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A theory for wind tunnel wall corrections Williams, C. D. (Christopher Dwight) 1973

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cl  A THEORY FOR HIND TUNNEL WALL CORRECTIONS. BY  CHRISTOPHER D. WILLIAMS  B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1967.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF H&STER OF APPLIED SCIENCE  in  the Department of  Bechanical  We accept  this  t h e s i s as conforming t o the  required  THE  Engineering  standard.  UNIVERSITY OF BRITISH COLUMBIA APRIL 1973  In p r e s e n t i n g an  thesis  in partial  f u l f i l m e n t of the  advanced degree at the  University  of B r i t i s h Columbia,  the  Library  this  s h a l l make i t f r e e l y  by  s c h o l a r l y p u r p o s e s may his representatives.  be  thesis for financial  written  permission.  Department of  A p r i l 16,  gain  1973  the  s h a l l not  Mechanical Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  g r a n t e d by  H e a d o f my  Columbia  be  for  that  study.  copying of t h i s  I t i s understood that  of t h i s  I agree  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 extensive for  requirements  thesis  Department' o r  copying or  publication  allowed without  my  Abstract  Wall c o r r e c t i o n wind  tunnels  with  theories  t h i n models o f  t h e o r i e s are  wall  angles of  attack,  An  exact  necessary  or  useless  low  for  for  theory  singularities  as  determined  by  represented i n the  high  pressure  distributions  indicates  that wall  around  certain wall corrections  for  Supervisor  models a t  high  source  of  airfoil  t e s t s on  the  airfoil  surface  configurations  Such  of  i n which  integration the  for  lift.  method o f A. l i . 0 . S m i t h ,  numerical  only  prediction  i s presented  by  valid  models,  in  Conventional  angles of a t t a c k .  the  large  airfoils  examined.  make t h e s e a s s u m p t i o n s . The are  negligible  camber a t  models d e v e l o p i n g  numerical  to  are  linearized theories,  corrections  wall sections  is  are  slight  shown t o be  required  two-dimensional  p a r t l y open w a l l s  wall c o r r e c t i o n theories small  for  it  and  is any  and  lifting  solid vortex  aerodynamic the  lift  calculated  c o n t o u r . The  will  not  require  theory  small  airfoils.  or  ii  TABLE  OF  CONTENTS  EMS I Introduction  1  II C o n v e n t i o n a l T h e o r i e s f o r Wall C o r r e c t i o n s . I I I Previous Experimental IV  Formulation  o f an E x a c t T h e o r y .  V Numerical Solution. VI R e s u l t s . VII C o n c l u s i o n s . References  Work.  2 4 6 12 15 20 45  iii  LIST OF FIGURES AND  TABLES Page  F i g u r e 1.  A i r f o i l i n a wind t u n n e l .  F i g u r e 2.  Comparison of l o n g i t u d i n a l s l o t t e d w a l l lift  21  theory with data f o r Clark-Y  airfoil. F i g u r e 3.  22  Comparison of p o r o u s - w a l l l i f t  theory  w i t h data f o r Clark-Y a i r f o i l t e s t e d between F i g u r e 4.  longitudinally slotted  walls.  Comparison of p o r o u s - w a l l l i f t  theory  with data f o r Clark-Y a i r f o i l slotted  with  f l a p t e s t e d between  longitudinally slotted F i g u r e 5.  23  walls.  24  P o r o s i t y parameter as a f u n c t i o n of openarea r a t i o .  25  Figure 6 .  Geometry  26  Figure 7 .  Source and vortex d i s t r i b u t i o n s f o r a  and n o t a t i o n f o r Smith's method.  two-dimensional a i r f o i l  between  solid  walls. F i g u r e 8.  27  P r e s s u r e d i s t r i b u t i o n s f o r Clark-Y airfoil.  F i g u r e 9.  P r e s s u r e d i s t r i b u t i o n s f o r NACA 23012 airfoil  Figure  10.  28  with f l a p .  Lift coefficient between  solid  f o r Clark-Y  walls.  29 airfoil 30  F i g u r e 11.  Ratio of l i f t airfoil  Figure  12.  c o e f f i c i e n t s f o r Clark-Y  between s o l i d  Streamlines  walls.  f o r a set of multiple  a i r f o i l s i n an i n f i n i t e Figure  13.  Streamlines  f o r an a i r f o i l  transversely slotted lower Figure  14.  15.  between  upper and  solid  walls.  V a r i a t i o n of l i f t w a l l geometry  Figure  stream.  c o e f f i c i e n t r a t i o with  f o r Clark-Y  V a r i a t i o n of l i f t  airfoil.  coefficient ratio  with  upper wall open-area r a t i o f o r Clark-Y airfoil. Figure  16.  Pressure d i s t r i b u t i o n  f o r Clark-Y  in correction-free l i f t  airfoil  test  configuration. Figure  17.  V a r i a t i o n * of l i f t w a l l geometry  Figure  18.  coefficient ratio  f o r Clark-Y  V a r i a t i o n of l i f t  airfoil.  c o e f f i c i e n t r a t i o with  w a l l geometry f o r NACA 23012 a i r f o i l slotted Figure  19.  1.  with  flap.  Relative error i n l i f t  coefficient for  10% upper w a l l open-area r a t i o . Table  with  Configurations  tested.  V  ACKNOWLEDGEMENTS  The advice  author  would  l i k e t o thank  D r . G. V. P a r k i n s o n f o r h i s  and g u i d a n c e i n t h e c o u r s e o f t h i s  The  computing  University contained  This Columbia  of B r i t i s h  facilities Columbia  of  research.  t h e Computing  were used  t o do t h e  Center of the calculations  herein.  research  was  and t h e D e f e n c e  s u p p o r t e d by t h e U n i v e r s i t y Research  Board  o f Canada.  of British  1  I  Introduction The  wind  problem  tunnel  occupied  the  from  those  of  a  complex  alter  a  on  flow  these  that  given  or  would  these be  exist  shape.  Any  field  measured  forces  and  I d e a l l y such  measured  in a  and  the old  years. the  walls one,  The  tunnel  these  test  problem  is  walls  to  the  model  free a i r .  e f f e c t s are  functions  corrections  test  is  in  moments, m u s t be  free-air  a has  q u a n t i t i e s , such  q u a n t i t i e s to  of  which  d i r e c t i o n ) near  i n a corresponding  flow  measured  forty  e f f e c t of  (speed  i n the  aerodynamic  to  the  e f f e c t of t e s t i s an  f o r more t h a n  would  point  the  under  since  wall effects.  directly they  model  conditions  m o d e l s i z e and  speed  determining  researchers  extremely  At  of  could  predict  as  corrected be  their  environment.  flow for  applied values  as  2  II  Conventional  Theories  Glauert(3), singularity  the  G o l d s t e i n (4) ,  image  interactions. airfoil  f o r Hall  techniques  The l i f t - p r o d u c i n g  each  and  Allen  to  model  doublets,  and  a suitable  in  the  tunnel  to  conditions  at  the  r e q u i r e s zero flow boundary  walls;  requires a constant  u s e f u l i f the induced  the  airfoil  individual  can  thin,  valid  only  slightly  satisfies  In  open  of  and/or  imaginary  a  technique  boundary  results  usually  at  of  the  linearized for  small,  of attack.  to transform  parts  I ti s  i s , these  terms  that i s ,  jet  images  that  requires  a t low a n g l e s  mappings  boundary  the a i r f o i l  value  problem  i s known; t h a t i s , a f u n c t i o n o f a  function;  f o r their  are  prescribed  in integral  numerical  complex on  equations  techniques  are  solution.  approach  disturbance flow flow  airfoils  Woods*  mapping  Either  this  an  flow  the boundary.  additive;  a geometrically simple  boundary.  required  equations,  wall  while  superimposed.  uses conformal  which a s o l u t i o n  the  be  be  required  a solid  o f each system  to  with  respectively.  the  along  f o r small wall effects,  v a r i a b l e whose r e a l  for  can  cambered  plus walls into  this  pressure  assumed  field  Woods ( 1 2 )  for  be  effects  corresponding  satisfy  example,  effect  o f the a i r f o i l ,  system o f "images" i s  normal t o t h e boundary,  very  theory,  for  airfoil-wall  sources  of singularity, walls,  the  used  wake a r e a s s o c i a t e d  type  found  and V i n c e n t i { 1 )  characteristics  t h i c k n e s s , and t h e a i r f o i l  d i s t r i b u t i o n s of v o r t i c e s , For  Correctiqns  velocity  Laplace's  can  be  formulated  potentialj equation,  in  terms  of  the  , which f o r i n c o m p r e s s i b l e  which  i s linear.  3  For a s o l i d flow  at  boundary which i s p a r a l l e l to  upstream  infinity  boundary; hence  there  the  undisturbed  i s zero flow normal to the  i s zero there.  dn For an pressure equation  open  at as  hence  the  For  linearized  condition  j e t boundary can be expressed  requiring  is  dx  j e t the  zero  streamwise  disturbance  velocity;  zero there.  slotted  a  constant  via Bernoulli's  or  perforated  walls  where  an  homogeneous boundary c o n d i t i o n " can be s t i p u l a t e d , al(2)),  of  linear  combination  of  "equivalent (Baldwin  et  t h e boundary c o n d i t i o n s f o r  s o l i d w a l l s and f o r an open j e t i s used. I n such details  of  equivalent the  slot  homogeneous or  particular their orientation bulk  properties  are p r e s e r v e d . longitudinal  such  boundary  perforation  geometry  as the p o r o s i t y or open area r a t i o  (OAR)  would  that  need t o be l e s s than  t h e OAR f o r 1 % to achieve a  from t h e open j e t case.  In p r a c t i c e , at such a s m a l l OAR, r e a l f l u i d  invalid.  lost; i n Only  boundary c o n d i t i o n a p p r e c i a b l y d i f f e r e n t  important,  are  ( l o n g i t u d i n a l or t r a n s v e r s e ) .  Wood (11) shows, f o r example, slots  conditions a l l  effects  would  so a p o t e n t i a l flow model f o r the c r o s s - f l o w would be Moreover,  Wood's  a n a l y s i s o f t h i s boundary c o n d i t i o n  i n d i c a t e s t h a t only c r o s s - f l o w v e l o c i t i e s of o r d e r l e s than of  the  involved  be  mean  flow  would be i n keeping  .5%  with the l i n e a r i z a t i o n s  i n the d e r i v a t i o n of t h i s boundary c o n d i t i o n .  4  III  P r e v i o u s Experimental Work The r e s u l t s of c o n v e n t i o n a l w a l l  shown  compared  airfoil,  of  with experiment  four  longitudinally  different  (streamwise)  test  section  sizes,  dimensional t e s t i n g Parkinson and Figure 2 its  only  or  the  presence  walls.  chord and equivalent  In H  these  is  the  height, f o r  (Figure 1). F i g u r e s 2,3,4,5 are  shows the r a t i o of measured l i f t  value m„ i n f r e e a i r , as a f u n c t i o n of model slotted  w a l l s of OAR  agreement with the theory f o r for  are  taken  of and wind twofrom  Lim(7).  longitudinally The  in  slotted  height,  theories  (Lim{6)) on a 14% t h i c k Clark-Y  subsequent f i g u r e s , C i s the a i r f o i l tunnel  correction  solid  walls;  curve s l o p e m to size  C/H  0.,5.6,11.1,18.5 and  longitudinal  t h a t i s , f o r zero OAR.  slots  is  for 100%. good  A l l other w a l l s  were p r e d i c t e d t o behave as i f they were almost completely open. Thus the theory f o r l o n g i t u d i n a l l y s l o t t e d w a l l s i s u s e l e s s p r e d i c t i n g the behaviour The  of a model under a c t u a l  same measured r a t i o of l i f t  Figure 3  for  test.  curve s l o p e s i s compared i n  with t h a t p r e d i c t e d by porous w a l l theory  (Woods (12)).  T h i s shows t h a t the p o r o s i t y parameter P ( d e f i n e d i n Ref. 7) f o r a g i v e n w a l l OAR the t e s t  so  good  agreement  with  data.  This airfoil  can be chosen t o produce  result  appears  in  Figure 4  to be true a l s o f o r an  with s l o t t e d f l a p , but the v a l u e s of p o r o s i t y  parameter  o b t a i n e d were not the same as f o r the b a s i c a i r f o i l ,  same w a l l c o n f i g u r a t i o n s .  f o r the  5  Figure the  wall  impossible theory.  5 portrays OAR,  which  situation  the p o r o s i t y also for  depends  the  p a r a m e t e r as a on  the model  practical  use  of  function under  of  test,  porous  an  wall  6  17 F o r m u l a t i o n of an Exact Theory What  then i s needed i s an exact  ( r a t h e r than a l i n e a r i z e d )  theory, where t h e net c o r r e c t i o n s t o measured f o r c e s and moments are achieved  d i r e c t l y , r a t h e r than a d d i t i v e l y . The  method  used  i n t h i s i n v e s t i g a t i o n i s an e x t e n s i o n of the s u r f a c e s i n g u l a r i t y d i s t r i b u t i o n theory of A.M.O. S m i t h ( 8 ) . In  Smith's  method,  the  airfoil  is  represented  d i s t r i b u t i o n of s o u r c e s and v o r t i c e s around i t s velocities and  at  p o i n t s i n the flow f i e l d  condition  of  zero  flow through  perimeter.  due to a l l such  v o r t i c e s a r e c a l c u l a t e d d i r e c t l y . The  usual  the a i r f o i l  by  flow  a  The  sources boundary  surface i s applied  and a f i n i t e - v e l o c i t y Kutta c o n d i t i o n i s a p p l i e d a t t h e t r a i l i n g edge. Again  <j> i s  satisfies  the  Laplace's  disturbance eguation,  velocity vanishes  potential at  which  infinity,  and  s a t i s f i e s the a p p r o p r i a t e boundary c o n d i t i o n s on t h e a i r f o i l . The  potential  at  a  point  P  due  to  a  single  three-  dimensional source s i n g u l a r i t y a t a point Q i s  <£> --m J_ p  where and  4TT  (D  rQ R  m i s t h e volume flow r a t e of f l u i d  emitted by the source,  PQ i s the d i s t a n c e between the p o i n t s P and  IT  potential  due to a l l such  surface S i s  Q.  The  total  sources d i s t r i b u t e d over an a r b i t r a r y  7  p  J  J C  r  PQ (2)  where cr \Q) 1/47TJ,  i s the source  of the source  Since  the  normal-velocity  strength  density,  i n c l u d i n g the  factor  e l e m e n t a t Q.  disturbance boundary  velocity  o f <f>\, t h e  i s the gradient  c o n d i t i o n a t a s u r f a c e c a n be  expressed  as  (3) where ix i s t h e o u t w a r d at take  upstream  infinity,  surface normal, and F t h e v a l u e  at the a i r f o i l surface. F i s zero  V» [the u n d i s t u r b e d  flow  the normal v e l o c i t y  must  for a solid  boundary, b u t nonzero f o r s u c t i o n o r blowing Analysis contributions "local"  (Smith (8))  shows  that  a t P on S by s u c h s o u r c e  term,  the  (impermeable)  there. normal  velocity  e l e m e n t s dS c o n s i s t o f a  ' 27TOr(P)  (4) due t o t h e s o u r c e  element a t P ,  plus  a "farfield"  term  (5) due t o t h e summation o f a l l o t h e r resulting  boundary  source  elements Q  on  S.  The  condition  27ro-(P) -fffa(j—)  a  (Q) -V»°n + F ds =  (6) comprises the  unknown  a  Fredholm continuous  integral source  equation strength  of the second density  kind, f o r  distribution  8  function  o~|(P) .  Existence  and  e q u a t i o n s may be found i n disjoint,  but  function In  of  a  distribution distributed applied  K e l l o g g (5).  outward  theorems  The  for  surface  normal v e c t o r  S  must be a  such  may  be  continuous  position.  practice  represent  is  the  uniqueness  an approximating  three-dimensional of  sources  H simultaneous  of  N finite  so  a  that  the  succession  linear algebraic  Defining the l i n e a r  to  continuous N  finite  The normal-flow boundary c o n d i t i o n  exact i n t e g r a l  s u r f a c e source  used  of  " c o n t r o l p o i n t " on•each s o u r c e  Thus the r e s u l t i n g of  body  becomes  s o u r c e elements. at a single  polygonal surface i s  element.  e q u a t i o n reduces to a  equations for  the  set  strengths  elements.  operator  (7)  the boundary  c o n d i t i o n a p p l i e d at  the i - t h c o n t r o l  point  ^ A j j o ^ - V ^ r i i + Fj(0)  indicates •i'  by  for  all  A  t o be the normal v e l o c i t y induced at c o n t r o l p o i n t  a unit strength  at  *j*.  Hence Ajj  is  2'TT  i=1,2,...N.  For becomes  a two-dimensional a i r f o i l a  polygonal  the two c o n t r o l p o i n t s fixes  source element  cylinder.  With  r e s p e c t to C a r t e s i a n  approximating  A Kutta c o n d i t i o n i s  a d j a c e n t to  the net c i r c u l a t i o n about  this  the axes  the  trailing  surface  applied  edge  -  at  this  airfoil. x  and Vj  fixed  to the  j-th  9  source element line  source  (Figure 6 ) , the i n t e g r a t i o n o f a into  a two-dimensional  produces a t a p o i n t ' i '  distributed  two-dimensional source  element  v e l o c i t y components  vax -t r  ._  ( x  (9)  ¥ j ) 2 + y /  and  i  y  L  x f +yji-(f f (10)  where Xj  and  yj  are t h e d i s t a n c e s from  the  j - t h to  the  i-th  element; the j - t h element has l e n g t h ASj With  reference  t o C a r t e s i a n "wind axes" X and Y, (X i s i n  the wind d i r e c t i o n ) , t h e j - t h source element i s i n c l i n e d angle  9] j to the X - a x i s . Thus  (11)  A=Vy.cosie.-op -Vx.sirKtffflj) }i  and  are  a t an  i t s orthogonal  the  respectively  normal  j  J  partner  and  tangential  disturbance  velocities  at c o n t r o l p o i n t *i» due to a u n i t s t r e n g t h  source  element a t ' j Corresponding  expressions  f o r the v e l o c i t y components due  to a d i s t r i b u t e d vortex element of c i r c u l a t i o n s t r e n g t h X(Q)  density  can be o b t a i n e d . Then the normal and t a n g e n t i a l v e l o c i t i e s  at c o n t r o l p o i n t ' i *  due t o a l l N source and vortex elements and  the uniform approach flow U are  10  N  N  VlA^-lBwji-Usinfl j=l  k=l  1  (13)  and N  N  V . Z B j i o j +JjA r +Ucos£, ki  k  (14)  Hence the normal-flow boundary c o n d i t i o n  V  at  N  all  control  n j  becomes, f o r  zero  =0  points,  (15)  while  the  finite-velocity  Kutta  c o n d i t i o n becomes V* •upper  at  the  two  =-V* "lower  (16)  c o n t r o l p o i n t s a d j a c e n t to the  In p r a c t i c e , a l l vortex  airfoil  elements on a c l o s e d  a Kutta condition  are chosen of e q u a l s t r e n g t h  about  polygonal  an N-sided  airfoil  requires  trailing  body r e q u i r i n g  Y , so  the  ^ j.  solution  method  obtained  directly  e l i m i n a t i o n , although i n d i r e c t  by  the  iterative  the  flow  s o l u t i o n of  e q u a t i o n s i n the N+1 unknowns O j , a j l , . . . Oj^jand is  edge.  Usually  procedures  of  N+1 the  Gauss-  are  also  possible. . In  the  distributed  present singularities  element a i r f o i l s walls.  On  condition On  investigation,  solid  i n the wall  is  extended  presence of s o l i d sections, only  a p p l i e s , hence only  the  above  to  include  method  of  multiple-  or s l o t t e d wind  tunnel  the zero norma 1 - v e l o c i t y  source elements are a p p l i e d  a i r f o i l - s h a p e d bodies with Kutta c o n d i t i o n s  applied  at  there. their  11  individual  trailing  required. Figure  7,  of  N  the  Thus  edges,  for  the  the  system  both  source  vortex elements  two-dimensional  o f N+M  equations  are  configuration  t o be  of  s o l v e d i s composed  equations N  M  R(k)  lAjior-I^iBmi-Usinfl j=l k=l m=l prescribing  zero  normal  N ?  and  M ( B  Ju  + B s  (17)  velocities,  and  the  M  equations  R(k)  JL H ? k? mUs k=l m=l +  i=i,2,-N  r  ( A  s  + A  ^L )  =  s  - U ( C O S 0 +C0S(9 u  )  L  s=l,2,-M  (18) for  the  The and  M bodies  subscripts lower  with D and  L  indicate  s u r f a c e s o f an  Thus there  and  airfoil  the c o n t r o l section,  trailing  p o i n t s on  a t the  edges.  the  trailing  upper edge.  are:  -a t o t a l airfoils  Kutta c o n d i t i o n s at t h e i r  solid  of H  source  elements  distributed  over  the  walls.  -a t o t a l  of M bodies r e q u i r i n g Kutta c o n d i t i o n s . M -a t o t a l o f ]^R(k) v o r t e x e l e m e n t s d i s t r i b u t e d o v e r t h e k=l airfoils; the k - t h s u c h body has R (k) s o u r c e e l e m e n t s and R(k) equal-strength  vortex  elements  distributed  over i t .  -N  unknown s o u r c e  s t r e n g t h d e n s i t i e s OJ j.  -H  unknown v o r t e x  strength densities  v  -N+M Of Utr r • • • n 2  equations  in  the  1  \ . N+M  unknowns  12  V Numerical S o l u t i o n A  program w r i t t e n i n FORTRAN f o r use on the UBC IBM 360/67  computer i s used t o c o n s t r u c t the m a t r i c e s the  coordinates,  lengths  and o r i e n t a t i o n s  vortex elements on the a i r f o i l then  used  to  H+H e q u a t i o n s .  and w a l l s .  be  Bji  and  partitioned  each c o n t a i n more than  source  and v o r t e x  describe  C  +C  c  where  + C  l,N+l i  C  2)2  °2  CT  +C  the r e l a t i v e  storage  on  geometry  of the  t o be s o l v e d i s w r i t t e n  2,N+I°2  +  + C  ""  + C  N+I,tf  C  +C  a  + C  N,N+l N A  + C  )  NH,N^ N+I,N*I^  '* N+M,I^M  +  N+I,2 I  a  " N,N N  +  a  + C  *" N,2 N  +  l,N i , 2 , N 2 a  and temporary  must  geometry.  +  ij2  of the  must be r e c a l c u l a t e d f o r each change  system o f equations  C oj  capacity  elements; t h a t i s , t h e i r r e l a t i v e p o s i t i o n and  o r i e n t a t i o n . These m a t r i c e s  The  100,000 nonzero,  f o r example, on magnetic tapes o r d i s c s . The  Bji  and  in relative  are  hence t h e  hence such l a r g e m a t r i c e s  f o r computation  devices;  matrices  matrices  N+M i s of order 300 t o 100,  i s o n l y 250,000 e n t r i e s ,  auxiliary  o f the source and  These  nonsymmetric e n t r i e s . The t o t a l computational 360/67  given  assemble the c o e f f i c i e n t s of the unknowns i n t h e Typically  Ajjj  matrices  and Bj  Ajj  +C  " N+M 2 M  +  +c  r  = d  l  +  = dN  " " N+M,N^M + C  + -  "  + C  2  (19)  N+M,N/M  l ^ + M ^ 2 ^ + M 2 ' " N ^ + M N N+l,N+M^ ' " N+M,N+M^M N+M j +C  the matrix  a  +  +C  CT  + C  CjiJ.and t h e column  a r e a s s e m b l e d from  +  + C  =  d  vector  in  t h e system  t h e m a t r i c e s Ajj  and Bji  by means of  13  (17)  equations  and (18) ; t h a t i s ,  AH  j=l,2,-N; i = l,2,-N  RM ~XB j  k=l,2,-M;i=N+k i=l,2,-N  JU  j = l,2,- N;s=l,2,-Mv i=N+s  m  m=l  ;  (20) B  +  B  S  Z(A  JL  S  +A ,  )  m  M U  k=l,2,-M-,i=N*k;s = l,2,-M;i=Nfs  and UsinSj  i=i,2,-N  -UC0So\j - U C O S 0  (21)  s=l 2,-M i=N+s  L  J  ;  The summations N'  M  Rjk)  Vn-lAjioj - Z r ^ B m i - U s i n S , j=l  (22)  k=l m=!  and N  M  V. = l B t  T  i  j=|  J  j j 0 1 J  R(k)  +Zr lAmi+  UcosSi  k  k=l m=l K  (23)  provide points points  the 'i*  net  normal  and  due to a l l sources  on  coefficient  solid  surfaces,  tangential velocities at control  and v o r t i c e s ' j ' . V  nj  i s zero,  i s c a l c u l a t e d from  At  a l l control  and the l o c a l  pressure  Vf, (25)  The r e s u l t i n g v a l u e s o f C around  the  airfoil  contour  r  may be to  integrated  determine the l i f t ,  numerically drag, and  14  p i t c h i n g momant c o e f f i c i e n t s , from the  expressions  C =-£ZC AXi LT  Pl  C , -c-£CpjAy, . =  D  1  (26)  where AXj = ASjCOS^i ,  Ay, =ASj sinc?j  (27)  and  summations are  performed  clockwise  around  the  polygonal  contours. Resultant the flow  field  throughout  velocities  may  a l s o be c a l c u l a t e d a t p o i n t s i n  not on the a i r f o i l or w a l l s , so t h a t  the f l o w f i e l d  may  be drawn as  isoclines.  streamlines  15  VI R e s u l t s The  agreement  of  pressure  distributions  Smith's method, f o r a i r f o i l s i n f r e e a i r , tests  is  well  established  calculated  with  (Smith (8))..  two-dimensional  Figure 8  shows  comparison of c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n s f o r a 14% Clark-Y  airfoil  i n f r e e a i r and i n the presence  (Wenzinger for  et  al{10)),  the same a i r f o i l  an  NACA  23012  with c a l c u l a t e d  i n f r e e a i r and  airfoil  pressure  i n the  thick  lift.  F i g u r e 9 shows a comparison of an e x p e r i m e n t a l l y for  a  of s o l i d w a l l s .  The w a l l s , f o r t h i s s i z e model, produce 30% h i g h e r  pressure d i s t r i b u t i o n  by  determined with  distributions  presence  w a l l s . E x p e r i m e n t a l l y , the p o t e n t i a l flow f r e e - a i r  flap  of  solid  p r e s s u r e s are  not a c h i e v e d because of boundary l a y e r e f f e c t s . For  both  the  above a i r f o i l s ,  that the u n d e r s u r f a c e presence  of  solid  pressure  ^walls;  The  lift  not  developed  much  in  the  lift.  by the a i r f o i l  terms  of  the  lift  i n the t u n n e l and  C|_F  i n free a i r .  F i g u r e 10 shows the v a r i a t i o n of a;for  change  observation i s  the upper or s u c t i o n s u r f a c e  c o e f f i c i e n t s are r e p o r t e d i n  coefficient developed  does  hence  p r o v i d e s most of the i n c r e a s e d  an important  models  of  differing  C/H.  with angle  of  attack  For s m a l l models, the  c u r v e s are concave downward as i s u s u a l , while f o r l a r g e  lift  models  they are concave upward. The  ratio  of l i f t  c o e f f i c i e n t s i s shown i n F i g u r e 11 as a  16  function  of  model  corresponding  size,  prediction  for of  three  conventional  (Woods (12)), which i s independent of w e l l with the t e s t data The high  very  lift,  wall  large  angles  angle  provide first  a i r f o i l s i n an (Figure 12).  that  would  exhibit  otherwise  This  unstalled  uniform  the  walls,  stream  configuration  tunnel  with  condition,  might  for a  known c a n c e l l i n g which  would  infinite  extent  (spanwise)  slats,  no flow  a  slotted  separation  would be o p e r a t i n g  in  c o n d i t i o n a p p l i e d at i t s  edge.  streamline  streamline, turbulent  is  at constant mixing  undesirable multiple  developing  be used t o r e p r e s e n t  lower s t r e a m l i n e  AB, which e n t e r s the  near the entrance t o the t e s t s e c t i o n .  lower  agrees  and  of  transversely  with a Kutta  Consider t h e l i m i t i n g tunnel  attack,  n e g l i g i b l e or s m a l l w a l l c o r r e c t i o n s .  With a i r f o i l - s h a p e d t r a n s v e r s e  trailing  theory  corrections for large a i r f o i l s  would occur a t the s l a t s , as each w i n g l e t an  wall  c o n f i g u r a t i o n i n v e s t i g a t e d was a s e t o f m u l t i p l e  two-dimensional walls.  of  The  (Lim (6)) .  e f f e c t s o f p a r t l y open, p a r t l y c l o s e d  The  solid  attack.  shown i n the above r e s u l t s , prompted a search  configuration  therefore  of  a  shear  zero  pressure,  the  tunnel,  e f f e c t . But  the  corresponding  stream  close  as which  this  a  free brings  to the a i r f o i l , an streamline  representation  i n the  i s not a f r e e  but merely one of the i n f i n i t e stream. Thus, i n t h i s  representation, streamline,  reference  idealized  into  air f o i l - i n f i n i t e  streamline,  layer,  Physically,  and  the  pressure  is  not  zero  e r r o r s would be i n t r o d u c e d  on  this  lower  i n representing  the  17  flow the  i n t h i s manner; i n p a r t i c u l a r , c l o s e to  the  underside  of  airfoil. S i m i l a r l y the upper l i m i t i n g  represented the  e a s i l y , but  airfoil  conditions pressure should  by  the  With  on  the  slats  flow,  secondary e f f e c t s on  only  one  p o s s i b l e to produce  the  cancelling  their  errors of  from  boundary  in incorrect  this  streamline  the main a i r f o i l .  s l o t t e d wall, Figure the  i s separated  with  l o c a t i o n i n the r e p r e s e n t a t i o n  have only  cannot be c o r r e c t l y  since t h i s streamline  intervening  impressed  and  streamline  13,  i t should  effects  of  still  partly  open,  p a r t l y c l o s e d w a l l s , s i n c e the upper s l o t t e d wall i s adjacent suction  lift  i s developed. Hence a combination of a t r a n s v e r s e l y s l o t t e d wall  of  with  effectively  a  the a i r f o i l ,  solid  where most of the  to  the  upper  side  be  lower  correction-free  wall  test  was  increased  envisaged  configuration, for a  as  an  lifting  airfoil. T h i s c o n f i g u r a t i o n should  be a c c u r a t e l y represented  d i m e n s i o n a l p o t e n t i a l flow t h e o r y , wall  slats  should  u n s t a l l e d , and only  For  the upper shear l a y e r w i l l  Clark-Y,  the r a t i o of l i f t 50%  and  solid  75%  OAR  w a l l s , and  a n g l e s at  the  re-enter  the  will  be  tunnel  airfoil.  i n v e s t i g a t i o n of t h i s c o n f i g u r a t i o n the  two-  be s m a l l enough t h a t these w i n g l e t s  w e l l downstream of the An  s i n c e the flow  by  at an angle  base  a  followed.  of 20° , F i g u r e  14 shows  c o e f f i c i e n t s as a f u n c t i o n of model s i z e f o r a  upper w a l l . A l s o shown are f o r the  airfoil  i n ground  the  curves  effect.  for  two  18  For of  lift  upper  t h e same a i r f o i l coefficients  w a l l OAR,  z e r o OAR the  airfoil  in  20°  C/H  corresponding the  the  forward  ground  of  at  70%  OAR  the  net wall  wall  of  to lift  correction;  this  70% OAR.  ^boge o f  a t an 70%  OAR,  i n F i g u r e 16 a l o n g f o r free  Although  of  corresponds ratio  c o n f i g u r a t i o n of  appears  cases.  upper  f o r the Clark-Y  pressure d i s t r i b u t i o n  there  a i r . The i s less  with  with the net  lift  suction  over  over  the  portion of the a i r f o i l .  results  a r e shown i n F i g u r e 17 f o r t h e same Q^ase  a different^ should  configuration. circular  Where  at approximately  .72,  An  upper s u r f a c e , t h e r e i s i n c r e a s e d s u c t i o n  Similar but  i s shown a s a f u n c t i o n o f  w a l l s ; 100% OAR  i s zero  distribution  same i n b o t h  rearward  15  effect.  a zero c o r r e c t i o n  size  is  t o two s o l i d  airfoil  pressure  f o r such  model  Figure  i s unity, there  occurs f o r t h i s  The  in  of a t t a c k , the r a t i o  f o r a r a n g e o f model s i z e s .  corresponds  coefficients  a t t h e same a n g l e  provide  o f 12 . A g a i n a  Comparison  arc a i r f o i l  a slotted  relatively is  i n ground  also  upper  wall of  correction-free  made w i t h  effect  airfoil  the theory  {Tomotika  et  test f o ra  al(9))  of  |  similar  5.3%  camber,  but  a t an  a  chon  j  jof 5 ° . T h e a g r e e m e n t i s !  favourable. Results Figure 70%  obtained  18 i n d i c a t e s OAR  with  correction-free  The  a  f o r t h e NACA  230 12 w i t h  flap  that a transversely s l o t t e d solid  lower  wall  provides  were s i m i l a r .  upper a  wall  of  relatively  test configuration.  relative  error  i n C|_  for  a 70% upper w a l l OAR  i s shown  19  in  Figure  19  and  the  OAR  is less  f o r the  Clark-Y  NACA 23012 a i r f o i l  Table  than  3%,  with  except  1 o u t l i n e s the  airfoil flap.  at  four  The  f o r extremely  details  angles  relative large  of  attack  error at  this  models.  of a l l c o n f i g u r a t i o n s  tested.  20  VII  Conclusion  An based  extension  on t h e  shown  that  configuration range  in  solid  singularity  relatively  by  (spanwise)  distribution  correction-free  for l i f t i n g airfoils sizes  p o t e n t i a l flow  a  70%  s l o t t e d upper w a l l ,  wall.  Small  or  i s , single airfoils  of experimental  undertaken. a  Where  linearized  configuration  wall  for  a  wide  area  ratio  i n conjunction  with c a n be  and f o r d i f f e r e n t combinations. A  of these r e s u l t s should  corrections  m i g h t be d e v e l o p e d .  test  open  or a i r f o i l - f l a p  perturbation  tunnel  negligible corrections  verification  the  has  c a n be a c h i e v e d  utilizing  thoery  procedure  wind  f o r a wide r a n g e o f a n g l e s o f a t t a c k ,  models, t h a t  small,  a  lower  achieved  program  surface  model  transversely a  of the two-dimensional  be  a r e not n e g l i g i b l e but  theory  based  on  this  21  cu  c c 3  4-1  •o c -H  rt c o t-l  CU  3 OC  ZD  22  .6  F i g u r e 2.  .8  Comparison of l o n g i t u d i n a l l y s l o t t e d lift  theory with data f o r Clark-Y  1.0  wall  airfoil.  SOLID CLARK-Y  14%  LONGITUDINAL  THEORY 1 2 3 4  .2  OAR% 5.6 II.1  18.5 100  .4  SLOTS  P .46 .83 1.25  EXPT  OD  na  A  n  V  .6  .8  c/,  H  F i g u r e 3.  Comparison o f p o r o u s - w a l l l i f t data f o r Clark-Y a i r f o i l longitudinally slotted  theory w i t h  t e s t e d between  walls.  CLARK-Y 14% • •SLOTTED LONGITUDINAL  SLOT FLAP SLOTS  O  •  THEORY  OAR %  P  EXPT  I  II. 1  .33  2  18.5  .67  3  29.6  1.54  V  4  100  00  na  .2  A  •  .6  .8  1.0  H F i g u r e 4.  Comparison o f p o r o u s - w a l l l i f t  theory w i t h  data f o r Clark-Y a i r f o i l w i t h s l o t t e d  flap  t e s t e d between l o n g i t u d i n a l l y s l o t t e d  walls.  v  AIRFOIL WITH FLAP  THEORY  OAR  Figure  5. P o r o s i t y parameter as a f u n c t i o n o f openarea  ratio.  X  F i g u r e 6.  Geometry and n o t a t i o n f o r Smith's method.  Figure 7.  Source and vortex d i s t r i b u t i o n s for a twodimensional a i r f o i l between s o l i d walls.  CLARK-Y 14% a = 20° C/H=.72 X \ \  \  \  XSOLID X WALLS FREE AIR  F i g u r e 8. P r e s s u r e d i s t r i b u t i o n s f o r C l a r k - Y  airfoil  F i g u r e 9.  P r e s s u r e d i s t r i b u t i o n s t o r NACA 23012 a i r f o i l .  Figure  10.  Lift solid  coefficient walls.  for Clark-Y  airfoil  between  t  I  t  I  • t  CLARK-Y  14%  SOLID  WALLS  2  figure  11.  Ratio  .4  of l i f t  between  solid  / /  .6  .8  coefficients  f o r Clark-Y  walls.  1.0  airfoil  32  C L A R K - Y  14%  a = 20° SLOTTED  UPPER  SOLID LOWER  8  .4 .6 .8  20  40  60  80  100  OAR-% Figure  15.  Variation wall  of  l i f t  open-area  coefficient  ratio  ratio  f o r Clark-Y  with  airfoil.  upper  i  F i g u r e 16.  CLARK-Y  14%  Pressure d i s t r i b u t i o n f o r Clark-Y a i r f o i l i n correction-free l i f t  test  configuration.  NACA  23012  a=8° 8 = 20° SLOTTED UPPER SOLID LOWER , SOLID  70%  OAR  GROUND EFFECT  6  .8  1.0  H Figure-  .13.  Variation  of lift  geometry  f o r NACA  coefficient 23012  ratio with  airfoil  with  wall  slotted  -2 '  9  H 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 f o r 70% upper w a l l open-area r a t i o . 7  F i g u r e 19.  .8  1.0  Table 1. AF  a  NSA  C-Y  20  50  Solid  50  50  C-Y  20  50  Solid  50  C-Y  20  -50  Solid  C-Y  20  50  C-Y  20  C-Y C-Y  WALL CONFIGURATION  Configurations Tested  NSU NSL NSLAT c/C  -  50  -  50  50  Solid  50  50  50  Solid  50  50  20  50  Solid  50  50  20  50  Solid  50  50  50  50  *•  t/c  NSS OAR NWC  -  -  -  -  -  -  -  -  -  C/H  CLF  CLT  CLT/CLF  6.3  1.0 3.091 5.360 1.734  -  -  6.3  .8 3.091 4.416 1.428  -  -  6.3  .72 3.091 4.121 1.333  -  6.3  .6 3.091 3.785 1.224  -  -  6.3  .4 3.091 3.378 1.092  -  -  -  -  6.3  .3 3.091 3.242 1.049  -  -  -  -  -  -• 6.3 - 6.3  .2 3.091 3.149 1.020  -  -  -  -  6  1.0 2.188 3.475 1.585  -  -  -  6  .8 2.188 2.982 1.360  -  -  -  6  .6 2.188 2.622 1.195  -  6  .4 2.188 2.372 1.080  -  -  -  -  6  .2 2.188 2.225 1.015 1.0 1.955 3.065 1.567  -  -  6.3  -  -  6.3  .8 1.955 2.648 1.355  C-Y  20  50  Solid  C-Y  12  50  Solid  100 100  -  C-Y  12  50  Solid  100 100  -  C-Y  12  50  Solid  100 100  -  C-Y  12  50  Solid  100 100  C-Y  12  50  Solid  100 100  -  -  C-Y  10  50  Solid  50  50  -  C-Y  10  50  Solid  50  50  -  C-Y  10  50  Solid  50  50  -  -  -  6.3  .6 1.955 2.337 1.195  10  50  Solid  50  50  -  —  6.3  .4 1.955 2.119 1.083  C-Y  10  50  Solid  50  50  -  -  -  -  6.3  .2 1.955 1.990 1.018  C-Y  0  50  Solid  50  50  -  -  -  C-Y  -  -  -  6.3  1.0 .7635 1.115 1.460  0  50  Solid  50  50  -  -  C-Y  0  50  Solid  50  50  -  -  -  C-Y  -  -  C-Y  0  50  Solid  50  50  -  -  C-Y  0  50  Solid  50  50  —  —  -  -  -  .1 3.091 3.100 1.003  -  6.3  .8 .7635 .988  1.293  -  -  6.3  .6 .7635 .889  1.163  -  -  -  6.3  .4 .7635 .818  1.071  _  —  —  6.3  .2 .7635 .775  1.013  AF  a  NSA  WALL CONFIGURATION  NSU NSL NSL  -  c/C  t/c  -  -  -  -  NSS OAR NWC  C/H  CLF  CLT  CLT/CLF  -.020 -1.39  6.3  1.0 .0415  6.3  .8 .0415  -.013  6.3  .6 .0415  -.004 -.290  6.3  .4 .0415  .0049 .338  6.3  .2 .0415  .0116  C-Y -6.2  50  Solid  50  50  C-Y -6.2  50  Solid  50  50  C-Y -6.2  50  Solid  50  50  -  C-Y -6.2  50  Solid  50  50  -  C-Y -6.2  50'  Solid  50  50  -  C-Y  20  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  1.0 3.091 3.768 1.219  C-Y  20  50  T.S.U.S.L.  80  15  .12  .33  9  .5  3.4  .8 3.091 3.524 1.140  C-Y  20  50  T.S.U.S.L.  80  15  .12  .33  9  .5  3.4  .72 3.091 3.465 1.121  C-Y  20  50  T.S.U.S.L.  80  15  .12  .33  9  .5  3.4  .6 3.091 3.348 1.083  80  15  .12  .33  9  .5  3.4  .4 3.091 3.218 1.041  80  15  .12  .33  9  .5  3.4  .2 3.091 3.131 1.013  80  15  .06  .33  9 .75  3.4  1.0 3.091 3.062  .991  80  15  .06  .33  9 .75  3.4  .8 3.091 3.037  .983  80  15  .06  .33  9 .75  3.4  .72 3.091 3.033  .981  80  15  .06  .33  9 .75  3.4  .6 3.091 3.035  .982  80  15  .06  .33  9 .75  3.4  .4 3.091 3.063  .991  -  -  6  1.0 3.091 2.676  .866  6  .8 3.091 2.733  .884  6  .72 3.091 2.762  .894  6  .6 3.091 2.811  .909  6  .4 3.091 2.919  .944  6.  .3 3.091 2.983  .965  6  .2 3.091 3.045  .985  100  -  -  6  .1 3.091 3.085  .998  80  15  .072 .33  9  .7  3.4  1.0 3.091 3.170 1.026  80  15  .072 .33  9  .7  3.4  .72 3.091 3.092 1.0  80  15  .072 .33  9  .7  3.4  .55  3.091 3.078  .996  80  15  .072 .33  9  .7  3.4  .39  3.091 3.087  .999  80  15  .072 .33  9  .7  3.4  .19  3.091 3.098 1.002  *  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  G.E.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  C-Y  20  50  T.S.U.S.L.  -  -  100 100 100 100 100 100 100  -  -  -  -  -  -  -  -  -  -  -.903  .80  CLT CLT/CLF C/H CLF 1.0 2.188 2.617 1.195  AF  a  NSA  WALL CONFIGURATION  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  t/c .33  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  .9 2.188 2.537 1.158  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  .8 2.188 2.463 1.124  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  .6 2.188 2.343 1.070  C-Y  12  50  T.S.U.S.L.  80  15  .12  .33  9  .5  3.4  .45 2.188 2.276 1.040  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  .26 2.188 2.221 1.013  C-Y  12  50  T.S.U.S.L.  -  80  15  .12  .33  9  .5  3.4  .16 2.188 2.199 1.003  C-Y  12  50  T.S.U.S.L.  -  80  15  .06  .33  9 .75  3.4  1.0 2.188 2.234 1.021  C-Y  12  50  T.S.U.S.L.  -  80  15  .06  .33  9 .75  3.4  .6 2.188 2.179  .996  C-Y  12  50  T.S.U.S.L.  • - 80  15  .06  .33  9 .75  3.4  .45 2.188 2.178  .994  C-Y  12  50  T.S.U.S.L.  -  80  15  .06  .33  9 .75  3.4  .26 2.188 2.187  .998  C-Y  12  50  T.S.U.S.L.  -  80  15  .06  .33  9 .75  3.4  .16 2.188 2.189 1.001  C-Y  12  50  T.S.U.S.L.  -  80  15  .06  .33  9 .75  3.4  .09 2.188 2.188 1.000  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  1.0 2.188 2.284 1.044  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .8 2.188 2.233 1.021  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .6 2.188 2.200 1.006  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .45 2.188 2.190 1.001  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .26 2.188 2.192 1.002  C-Y  12  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .16 2.188 2.190 1.001  C-Y  12  50  T.S.U.S.L.  80  15  .072 .33  9  .7  3.4  .09 2.188 2.188 1.000  C-Y  12  50  G.E.  -  100  -  C-Y  12  50  G.E.  -  100  -  C-Y  12  50  G.E.  -  100  -  C-Y  •12  50  G.E.  -  100  -  C-Y  12  50  G.E.  —  100  -  -  NSU NSL NSLi  c/C  -  -  _  _  -  NSS OAR NWC 9 .5 3.4  6  1.0 2.188 2.060  .940  6  .8 2.188 2.068  .945  6  .6 2.188 2.087  .954  6  .4 2.188 2.121  .968  6  .2 2.188 2.170  .990  A I' C-Y  6  NSA 50  C-Y  6  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  C-Y  6  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4 .45 1.483 1.485 1.002  C-Y  6  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4 .26 1.483 1.484 1.001  C-Y  6  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4 .16 1.483 1.483 1.000  C-Y  6  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4 .09 1.483 1.482 1.000  C-Y  0  50  T.S.U.S.L.'  -  80  15  .072 .33  9  .7  3.4 1.0 .7535  C-Y  0  50  T.S.U.S.L.  -  80  15  .072 .33  9  .7  C-Y  0  50  T.S.U.S.L.  -  80  15  .072 .33  9  C-Y  0  50  T.S.U.S.L.  -  80  15  .072 .33  C-Y  0  50  T.S.U.S.L.  -  80  15  C-Y  0  50  T.S.U.S.L.  -  80  15  U  WAL L CONFIGURATION T.S.U.S.L.  NSU NSL NSLAT c/C t / c .072 .33 80 15  NSS OAR NWC C/ll CLF CLT CLT/CLF 3.4 1.0 1.483 1.557 1.050 9 .7 .6 1.483 1.495 1.008  .772 1.001  .6 .7635  .758  .992  .7  3.4 .45 .7635  .757  .992  9  .7  3.4 .26 .7635  .761  .996  .072 .33  9  .7  3.4 .16 .7635  .762  .998  .072 .33  9  .7  3.4 .09 .7635  . 762 .998  .3-4  23012 8 -20 81=46+35  Solid  100 100  -  -  -  -  -  6  1.0 2.442 3.300 1.392  23012 8 -20  81  Solid  100 100  -  -  -  -  -  6  .8 2.442 3.009 1.233  2 3012 8 -20  81  Solid  100 100  -  -  -  -  6  .6 2.442 2. 770 1.135  23012 8 -20  81  Solid  100 100  -  -  -  6  .4 2.442 2.587 1.061  23012 8 -20  81  Solid  100 100  -  -  -  6  .2 2.442 2.472 1.012  23012 8 -20  81  G.E.  -  -  -  6  1.0 2.44 2 2.176  .892  23012 8 -20  81  - •  -  6  .8 2.442 2.212  .905  23012 8 -20  -  -  6  .6 2.442 2.261  .926  6  .4 2.442 2. 330  .955  2.442 2.4.1.3  .988  100  G.E.  -  -  100  -  81  G.E.  -  100  -  -  -  2 3012 8 -20  Kl.  G. I'.  100  -  -.  -  -  -  2 301 ? 8 -20  81  c;. If..  -  _  _  _  _  100  -  -  v  6  9  AF  a  NSA  WALL CONFIGURATION  NSU NSL NSLAT c/C  t/c  NSS OAR NWC  C/H  CLF  CLT  CLT/CLF  23012 8-20  81  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  23012 8-20  81  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  23012 8-20  81  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .45  2.442 2.416  .989  23012 8-20  81  T.S.U.S.L.  80  15  .072 .33  9  .7  3.4  .26  2.442 2.439  .999  23012 8-20  81  T.S.U.S.L.  -  80  15  .072 .33  9  .7  3.4  .19  2.442 2.443 1.000  23012 8-20  81  T.S.U.S.L.t  —  80  15  .072 .33  9  .7  3.4  .09  2.442 2.442 1.000  1.0 2.442 2.413  AF - A i r f o i l c o n f i g u r a t i o n , a  - Angle  of attack-degrees.  NSA - Number o f source and v o r t e x elements on a i r f o i l . T.S.U.S.L. - T r a n s v e r s e l y s l o t t e d upper and s o l i d G.E. - A i r f o i l  i n ground  lower w a l l s .  effect.  NSU - Number o f source elements on upper s o l i d  wall.  NSL - Number of source elements on lower  wall.  NSLAT - Number o f a i r f o i l - s h a p e d  slats.  c/C - S l a t chord  ratio.  t/c  : airfoil  chord  - S l a t t h i c k n e s s : chord  solid  ratio.  NSS - Number o f source and v o r t e x elements p e r s l a t . NWC  - T o t a l extent o f w a l l i n a i r f o i l  chords.  .8 2.442 2.402  .988 .984  45 REFERENCES 1.  Allen,H.J.  "Wall  I n t e r f e r e n c e i n a Two-Dimensional  V i n c e n t i , W . G.  Wind T u n n e l , Effect  Flow  with C o n s i d e r a t i o n o f the  of Compressibility".  NACA TR 782, 1944  2.  Baldwin,B.S.  "Wall  I n t e r f e r e n c e i n Wind T u n n e l s  T u r n e r , J . B.  Slotted  Knechtel,E.D.  Subsonic  with  and P o r o u s B o u n d a r i e s a t Speeds".  NACA TN 3176, 1954.  3.  Glauert,H.  4. G o l d s t e i n , S .  "Wind T u n n e l  I n t e r f e r e n c e on Wings,  and  Airscrews".  RSM  1566, B r i t i s h  Bodies  ARC 1933  "Two-Dimensional Wind-Tunnel I n t e r f e r e n c e , Part I I . " RSM  5.  Kellogg,O.D.  1902, B r i t i s h  "Foundations  ARC 1942.  of P o t e n t i a l  Theory".  Dover  6. Lim,A.K.  "Effects  of Porous Tunnel  A i r f o i l Testing". Thesis,  UBC 1970.  Walls  on High  Lift  46 7.  Parkinson,G.V. Lim,A.K.  "On t h e Ose o f S l o t t e d Dimensional  Testing  Walls  i n Two-  o f Low-Speed  Airfoils".  CASI T r a n s . 4, S e p t . 1971.  8. Smith,A.M.O. H e s s , J . L.  "Calculation  of Potential  Arbitrary  Bodies".  Progress  in  Flow  Aeronautical  about  Sciences  Vol.8,  1967.  9. T o m o t i k a , S . Tamada,K.  "The  Lift  Airfoil  and Moment on a  in  a  Stream  Circular-Arc  Bounded  by  a  Plane  Wall". U memoto,H.  10.  Wenzinger,C.J. D e l a n o , J . B.  QJMAH  "Pressure Airfoil NACA  11.  Wood,W.W.  Woods,L.C.  "The  D i s t r i b u t i o n over  with  a Slotted  an NACA 23012  and a P l a i n  Flap".  TR633, 1938.  "Tunnel QJMAM  12.  V o l . 4 , 1950.  I n t e r f e r e n c e from  Vol.17,  Theory  Slotted  Walls".  P a r t 2, 1964.  of Subsonic  C a m b r i d g e , 1961.  Plane  Flow".  

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