  th the j  v o r t e x element and the element ' i '  0j and 0^ to the X-axis r e s p e c t i v e l y .  V  V  m  T  = u.. cos (9.-0.) + v..  ij  N. .  11  =  V  i j  C  i  O  S  are the t a n g e n t i a l  (  9  i~ j e  l  "  )  i l  u  are i n c l i n e d at angles  Thus,  sin(0.-0.) I D  (19a)  i j sin(e -6 .) i  (19b)  ;  and the normal v e l o c i t i e s induced a t the element  due t o the vortex element ' j ' o f d e n s i t y y.. The l o c a l t a n i g e n t i a l v e l o c i t y V_ . . i s -=— while the l o c a l normal v e l o c i t y V\ . . ^ Tn 2 Nn 'i'  Y  3  J  i s zero.  J  The d e t a i l s o f the c a l c u l a t i o n of equations  T  (18) and  (19) a r e provided i n Appendix 2. T h e r e f o r e , the t a n g e n t i a l at t h e c o n t r o l  point  and the normal v e l o c i t i e s  on the element ' i '  due t o a system o f  v o r t e x elements immersed i n an i n f i n i t e uniform to the X - d i r e c t i o n ,  are  induced 1  N'  flow U, p a r a l l e l  - 17 -  V  V  N = I { , . cos (9 -0 ) + v j=l J D 1 J u  i  N.  =  E  N  s i n (0.-0.)} + Uoocos 0. 1 3  1  1  (20a) {  j=l  l  X  v  i i cos(8.-8.) J  J  - u  X  s i n (9.-9.)} - L U ^ s i n 1  J  1  0. i (20b)  Hence, a t any f r e e p o i n t  i n the f i e l d of the above  system the t a n g e n t i a l and the normal v e l o c i t i e s of the flow there are given by, N V  T.  =  V  l  V  M  N  X  - V ~ Y V  ±  ._ i j 1-1  =  v  {  =  =  U  N  l  E  i v  ±  C O S  V  J  {v, _. co ij j c o s  9  +  i i  u  s i n  i j  9  J  s i n  i> +  J  (21a)  Uoo  9j>  (21b)  j = 1,2, . . . , N and the p r e s s u r e c o e f f i c i e n t o f the flow at t h i s p o i n t p^, i s i s g i v e n by, C  P  ±  = 1 "  (V^  i  +  i  )  ( 2 2 )  IV-3 - Free Streamline T r a c k i n g The r o u t i n e d e s c r i b e d i n the p r e v i o u s chapter must be c a r r i e d out to c a l c u l a t e N v o r t e x d e n s i t i e s , yj»  f°  r  the N elements  which r e p r e s e n t the t e s t a i r f o i l , the w a l l s l a t s and the  solid  w a l l s . The values o f the v o r t e x d e n s i t i e s can then be i n s e r t e d i n equations  (21a) and  (21b) to get the induced t a n g e n t i a l and normal  v e l o c i t i e s , r e s p e c t i v e l y , at a s p e c i f i e d p o i n t flow f i e l d . to  Thus from a s t a r t i n g p o i n t  (x^, y^) i n the  (x^, y ^ ) , which i s supposed  be at the downstream edge of the s o l i d w a l l but, to a v o i d the  -  singularity flow  there  direction  9  where V  1  =  m  T  A  0^ i s  N  _  1  and V  >T  the  located  -  instead  calculated,  very  given  point  are  (x^,  the  Yj_) /  x  2  = x  1  + Ax  y  2  = y  1  + Ax  flow  induced  co-ordinate  is  x  2  = x  y  2  = y  to  include  0  changed  velocities  and" t h e y  1  to  next  given  by  point  (x ,  y )  2  the  + Ax  *  tan  0^ i s  (24b)  2  is  then are the  calculated averaged  next  there  to  point  give  is  in  the  y-  now  (25b)  the  geometry  calculate  the  Then each  streamline  effect  of  and M components, get  N'  to  these  once more  there, of  stream  elements  elements  to  same  0, a n d t h e  0  calculated  we s h o u l d  simultaneously  where  2  (25a)  16 v o r t e x  .  the  are  1  so t h a t  analysis  functions  and n o r m a l  + Ax  1  (8).  adds  6  * tan  direction  direction  can proceed  which  the  (24a)  Having defined  equation  edge,  by,  tangential  respectively,  The f l o w d i r e c t i o n s  0^..  flow  the  1  The  The  to  (23)  e q u a t i o n s ' ' (21a) and ( 2 1 b ) . T h e n 0^ i s u s e d t o c a l c u l a t e  way as  close  ^ N / ^  1  at  is  18  solve  is  divided  previous  free (N  prescribed  vortex,  free  functions  the  two  the  a n d so  1  on.  streamlines on each, into  by  elements, Thus  in  the  equations,  (N  equations  densities,  using  N elements.  streamlines + M)  8  (8)  , and M  one  1  and  present =  N+16)  (10),  stream  -  CHAPTER V -  RESULTS AND DISCUSSION  The t h e o r e t i c a l t h e method o f isons  of  slotted  Ref.  NACA-0015 flap  free  air  test  section  at  curves  (6).  theoretical  c h o r d C,  19 -  Figures  pressure  at  (12),  (13)  and  distributions  for  6 = 2 0 ° a n d CLARK-Y  of  height are  here were  a = 10° NACA-23012  and between t h e  walls  of  with  C/h =  .8.  C  1.6873,  30.36% and  T 41.11% h i g h e r  3.6448  than  and  the  calculated  (14) the  a = 10°  The t u n n e l  l i f t  compar-  airfoils,  of  2 5 . 66%  a conventional  air  by  respectively,  3.0131 which  free  show  at. a = 8 w i t h  14% a t  solid  h,  efficients  T  presented  in  windtunnel  l i f t  are  co-  37.66%,  coefficients  C  T  L  of  1.2257,  2.7959  The v e r y developed results, exhibit  tunnel  with for  airfoil  solid  a wall  effects  above r e s u l t s  values  suction step  was t o  upper  NACA-0015 solid  at  over  wall.  the  the  l i f t  walls  coefficients  shown by t h e  configuration of  partly  that  open,  therefore  provide  it  seen t h a t  can be  tunnel  top  surface  wall  above would  partly  negligible  walls,  pressure  some i m p r o v e m e n t  of  here  by u s i n g  appears  in  distribution in  the  the of  are  airfoil. the of  or  all  the  of  upper  wind-  solid  the  airfoil  59% UOAR,  Fig.  (15)  along with  the  for  free  air.  figure  pressure  distribution  the  Therefore,  transversely-  for  configuration  of  due t o  conventional  a uniformly  distribution  such a w a l l  nearly  coefficients  configuration  The p r e s s u r e  a =lQ?for  l i f t  the modification  the  was d o n e  lower  corresponding  in  towards change  and t h i s  slotted  shows  a search  the  errors.  first  with  windtunnel  in  and w h i c h w o u l d  increased  wall,  errors  respectively.  known c a n c e l l i n g  walls,  greater the  large  the  From t h e the  2.1353  prompted the  closed small  in  and  F  This  opposite  to  the  upper  uniformly  differences suggests in  the  that  the the  presence  possible as  in  lower  pressure  is  the  lower  solid wall  to  this  also  lower  than  to  there  distributions,  experiencing  problem i s  shows t h a t  in  and  induced  free  air,  use a s l o t t e d  are it  velocities and  lower  a  wall  well.  slotted  Fig.  wall  (16),  closer  to  suctions  the windtunnel  has been  show t h a t  a = 10 a n d o f  the  C/h=.8,  the  free  still the  air  tend  pressure opposite  values  to  be  near  this  problem would appear  and narrow wall  ones  the  slotted wall actual  test  forward,  to  and t h e  the  lower  than before,  for  to  here  the  the  be t o  rather  uniformly  distributions  low near  edge,  double  than  results,  for  while  NACA-0015  the  edge and  graded,  in at  is  upper  same UOAR 5 9 % .  uniform  as  slotted wall  leading  use  transverse-  surface  slightly  A solution  wider  gaps  to  rearward  spacing between  the  slats. All  the  trailing  with  investigated  high  as  it  there  Accordingly, ly  surface  but  flow  of  solution  slotted wall,  test  the  streamlines.at  (9a), shear  using  free  streamlines,  they  can be  of  nonuniform  upper  layers  motivation  the  should  constant  but,  the  section with  limiting  Fig.  streamlines.are  flow  in  for  this  and l o w e r  pressure.  double  has  their  included  wall  in  average  geometries the  been  For wall  AB a n d CD, the as  free  attempted,  here  for  tracking  c a n be d e s c r i b e d  and  consequently  representation.  OAR h a v e b e e n e x a m i n e d t o  configuration  slotted  idealized  not  double  (9b).  Physically,  c o u l d be  This  the  Fig.  streamlines  be c o n s i d e r e d . and t h e y  using  spacing,  t h e method w h i c h has been d e v e l o p e d  Several suitable  support  representation  section  shown i n  above  which would  develop  look the  for  least  the  most  errors  -  in  the  tunnel  different  l i f t  found  linearly  increased  ation  for  Figure  mentioned  that  this  -  and p i t c h i n g  airfoils,  has been  21  before,  AOAR=59% w i t h downstream,  wide (17)  moment  range  of  coefficients and o f  the  airfoil  shows c o m p a r i s o n s  most  the  different  2% i n c r e m e n t ,  is  for  three  sizes.  It  the  gaps  are  suitable  wall  configur-  shapes  and  of  effect  the  sizes. on the  theoret-  C ical  l i f t  test  section  of  attack  either be  coefficients wall  or  seen t h a t  ratio  used b e f o r e ,  double  with  the  configuration.  are those  solid  of  the  uniformly solid  ^  for  different  The a i r f o i l s  airfoils  and t h e i r  and t h e  test  slotted  w i t h AOAR=59%.  walls,  for  the  section  and  angles  walls  calculated  are  It  can  range  of  C ^,  the  with  corrections  a suitable  errors Also  can exceed  AOAR o f  can be k e p t  Fig.  quarter  (18)  the  within  4 0% o f  slotted  4% f o r  shows c o m p a r i s o n s  chord  pitching  moment  same a i r f o i l s  and t e s t  section  seen t h a t the  with  free-air  59% t h e  and  comparisons  for  of  test  £  there  tions  the  is  three  of  quite  the of  the  effect of  and  on t h e the  It  1.8%  for  in  again with  airfoils.  and  air C ^  between  =  in  .8.  the  two  the  3%  of AOAR=  three  (21)  and  .8.  be  of  the  distributions , free  for  exceed  within  (20)  p.s?  can  section  pressure  attack,  ^  test  (19),  for  theoretical C  ratio  can  while  predicted  slotted  good agreement  different  the  airfoils,  corrections  the  AOAR=59%,  values,  configurations.  Figures  of  section  three the  can be k e p t  theoretical  sections  test  wall  with  .8.  and a n g l e s  seen t h a t for  ^  free-air  coefficients  walls  while  errors  the  airfoils  slotted  solid  values,  predicted  airfoils,  above  the  the  show for  the  the It  double  can  be  distribu-  - 22 Figure  (22)  shows comparisons of the e f f e c t on the  C c o e f f i c i e n t of the r a t i o t - f o r NACA-0015 a i r f o i l h  ical l i f t  d i f f e r e n t angles of a t t a c k , a=10° and a=20°, and wall conditions; 59% without and here by f r e e The  w i t h three d i f f e r e n t  streamlines.  f i r s t two  wall conditions  e x h i b i t the known blockage c o e f f i c i e n t s as the angle of  i n c r e a s e s , however the t h i r d wall: c o n d i t i o n shows the  separately.  The  should  study the l a s t two  wall  co-  i n c l u d e the flow i n the plenum i n the  a n a l y s i s f o r the same angle of a t t a c k and the a n a l y s i s of s t r e a m l i n e  f o r a l l r a t i o s of ^.  i n the upper f r e e streamline  tends to reduce the e f f e c t of t h i s s t r e a m l i n e negative  pressure  c o e f f i c i e n t at the a i r f o i l  have two  opposite  effects.  The  i n increasing l e a d i n g edge.  above f i g u r e shows t h a t the  e f f e c t i s weaker than the e f f e c t of the i n c l u s i o n of the  e f f e c t s are combined.  t h a t the s l o t t e d w a l l s w i t h AOAR=59% s t i l l  Also, F i g .  the So  we  blockflow  (22)  shows  i s the most s u i t a b l e  w a l l c o n f i g u r a t i o n f o r a wide range of angles of (23)  which  because of t h a t the e r r o r i n l i f t c o e f f i c i e n t s  improves when the two  Figure  Also  geometry shows t h a t an i n c r e a s e i n angle  of a t t a c k causes more curvature  i n the plenum, and  opposite.  conditions  above f i g u r e shows t h a t the e r r o r i n l i f t  e f f i c i e n t s i n c r e a s e s as we  attack.  shows the geometry of the i d e a l i z e d flow i n the  plenum chambers, the two the new  two  with the flow i n the plenums, which i s i d e a l i z e d  To e x p l a i n t h i s trend one  age  at  s o l i d w a l l s , double nonuniformly s l o t t e d of AOAR=  e f f e c t of i n c r e a s i n g the e r r o r i n l i f t attack  theoret-  free streamlines,  of a t e s t s e c t i o n with  w a l l c o n f i g u r a t i o n of AOAR=59%, and  the t e s t a i r f o i l NACA-  C 0015 ips,  at a=10°, and  line.  I t a l s o shows the values  the average p r e s s u r e  of the stream f u n c t i o n ,  c o e f f i c i e n t C , f o r each f r e e streamP  - 23 CHAPTER V I  -  CONCLUSIONS  A two-dimensional correction-free The t h e o r y theory  is  of  windtunnel  airfoil  loadings  sizes  of  the  takes  different  windtunnel  wall  the  flow  in  the  in  the  test  section  satisfactorily  has been  the  not  flow  singularities. only  effect  developed.  potential  surface  consideration also  for  flow  distributed  flow  a  two-dimensional  but  potential  predicts  configuration  the  into  and shapes,  The above account  of  method of  theory  which  test  an e x t e n s i o n  based on t h e  The e x t e n d e d  theory  a wide  on t h e  range  airfoil  configurations.  theory  was t h e n  plenum chambers, as c l o s e  in  developed order  as p o s s i b l e  to  to  to  represent  the  physical  situation. The r e s u l t s foil  testing,  uniform slats  the  theoretical  a windtunnel  linearly  increased  yield  at  zero  walls,  pressure  with  the  gaps b e t w e e n  and t h e  with  average  which  the  values.  predicts  wall  configuration  wide  range In  sizes  the  present  analysis,  two  layers.  flow ation  exists  outside for  by  in  each  and  two bounding  the  test  the  plenum.  shear  layer  the  flow  the  l i f t  ratio  59%,  coefficients  a few p e r c e n t  this  low  of  correction  correction-free  for  a  in  the  plenum chambers  streamlines,  but  actually  two  section, So t h e  does  non-  shapes..  Each one d i v i d e s in  that  air-  symmetrical  open area  are w i t h i n  relatively  airfoil  here  flow which  remain  of  idealized shear  will  slats  distributions,  The t h e o r y  for  symmetrically  moment c o e f f i c i e n t s  air  that  two  and p i t c h i n g free  indicate  of  downstream,  incidence  uncorrected  study  consisting  transversely-slotted  airfoil-shaped will  of  not  flows, arid t h e  single model  the  there  are  high-energy  low-energy  free-streamline exactly  is  the  stagnant represent-  division  of  the done  two to  flows  different  represent Also  so t h a t  of  it  the  these  present  would account  energy  two  shear  potential for  the  level  and more work  layers flow  more  be  accurately.  analysis  viscous  should  effects  s h o u l d be that  developed  occur  exper-  imentally. Finally wall  the  configuration  performance needs  to  of  the  test .section with  be examined  experimentally.  the  new  -  25  -  REFERENCES (1) i i i (2)  (3)  Pope a n d H a r p e r , Pankhurst  G.V.  "Windtunnel  and L i m ,  in  Two-Dimensional  4,  Sept.  A.K.,  Testing  of  Testing,"  Wiley,  1966.  Technique,"  Pitman,  1952.  "On T h e Use o f  Slotted  Walls  Low-Speed A i r f o i l s , "  CASI  Trans.  1971.  Williams,  Ph.D.  (5)  and H o l d e r ,  Parkinson,  Boundary  (4)  "Low Speed W i n d t u n n e l  CD.,  " A New S l o t t e d ^ - W a l l  Corrections  Thesis,  Oct.  in  Two-Dimensional  1975,  University  Theodorsen,  T.,  "Theory  NACA R e p o r t  No.  411,1931.  Hess,  J.L.  and S m i t h ,  About  Arbitrary  Method  of  British of  "Calculation  Prog,  in  Producing  Airfoil  Wing S e c t i o n s  A.M.O.,  Bodies,"  of  for  Aero.  Testing," Columbia.  Arbitrary  of  Sci.,  Low  Shape,"  Potential 8,  Flow  Pergamon,  1966. (6)  Kennedy, Ph.D.  J.L.,  Thesis,  (7)  Milne-Thomson,  (8)  Prandtl,  L.  dynamics," (9)  Design  1977, L'^M.,  Model  the Analysis  Hess,  and B r a d l e y ,  AGARD-CP-102,  J.L.,  "Higher  Integral  Equation  Computer  Methods  1973.  O.G.,  of  Airfoil  Sections,"  Alberta. Aerodynamics,"  "Applied  Hydro-  Dover, and  1973.  Aero-  1957.  T.C.  for  of  "Theoretical  and T i e t j e n s , Dover,  and A n a l y s i s  University  Bhateley,  Stall," (10)  "The  Multi  "A S i m p l i f i e d  Element  Mathematical  Airfoils  Near  the  1972.  Order  for in  of  R.G.,  the  Numerical  Solutions  Two-dimensional  Applied  Mechanics,  of  the  Neumann  Vol.  2,  pp.  Problem," 1-15,  -  26  -  APPENDIX .1 EVALUATION OF THE INTEGRAL I N EQUATION ( 5 ) The m o s t in  equation The  straightforward  (5) i s  to  surface  do  so u s i n g  element  is  distribution  can v a r y  distribution  on one e l e m e n t  is  calculated  a numerical  chosen t o  the  evaluating  on t h e  point  procedure.  and t h e  The i n f l u e n c e  control  integral  integration  be c u r v e d  element.  the  of  of  velocity this  another  element  integral  S2  1  I  over  by t h e  method o f  = 27  /  Y(SHn  r(C ,  S)  i  dS.  SI A coordinate control The  point  of  influenced  system.  The  hood o f  the  convenient  the  surface origin 2  {  1  on e x p a n d i n g  up w i t h  element  as  origin  shown i n  located  at  element  is  defined  by n=n.(£).  + is  series  (b,a)in  expansion  is  at  the  Fig.  this In  (11). coordinate  the  neighbour-  used,  ...  taken  ( )* 2  over  the  surface  distance  and i t  is  use:  - i i j. d n - ,  dT "  set  is  + e  2  d s  is  point  a power  integral to  (E,, n)  influencing  control  n = c C The  system  +  k  ~^" ,dr -  (3)*as  a series  about  density  can a l s o  .  5=0,  ^ • = l + 2 c ? + 6 c e 5 + . . . The v o r t e x  *  (3)  }  (4)  be w r i t t e n  as a s e r i e s  + Y  ...  defined  by, Y(S)  = Y<°»  +  y  ( 1  »S  +  Y  (  2  )  S  2  (  3  )  S  3 +  |  *  Applying  (4f  to  TIE)  (5)*,  = Y  Y  < 0 ) +  U  5  )  Y  +  ( 2  »C  (|c Y  2  2  +  Y  < 1 ) +  (  3  )  )  C  3 +  * The d i s t a n c e  r(C.^,S)  r  (c  S)  At  this  i f  =  (5).  modification  is  appear  the  distance  expanded r  is  first  point  to  the  surface  is,  (b -0 } 2  (7)*  H  necessary  Instead  O  of  to  employ  expanding  term of  this  the  the  technique  term d i r e c t l y  basic,  series.  the  flat  By  a  element,  term  writing *  O  = a  z f  it  +  2  control  used w h i c h p e r m i t s  as t h e T  r  {(a-n)  point  used by Hess  to  from the  (6)  (b -K)  +  to  the  (8)  flat  surface  n=0,  the  remaining  terms  are  and, 2  2  = i  -  2  Substituting  3  -  2ae  £  (9)*in  the  l o g a r i t h m t e r m and e x p a n d i n g  E,  2ac  +  ...  (9) all  2 but  the  r  term about  f  r  in  = \  In  yields,  5=0  ( r  2 f  ) - ^ § r  The of are  integral  accuracy  as  retained,  Equation  (1)  is  then  I = ^ f u  f  (l)*can  order  r  first  term  second t e r m third  is  In  this  terms  (10)  to  case o n l y  being  of  '  r  as h i g h  the  a  first  diminishing  degree  few  terms  importance.  becomes, *n r / . Y '  W  1  in  2 f  K -  2  ^  W  5  2  +  . . . . , «  f the  straight  introduces  introduces  . . .  f  -A the  +  3 5  now b e e v a l u a t e d  desired.  higher  -  2 ?  the  element,  a linear  surface  velocity  curvature.  constant  velocity  distribution  and  case, the  *  Each t e r m  in  -  28  (11)*can  be  integrated  separately,  as  follows:  A  j  Zrx r  de; =  2 f  In r  (b+A)  -  2 ±  (b-A)  £n r  -  2 2  4A  —A +  Jin r  2 a tan"  (  1  , ) a + b -A  K & K = ^  2 f  ^  2  £n  2ab  tan"  (^-)  (  1  2  a  -  .2 -Is- d r, f  /* / •A  C =  (b -a ) 2  tan  2  + a b  (  1  Jin  —  A 2  a + b A  (12)*  2 a A  2  2  2bA  2  )  (13)  *  t-A  a 2  2  a +b  -A  )  +  2aA  (—) l  (14)  *  r  For the  the  X-wind  straight  axis,  the  element  'a'  and  ' j ' , w h i c h makes  'b'  are  given  a =  (x.-x.)  sin8.  +  (y.-y.)  cos9.  b  (x.-x.)  cos9 . +  (y.-y.)  sin9 .  an a n g l e  9_.  with  by,  (15) =  D  I D where  (x.,y.) 1  points  on the In  Yj K^j  the  and  1  elements  case  of  integral  and i t  is  (x.,y.) 3 3 ' i '  straight  (l)-*,is  given  D  I  D  are  the  and  co-ordinates  ' j '  elements  called  by e q u a t i o n  the  of  the  control  respectively. with BASIC  (12)*.  constant influence  vortex  densities  coefficients  - 29  -  For higher order terms i n v o l v i n g s u r f a c e c u r v a t u r e the constant C must be determined. f i t t i n g parabolas through points. airfoil  In t h i s case t h i s was  done by  s e t s of t h r e e adjacent element end  The c u r v a t u r e thus determined  was  assumed t o be  the  s u r f a c e c u r v a t u r e at the c e n t r e of the t h r e e p o i n t s .  c u r v a t u r e at the c o n t r o l p o i n t s were then found by  interpolation.  The values at the elements adjacent to the t r a i l i n g found by e x t r a p o l a t i o n .  Having determined  The  edge were  the c u r v a t u r e of the  element the l o c a t i o n of the c o n t r o l p o i n t can be c a l c u l a t e d  as  t h i s p o i n t i s no longer on the s t r a i g h t l i n e j o i n i n g the element end p o i n t s . In employing v a r i a t i o n s i n the s i n g u l a r i t y s t r e n g t h the term  i  sa  n  unknown and must be r e l a t e d to the y ^  •  Various  schemes are a v a i l a b l e to do t h i s and the technique used by Hess i s f o l l o w e d here.  The d e r i v a t i v e s of the  d i s t r i b u t i o n on  (10)  the  th j  element are determined  through  by assuming a p a r a b o l i c d i s t r i b u t i o n  the t h r e e s u c c e s s i v e v a l u e s y ^ ^ j _ , 1  Y ^ j ,  l i n e a r v o r t e x d e n s i t y term, u n l i k e the other two comprised  Y  ^ j + i *  T  h  e  terms, i s t h e r e f o r e  of terms t h a t i n v o l v e the v o r t e x d e n s i t i e s of adjacent  elements. The a p p l i c a t i o n of the higher order methods t o the i n v o l v e s the c a l c u l a t i o n of the e x t r a terms  (13)* and  solution  (14)*.  c u r v a t u r e terms are simply added t o the i n f l u e n c e c o e f f i c i e n t c a l c u l a t e d f o r the b a s i c case. added to the c o e f f i c i e n t s K. not d i f f i c u l t  The  K^^  l i n e a r v e l o c i t y terms must be  . ,, K..,  K.  Although  to do, the e x t r a c a l c u l a t i o n s i n v o l v e d do  c o n s i d e r a b l e amounts of time to  The  perform.  this i s take  -  30  APPENDIX  2  CALCULATION OF THE V E L O C I T Y COMPONENT INDUCED AT A POINT I N THE F I E L D OF VORTEX The element is  DISTRIBUTION:  stream  function  at  ' j ' , with  constant  density  given  a point  P^ d u e t o  a straight  distributed  y.  vortex  over  that  element,  point  Q on  the  by  -i where  r  is  surface,  the  as  distance  from the  i n F i g . (10a),  r.. =  i(x\-V  and  point  it  P^ t o  the  can be w r i t t e n  as  + Yp  2  (2)*  k  Thus ^1  i n { ( x ! - a  y  -A The v e l o c i t y to  the  element  ' j  i^ j  v, , ^  =  +  components  in  —y .  - r8dy ^-  2TT -*k f  =  = - ^ - J - = 97  /  } ^  (3)*  induced  at  parallel P^  and  normal  are:  y'.  A ,  y 7 -A  2 l T  2  directions  respectively,  1  dty • . u  y !  2  ,  . 5 2  r  : L  9  ,2  L  (x'.-5) +y! 3 ^ 2  (4a)  5"  **  (4b)  S  d  2  S  Therefore Y  and  D  f  ,  "I  u . . = TJ-*- { t a n ii 2TT v.. ij  =  4TT  £n{  (  X  i "  (referring)  to  )  .—-4 y: 3  ,  "I  (  i  X  tan  +  A  )  ,  }  (5 ) v~><=w a  v! 3  y'^+fx'.-A) j j  Where x . and y . a r e 3 3 respect  A  the  2  /  y'. + ( x ' + A ) j j 2  co-ordinates  a system of  2  of  } the  (5b?* \-> > u  point  P.  with  1  axis  its  origin  is  the  control  - 31 p o i n t on the element  'j'  which has the l e n g t h 2A and makes an  angle 0j w i t h the wind a x i s , and they are given  by k k  x! = (x.-x.) cos0. + 1 1 1 D  (y.-y.) s i n 8 . i 1 1  (6a)  y! = 1  (y.-y.) cos0. i l 1  (6b)  (x.-x.) s i n 0 . + l i l  k ic  J— - u  To express the v e l o c i t i e s t a n g e n t i a l and normal c o n t r o l s u r f a c e we c o n s i d e r  to the i  the f o l l o w i n g :  From F i g . (10b) and from the v e c t o r a n a l y s i s of the 'j',  element  the v e l o c i t y v e c t o r V.. can be w r i t t e n as V. . = u. . t . + v. . n . il i l 1 i l 1  (7)  also V. . = X. . i + Y. . il i l i l  (8**  j  thus x. . = v. . i  ii  (9) :•  i i  then from equations  (?)•** and  .  ( 9 ) * * , we get  X. . = u. . cos0 . - v. . sin9 . il i l 1 i l 1 similarly  (10)  Y. . = u. . s i n 0 . + v. . cos0 . il i l 1 i l _1 A l s o the v e l o c i t y v e c t o r ^, r e f e r r i n g t o F i g . (10b) from the v e c t o r a n a l y s i s o f the element  'i',  ~  and  can be w r i t t e n as A  m  11  11  —  v  m  T. .  m T  J  ( 8 ) * * and  (13)**, we  = X . . cos0. + Y.. s i n 0 . i l i i l i  ij  kk  ,(13)  l] l  then from equations V  (12)* '  •  - v. . t i  11  k k  (11)  V.. = V t . + V„ n. l] T. . l N. . l thus  ••.  get (14**  similarly V  N.. 11  =  V  i j  H  J  = Y..  i cos0. - X..  ( sin0.  1 5  "  -  substitute  (10)** a n d  32  -  (11)** i n t o  equations  (14)** a n d  (15)**,  hence  V .  V „  V  =  ID  eos(e.-e.) + v  i  are  the  1  ±  j  sin(6 -e ) i  j  ( 1 6 a )  - f U i bf tD h ; )  I D  tangential  and t h e  normal  at  ' i ' , tangential  velocities,  ID  due t o  induced  a straight  element  vortex  element  ' j '  with  **  **  + v . . c o s (6.-0.)  ID  ij  respectively, it,  i j  = - u . . s i n ( 6 . - 0 .)  j  and V „  m  u  and n o r m a l  constant  to  density  - 33 -  FIGURE  1 - C O M P A R I S O N OF A I R F O I L P R E S S U R E C O E F F I C I E N T S : Ref.  (3)  THEORY  -  VORTEX REPRESENTATION  34  -  OF TWO  COMPONENT  AIRFML  -  35  -  GURE 3- NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS  FIGURE 4- STREAMLINES CONTOURS AROUND AN AIRFOIL Ref.  (8)  -  37  -  LOCATION OF TRAILING CONTROL POINT  -  38  F I G U R E 6- L O C A T I O N OF E L E M E N T S  -  ON A I R F O I L  SURFACE  -  39  -  3.0  2.5  2.0  Karman-Trefftz Aerofoil  1.5  1.0  a = 0°  Element Shape  Vortex Density  O  Straight Line  Constant  •  Curved  Constant  V  Straight Line  Linear  A  Curved  Linear  — « Exact Analytic  FIGURE  7- C O M P A R I S O N O F H I G H E R O R D E R Ref. (6)  METHODS  -  0  0.2  "  40  0.4  -  0.6  0.8  1.0  x/c FIGURE  8-  C O M P A R I S O N OF A I R F O I L V E L O C I T Y Ref. (6)  DISTRIBUTIONS  -'41  -  (a) THEORY  f  (  ('  f  (  (  (  / / / / • / /  /  / /  //  / / i /f  "y (b) EXPERIMENT  /'v  A ;  —}  > ) )—7—7—7—J—*  i  } t i  FIGURE 9- AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL  - 42 -  -  FIGURE  1 1 - GEOMETRY  43  -  FOR C A L C U L A T I O N OF H I G H E R ORDER  TERMS  -  44  -  NACA-0015 a =10  C'/h=. 8  — - FREE AIR  C.  =1 . 2 2 5 7 I"  S O L I D W A L L S C,  i  N c / C -  0  1  8  9  =1.6873  E =37.66% L  C  E =-2.92% M  FIGURE  1 2 - C O M P A R I S O N OF P R E S S U R E AIRFOIL SECTION  IN F R E E A I R AND  C O E F F I C I E N T S FOR BETWEEN S O L I D WALLS  NACA-0015 TEST  -  45  -  i NACA-23012  .2  WITH 2 5 . 6 6 % S L O T T E D  A  .6  FLAP  .8  X/  FIGURE 13- COMPARISON OF PRESSURE C O E F F I C I E N T S FOR IN F R E E A I R AND  c  NACA-23012  B E T W E E N S O L I D WALLS T E S T  AIRFOIL  SECTION  -  46  CLARK-Y  14%  a =10  C/h=.8  ---FREE  AIR  SOLID  WALLS  C  L p  = 2.135.3  C _=3.0131 L  E =41.11%' L  C  M  C  4  = -.1331  M c / 4  =~.2035  c  /  E =-3.3% M  FIGURE 15- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN F R E E A I R AND WALL T E S T  BETWEEN S I N G L E  SECTION  UNIFORMLY  SLOTTED  -  48  -  FIGURE 16- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN F R E E A I R AND WALL T E S T  BETWEEN  SECTION  DOUBLE UNIFORMLY  SLOTTED  - 49 NACA-0015 a =10 NACA-23012 WITH 25.66% SLOTTED FLAP a =8 .  5 =20  /  CLARK-Y 14%  /  a =10 A0AR=59%  /  /  //  c/C=.15  /  //  /  '/ *  SOLID WALLS  DOUBLE SLOTTED WALL WITH FLOW IN THE PLENUM  I  .2  /  A  C/h GURE 17- COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN L I F T COEFFICIENTS  -  50  -  C/h FIGURE 18- COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS  FIGURE 19- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN F R E E A I R AND B E T W E E N D O U B L E N O N - U N I F O R M L Y WALL T E S T S E C T I O N  WITH T H E FLOW  IN T H E P L E N U M  SLOTTED  - 52 -  NACA-23012 WITH 25.66% SLOTTED FLAP a =8 —  6 =20  C/h =.8  FREE AIR DOUBLE SLOTTED WALL WITH FLOW IN THE PLENUM  AOAR =59%  V2  7417  C, =2.7417 Ei —  1.  jt/o  w-  3031  C _ =-.3282 M  //1  E  M=-  X/C FIGURE 20- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN F R E E A I R AND B E T W E E N D O U B L E NON-UN I F O R M L Y S L O T T E D WALL T E S T S E C T I O N WITH T H E FLOW  IN T H E P L E N U M  - 53  -  CLARK-Y 14% lj  a =10  |  - F R E E AIR  I  C/h=.8  A0AR=59% " ^=2.1353  ^=-.1331  DOUBLE SLOTTED WALL  .2  .4  .6  • *8  t. x/c  FIGURE 21- COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN F R E E A I R AND B E T W E E N  DOUBLE NON-UNIFORMLY  WALL T E S T S E C T I O N WITH T H E FLOW  IN T H E P L E N U M  SLOTTED  - 54  -  NACA-0015  /  A0AR=59% —  a =10  ---  a =20  /  /  / / /  /  / SOLID  WALLS  /y  D O U B L E S L O T T E D WALL WITH FLOW IN T H E P L E N U M  D O U B L E S L O T T E D WALL W I T H O U T FLOW IN T H E PLENUM  1.  .6' C/h  F I G U R E 2.2- C O M P A R I S O N OF T H E E F F E C T OF TWO  D I F F E R E N T A N G L E S OF  A T T A C K ON T H E R E L A T I V E ERROR IN L I F T FOR N A C A - 0 0 1 5 A I R F O I L  COEFFICIENTS  C/h  =  .8  c/C  =  .15  a  p  >  o  o  'o  o>  10  ^ =.6822  c =-.i  *JJUJ*dJ~LLU Q  =  s  o>,  o-  o  h-c  -H  h  ^Yj j fTT7T  C>  B :  e  51%  O  53%  O  55%  F I G U R E 2 3 - GEOMETRY AIRFOIL  0>  57%.  O  59%  OF THE I D E A L I Z E D  0>  61%  FLOW  0>  63%  TTTT  O  6 5%  I N T H E P L E N U M CHAMBER  6 7%  FOR N A C A - 0 0 1 5  

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