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A new slotted-wall method for producing low boundary corrections in two-dimensional airfoil testing Williams, Christopher Dwight 1975

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A NEW SLOTTED-WALL METHOD FOR PRODUCING LOW BOUNDARY CORRECTIONS IN TWO-DIMENSIONAL AIRFOIL TESTING by  CHRISTOPHER DWIGHT WILLIAMS B.A.Sc., University of B r i t i s h Columbia 1967 M.A.Sc., University of B r i t i s h Columbia 1973  THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f M e c h a n i c a l E n g i n e e r i n g  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY  OF B R I T I S H  O c t o b e r 1975  COLUMBIA  OF  In p r e s e n t i n g an  this  thesis  advanced degree at  the L i b r a r y  shall  I f u r t h e r agree for  in p a r t i a l  the U n i v e r s i t y of  make i t  freely  that permission  available  his  of  this  representatives. thesis for  It  financial  is understood gain s h a l l  Department of University of  British  Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  a  t  e  for  the  requirements  Columbia, reference  for e x t e n s i v e copying o f  w r i t ten pe rm i ss ion .  D  British  of  I agree  for  that  and study. this  thesis  s c h o l a r l y purposes may be granted by the Head of my Department or  by  The  fulfilment  t.]>  Qdi9kf  1 9 1 6  that  not  copying or p u b l i c a t i o n  be allowed without my  SUPERVISOR:  D r . G.V.  Parkinson  ii  AB3 TRACT  This  thesis  wiadtunnai  deals  with  a  new  wall corrections in a i r f o i l  transversely-slotted airfoil,  and  elements  of  incidence.  wall  a solid the  wall opposite  slotted  T h i s geometry  streamline  opposite  pattern  permits  for  the  the  flow flow,  quality  through  The  theory  uses  the  Kutta  conditions satisfied  method, w i t h the  wall  airfoils  slats. of three  predictions  of  uncorrected  lift  accurate of  wall  In  theory  has  coefficients percent,  shapes, s i z e s ,  of open-area r a t i o  and  side.  assuaa  the  on  good  the a  can  lift  between 60  zero  closely  the  degrading  the  airfoil.  airfoil  and  of s i z e s  agreement  with  the  I t appears  that  obtained  70  solid  range  coefficients, and  the  source-element  pressure be  a  at  test  test  been o b t a i n e d . and  of  The  airfoils  surface  using  profiles,  side  without  near  flow  experiments  different the  mixing  potential  t o w i t h i n one  airfoil  layer  to  reducing  employing  suction  pressure  flow  to  by  symmetrical  unconfined  shear  testing,  the  w a i l are  approaca  of  distributions, f o r a wide using  percent.  a  range  slotted  i i i  Cette les  flanc  methods de  de  zero.  le  e s t obtenu  d'essai-  de  La  theorie  experiences  ete  l e s hypotheses  pres;  variete  de  utilisant  surface  mux  des  a,  source  totale,  de  d'essai  farts  et i l s  mur  de  en  Les  de  eaployant.  etre  dimensions  soixante  de a  de  obtenus  at de  quotient  sont  d'incidence suivie  l'ecoulement  par  pres  da  a  l'aile  potentiel  satisfaisant  ailes  de  soixante-dix  Les  corde  pour  exacts une  non a  un  qrande  portances;  paroi  pourcent.  de  assez  portances  sont  la  les  murales.  coefficients entre  face  ce  coefficients pressions  du  libre;  ailes des  en  t h e o r i e s obtenuas r e s p e c t e n t  peuvant  a fente,  aux  face  d'air,  l'ecoulement  surface, et  ailes.  fente  l'ecoulement  discontinuite,  distributions  profils, un  de  diminuer  solide,  l'ecoulement  de  p r a e t a b l i e s . Les  les  pourcent  en  la  differents.  et  permet  demande d ' u t i l i s e r  a l'aile  ont  profils  verifies,  de  t r a n s v e r s a l e , en  s o l i d e s du' mur  courant.  couche  de  c o n d i t i o n s Kutta  bien  vue  essais  d ' e s s a i ; e t un  elements  en  s a n s d i i a i n u e r l a g u a l i t e de  la  elements  trbis  Les  aux  taur a. f e n t e  l'aile  l e s l i g n e s de  melange  des  de  un  inethode  soufflerie  f i g u r e geotuetrique  par  resultat  nouvelle  aerodynamiques et syraetriques a 1'angle  Cette  pres  einploie  pression.  profils  une  p a r o i s en  depression  flanc  de  de  d_crit  c o r r e c t i o n s da  Cette  du  these  et  la  IV  ACKNOWLEDGEMENT  This Dr.  research  G. V. P a r k i n s o n ,  gratefully  In  the design  This Columbia  whose  under  expert  the  advice  supervision  and  guidance  and c o n s t r u c t i o n o f t h e v i a d t u n n e l  t h e work done and t h e a d v i c e  the Mechanical  All  out  of i s  acknowledged.  equipment, of  was c a r r i e d  Engineering  t h e computing  research  contributed  D e p a r t m e n t was e x t r e a e l y  was s u p p o r t e d  and  more t h a n  by t h e t e c h n i c i a n s  was done a t t h e U.B.C. C o m p u t i n g  and t h e D e f e n c e  Encouragement  given  models and  by t h e  University  Research  Board  of Canada.  support  were  provided  her share  i n our j o i n t  of  valuable.  Center.  British  by F.M.W. , who  effort.  Table  of Contents  Abstract Resume Acknowledgement List  of  Figures  List  of Plates  List  of  Tables  Notation Introduction Survey o f Windtunnel Wall §2.1  Conventional  §2.2  Results  §2.3  Low  A New  Linear  Theories  of Conventional  Correction Test  Slotted-Wall  A Physical Basis  §3.2  Formulation  §3.3  Other A i r f o i l - W a l l  Assembling  §4.2  Solving the  Testsection  §6.2  Airfoil  §6.3  Test  Theory  Numerical  Configurations  Equations  Equations Theory  Experiments to V e r i f y §6.1  f o r t h e New  Solution  the  o f t h e New  Theories  Configurations  o f an E x a c t  Methods o f N u m e r i c a l §4.1  Linear  Theory  §3.1  Results  C o r r e c t i o n Methods  t h e New  Design  Models  Procedures  Tested  Theory  Theory Examined  vi  54  7. E x p e r i m e n t a l R e s u l t s 8. E x t e n s i o n s  t o t h e New  §8.1 P o t e n t i a l  Theory  60  Flow C o n s i d e r a t i o n s  §8.2 The F l o w i n t h e P l e n u m : The  of Viscous E f f e c t s  Bounding  Shear L a y e r  §8.3 Summary  1.  The  77 Integration  of a Three-Dimensional Point  Source t o a Two-Dimensional Source Appendix  2.  3.  Flat  Distributed  A Procedure f o r Block Computation o f  Two  79  Element  Matrices Appendix  69 76  9. C o n c l u s i o n s Appendix  62  A, B,  and  84  C.  Methods o f S o l v i n g  the Systems  of  91  Simultaneous Linear A l g e b r a i c Equations Appendix  4.  A Streamline Tracking Algorithm  96  Appendix  5.  Design o f the Two-Dimensional N o z z l e I n s e r t  99  Appendix  6.  An A n a l y t i c  Representation of a L i f t i n g  V o r t e x Between a S o l i d , Slotted 7.  Standard S o l i d  Appendix  8.  A Reduced  9.  Lift  A Reduced  Circulation  Airfoil  11. L i s t  Figures  the  111  Determined  113  Determined  116  Profile  Computer P r o g r a m  Numerical Appendix  Circulation  from the Measured  10. The  Boundary  Wall Corrections  Airfoil  by M o d i f y i n g Appendix  Transversely-  and a C o n s t a n t P r e s s u r e  Appendix  Appendix  a  10 3  f o r the Exact  118  Theory  o f Equipment  Used  152 153  Plates Tables References  viii  List  of  Figures Page  Figure  2.1  Porosity  p a r a m e t e r as  open-area r a t i o Figure  2.2  Ratio  of  Figure  2.3  Variation straight  Figure  3.1  An  3.2  Wall  slotted  boundary:  u p p e r and Figure  l i f t - c u r v e slopes  of pressure  airfoil  wall  for longitudinal slots  airfoil  longitudinally  a f u n c t i o n of  walls:  [12]  for  Experiment[12]  coefficient  along  3.3  155  Theory  between t r a n s v e r s e l y - s l o t t e d solid  effect  on  lower w a l l s : airfoil  156  Theory  pressure  coefficient:  Figure  3.4  157  G e o m e t r y and source  5.1  Surface  Effect  notation of  velocity  5.2  variations for a  source  of a i r f o i l  coefficients: Figure  two-dimensional 158  e l e m e n t s f o r S m i t h ' s method  dimensional Figure  two-  element size  on  159 ratio  of  lift  Theory  Comparison o f a i r f o i l  160 pressure  coefficients: 161  Theory Figure  5.3  Variation straight airfoil  Figure  6.1  154  a  Theory Figure  153  of pressure  coefficient  boundary f o r a with  zero  a  two-dimensional  lift-correction:  U.B.C. M e c h a n i c a l E n g i n e e r i n g closed-circuit  along  windtunnel  Theory  162  low-speed 163  Figure  6.2  Variation  o f mean w i n d s p e e d i n  two-dimensional  Figure  6.3  6.4  p i t o t s t a t i c traverse  Velocity  p r o f i l e i n f l o o r boundary  E f f e c t o f endplate and p i t c h i n g  loadings moment  two-dimensional a i r f o i l 6.5  164  two-dimensional t e s t s e c t i o n  drag  Figure  i n s e r t on  vertical  in Figure  testsection  Calibration  of nozzle  layer  insert  on  165  lift,  coefficients for tests  and  166  testsection  dynamic p r e s s u r e s Figure  6.6  Error  bars  167  f o r measured a i r f o i l  lift  coefficients Figure  6.7  Variation  o f measured a i r f o i l  coefficients Figure  7.1  Variation  7.2  lift  coefficient  169 with  open-area r a t i o : Experiment open-area  171 open-area  ratio  r a t i o of l i f t - c u r v e slopes f o r  NACA-0015 a i r f o i l Figure  7.4  Effect of a i r f o i l for  Figure  7.5  172 s i z e on l i f t - c u r v e s l o p e  NACA-0015 a i r f o i l :  Effect of a i r f o i l for  170  ratio  airfoil  Effect of slotted-wall on  runs  r a t i o of l i f t - c u r v e slopes f o r  Clark-Y 7.3  lift  consecutive  Effect of slotted-wall on  Figure  on t h r e e  of a i r f o i l  slotted-wall Figure  168  Clark-Y  Experiment  173  s i z e on l i f t - c u r v e s l o p e  airfoil:  Experiment  174  X  Figure  7.6  Effect  of slotted  coefficient: Figure  7.7  w a l l on a i r f o i l p r e s s u r e  Experiment  Comparison o f a i r f o i l  175  pressure  coefficients:  Experiment Figure  7.8  Variation  176 of a i r f o i l  moment c o e f f i c i e n t  Figure  7.9  Figure  8.1  ratio:  Variation  of a i r f o i l  Effect  8.2  8.3  Modification  8.4  Figure  8.5  8.6  8.7  Theory  of a i r f o i l circulation  (Appendix  8)  value:  9)  180 airfoil 181 upper  lower w a l l w i t h a plenum i n the plenum  the s l o t t e d  of different  Variation streamline  179  to reduce  t o measured  between a s l o t t e d  The s h e a r l a y e r  178  airfoil  coefficient  An a i r f o i l  Effect  with  Experiment  profile  o f m o d i f i e d p r o f i l e on  on r a t i o o f l i f t Figure  ratio:  Effect  surrounding Figure  open-area  (Appendix  solid  177  drag c o e f f i c i e n t  Theory  pressure Figure  slotted-wall  Experiment  coefficient:  theoretical  Figure  pitching  o f r e d u c e d c i r c u l a t i o n on  pressure Figure  with  open-area  slotted-wall  midchord  and a  chamber  182  chamber 183  wall  types of w a l l boundaries  coefficients:  Theory  of pressure c o e f f i c i e n t i n p l e n u m chamber:  184  along a  Theory  185  xi  Figure  8.8  Effect  on  airfoil  lift  coefficients  assumed p r e s s u r e  coefficients  representing  plenum shear l a y e r :  the  Figure A l . l  Geometry f o r i n t e g r a t i o n  F i g u r e A5.1  The  F i g u r e A6.1  A  Figure  A6.2  vortex  and  a constant  The  image s y s t e m f o r a l i f t i n g  between a s o l i d  and  streamline  a  Notation  f o r the  Appendix  10  187  slotted,  boundary: Theory  a constant  186  188  189  vortex pressure  boundary: Theory F i g u r e A10.1  Theory  insert  between a s o l i d ,  pressure  a  of a p o i n t source  two-dimensional nozzle  lifting  on  of  . 190  computer program  of 191  X X X  L I S T OF PLATES Page Plate l a .  The  U.B.C. M e c h a n i c a l  closed-circuit  Engineering  low-speed  windtunnel  192  Plate l b .  The  octagonal t e s t s e c t i o n i n the windtunnel  192  Plate  2.  The  airfoil-shaped wall  193  Plate  3.  The  wall  Plate  4.  The  616mm NACA-0015  Plate  5.  The  354mm C l a r k - Y a i r f o i l  Plate  6.  The  616mm NACA-0015 a i r f o i l  slats  slats  i n the side wall  frame  airfoil  193 194  f i t t e d w i t h e n d p l a t e s 194 i n the testsection  195  xixi  List  of Tables Page  Table  1.  Airfoil  Table  2.  Free  Table  3.  Airfoil  profile  a i ra i r f o i l  coordinates  196  coefficients:  and w a l l c o n f i g u r a t i o n s  Theory  197  examined  198  theoretically Table  4.  Airfoil  and e n d p l a t e  loadings  203  Table  5.  Windtunnel balance  results  - Clark-Y  Table  6.  Windtunnel balance  results  - NACA-0015  219  results  - Joukowsky  234  airfoils  204  airfoils Table  7.  Windtunnel balance airfoil  Table  8.  Quantities  Table  9.  Pressure  coefficients  f o r NACA-0015 a i r f o i l  243  Table  10.  Pressure  coefficients  f o r Joukowsky a i r f o i l  246  Table  A10  Equations Appendix  derived  from  balance  results  f o r t h e computer program o f 10  238  249  xiv  Notation  A..,  B.., C.. m a t r i c e s  Di  Di  of disturbance  c  airfoil  c. . Ji  element o f t h e matrix  C_  = D/qc  drag  coefficient  C  = L/qc  lift  coefficient  C  velocities.  Di  = Mo/qc  2  chord  midchord p i t c h i n g  C.. 31  moment  Mo C  Mc/qc  = M  C  t  v  n  =  '  quarterchord  2  *  measured  q  C_  pressure  average pressure = 1-(v  C  pitching  /u)  t  2  calculated  moment  coefficient coefficient  pressure  on s t r e a m l i n e  coefficient  i d.  element o f column  1  boundary  v e c t o r o f approach  flow  conditions  ds_.,dXj,dy^.  s u r f a c e element  length  H  windtunnel t e s t s e c t i o n  K(s)  p o r o s i t y parameter  differentials height  (or t o t a l  forlongitudinally  head)  slotted  walls m =  lift-curve  it  outward  OAR  transversely-slotted  p  local  P(s)  p o r o s i t y parameter  q = -ipU  2  slope  surface  static  dynamic  normal w a l l open  area  ratio  pressure f o r porous  or perforated walls  pressure  r . .,r(PQ)  d i s t a n c e between  Re  Reynolds  number  surface  elements  XV  s  distance  along  the  surface  As.  length of surface  U  magnitude o f approach flow  3  element  V.,V ,V. ,V ,V : v e l o c i t y induced l n . ' t . x . y . _^ x x 1 1 approach flow v e l o c i t y , 2  x.,y.  C a r t e s i a n axes  X,Y  wind  3  x x  3  •  axes, with  aerodynamic  ac  midchord  0  airfoil  r  circulation ,v  , ;••  element  magnitude U  t o the j - t h surface, element  X-axis  center  by a s u r f a c e * .  i n the flow  direction  distance  distance  a  Y  fixed  velocity  incidence  surface  vortex  element  strength  density  ° y V y  surface  source  element  strength  density  <j>  disturbance  8  i n c l i n a t i o n o f s u r f a c e element w.r.t.  3  p  fluid  velocity  potential X-axis  density  dC T = ^-Mq  midchord p i t c h i n g  \\)  stream f u n c t i o n  S u b s cf rr i e ep t sa:i r v a l u e F L  lower  surface  n  normal  s  streamwise  oo  upstream- u n d i s t u r b e d - f l o w  direction direction  moment c u r v e  slope  S  solid  t  tangential  direction  T  windtunnel  value  U  upper  condition  wall  value  surface —•-  1  __.  In  the  subsonic  existing  theory  effects  of  windtunnels small  for  However, requires  to  small  wall walls  the  l i f t  at  models  numbers.  Under  these  unless  availableso  a  that  Such  method would  would  solid  windtunnels  be  most  the  with  solid  walls  are  with  open  jets,  an  Two  years and  forms  for  walls  major  presented  of  for  data  the  for  the  satisfactory  test  airfoils  crosssections,  and  for are  develop  walls  may  very are  and  sufficiently  high  the  wall  become  large  the  use  corrections  build  and  smaller  large  of  Reynolds in  large,  crosssections  to  existing  these  sections  unacceptably  test  expensive of  airfoil  coefficients,  conditions,  corrections  to  of  sign  a  opposite  obvious  are  operate,  windtunnels,  wall  walls,  purpose, pattern  of  prediction  of  Unfortunately,  measured from  possibility  windtunnel  this with  give  eliminate  s o l i d - p a r t l y open  such  the  sections,  corrections,  desirable.  Since  partly  to  with  or  is  high-lift  lift  modification  reduce  measured  which  on  high  windtunnels of  in  testsection  very  large  with  the  constraints  research  relatively  windtunnels  to  airfoil  coefficients.  current  testing  tastin.g o i  corrections  solid  relative  relatively  windtunnel  windtunnel with  Introduction.  to  wall  walls  the  experiments  those  with  been  for  windtunnels  i s  the  use  of  effects.  considered  in.recent  longitudinal Theories  corresponding  have  windtunnels  cancelling  narrow  holes.  in  explore  produce have  small  to  data  shown  wall  that  have  slots, been  corrections.  the  existing  2  theory present  for walls purposes,  to  apply.  on  the w a l l In  theory wall  with l o n g i t u d i n a l and  An e m p i r i c a l  on  thesis,  developed  s y s t e m , and  theory are  porosity  g e o m e t r y and  the present is  the t h e o r y  for  the r e s u l t s  presented.  is  useless  f o r porous w a l l s factor  the t e s t a  slots  is  for  the  impractical  i s needed, which  depends  model.  two-dimensional  a different partly of experiments  potential  flow  s o l i d - p a r t l y open  designed  to test  the  3  2-. Survey, of Windtunnel Wall C o r r e c t i o n Met hod s.. 2^1_ C o n v e n t i o n a l L i n e a r T h e o r i e s . In windtunnel t e s t i n g at  subsonic  the  measured f o r c e s ,  current  of  two-dimensional  airfoil  speeds , the windtunnel w a l l c o n s t r a i n t  theories  measured values  moments,  for  the  and  pressure  corrections  to account f o r  wall  crosssection,  and  develop  p r a c t i c e f o r such cases i s  to  be  effects  feature  the s e l e c t i o n  are  to the  Current  w e l l summarized i n a r e p o r t by Garner  of c u r r e n t w a l l  correction  together with images of  the  such t h a t the flow  the  boundaries are s a t i s f i e d .  The l i f t - p r o d u c i n g  of  the  airfoil  the a i r f o i l  (incidence,  wake, are a s s o c i a t e d  windtunnel  are  appropriate set example,  a  solid  walls;  by  sources,  direction, alternating  wall  normal t o  The  simulated  boundary  at  by  vortices, of  associating in  and  .the  the an  field.  condition requires  zero  the boundary. T h i s c o n d i t i o n  may  an  infinite  set  and  doublets  oriented  For  thickness,  characteristics  of images i n the in  the  have images of the same s i g n ; v o r t i c e s sign.  singularities  characteristics  of images with each s i n g u l a r i t y  disturbance v e l o c i t y simulated  then  v o r t e x and  with d i s t r i b u t i o n s of  respectively.  walls  theories  conditions  camber), the a i r f o i l  d o u b l e t s and sources  be  windtunnel  coefficients.  the windtunnel b o u n d a r i e s ,  For  the  satisfactory  of an a p p r o p r i a t e system of s o u r c e ,  doublet s i n g u l a r i t i e s , in  lift  applied to  The  a l [ 1 ]. The e s s e n t i a l  is  small  influences  distributions.  only when the t e s t a i r f o i l s are s m a l l r e l a t i v e  et  sections  further d e t a i l s ,  see  windtunnel streamwise  have images of  A l l e n and V i n c e n t i  4  [2],  and G o l d s t e i n  When t h e images the  has  net  airfoil  all  the  effect  been  be  and  of  the  field  equations,  turn  are  implies  a  systems the  application, the  direct  a  For it  small,  is  In  for  thin,  the  is  if  of  induced  the  the  cambered  net  at in  effect  effects  of  corresponding if  The l i n e a r induced  and  both  induced  possible  small  slightly  thereby  useful  terms  may be l i n e a r i z e d . only  singularities  simplicity  sum o f  superposition  valid  of  velocities  i n d i v i d u a l systems.  boundary c o n d i t i o n s in  the  determined.  systems i s  each  of  calculated,  may  interpretation of  [3].  the  exact  approximations  velocities, airfoil  which  at  low  incidence. An similar  alternative results,  Woods [ 4 ] , complex  domain  thus reduced to transformed prescribed equations  the  for are  theory  real Woods'  mapping  with  which  to  value  their results  potentials  an  of,, an a n a l y t i c  technique  solution. to  is but  on  parts in  usually  is a are  integral numerical  A linearized  be f o u n d  walls  function  results  and  a  The p r o b l e m  imaginary  by  in  equivalent,  problem-  and/or  yields  developed  and w i n d t u n n e l  functions,  for the  velocity,  an a i r f o i l  boundary  whose  required  technique,  mappings,  boundary.  the  agrees  by  determination  domain on t h e  finding  conformal  simpler,  images  c o n f o r r a a l mapping t e c h n i q u e  bounded  by.  geometrically  his  the  the  The p r o b l e m . o f  transformed,  methods  is  to  i n Garner  form  of  et  al  [1]. For  the  boundaries,  the  case  of  a  s o l u t i o n s of  thin  flat  plate  Havelock [ 5 ]  between  and T o m o t i k a  parallel [6]  are  5 available uses  Havelock  t h a t of  uses  the  method o f  t r a n s f o r m a t i o n s . The  conformal  In r e c e n t p u b l i c a t i o n s , de de  V r i e s and  with  hinged For  [9]  Jager  S c h i p h o l t [ 8 ] , use flaps,  porous  p r o p o s e an  and  while Tomotika  results  van  de  are s i m i l a r .  Vooren [ 7 ] ,  image methods f o r t h i n  between s o l i d or  images  windtunnel  and  airfoils  walls.  p e r f o r a t e d windtunnel  w a l l s , Baldwin  et  al  " e q u i v a l e n t homogeneous w a l l b o u n d a r y c o n d i t i o n " . dary for  is  condition  a  w a l l s and  solid  combination  for  an  ace's ay  which  equation.  T he  the  open j e t .  ndary c o n d i t i o n s ar e expressed o c i ty p o t e n t i a l  of  for  i n terms of  a  incompressible  general  linear  wall  be w r i t t e n :  «(»>!!•+ »!.>!!• cowsS - o. For  a  solid  normal to the  boundary  where t h e r e i s z e r o  boundary, e q u a t i o n  !*• = 8n  For from  an  the  test  linearized using  open j e t b o u n d a r y  condition  Bernouilli's  disturbance (2.1)  airfoil  becomes  at of  the  flow  form  0• .  (2.2) .  the  jet  constant as  Hence f o r an '  has  i t i s assumed t h a t t h e  equation,  velocity.  (2.1)  disturbance  boundary  pressure requiring open  is  can  disturbance small.  The  be i n t e r p r e t e d ,  zero  streamwise  j e t boundary, c o n d i t i o n  6  3*  wall  For  porous or perforated  due  t o the cross-flow  normal  disturbance  relation  between  velocity  A is  „  similar  used  slats.  components  for a  constant  expression,  slats,  the  spacing;  For of  Maeder  slots.  Hence  »a'  with  the cross-flow to  resulting  streamwise  consists  length testsection and  Hood  the  linear  disturbance  value of  of P { s ) ,  transverse  w a l l of c l o s e l y  spaced  [ 1 0 ] , a n d Woods [ 4 ] d e d u c e  .  i s the slot  i s t h e open  walls  potential  The  the  a  f o r P (s) :  (2.5),  a/d  across  proportional to  a different  porosity  •2d  equation  and  but with  P = tang)  In  t o be  at the wall.  cross-flow  w a l l where  value  i s assumed  drop  requires  F o r an i n f i n i t e  transverse  walls the pressure  velocity the  '  area  vanish  and  width  ratio  longitudinal  through  (2.5)  the the  and  'd»  i s  the  slat  flow  model  (OAB).  slots,  slots  a  potential  requires  pressure  the  disturbance  t o be c o n s t a n t ,  i n the  7  |8si + K ( s3)s 8^n =• 0 . • •'• Maeder wall  a n d Wood [ 1 0 ]  of uniformly  give,  spaced  K = £  where a/d  i s again  the  f o r an  infinite  longitudinal  l o g csc(|§]  OAS.  slots,  ,  '  • (2.6).' length the  testsection  value  (2.7)  8  2.Z.Z.  E ^ s u l t s of Convantional In  the  theoretical  Linear  Tneories_.  determination  of homogeneous boundary  c o n d i t i o n s , t h e d e t a i l s o f the s l o t or h o l e geometry can be, used t o c o n s t r u c t an e x a c t boundary c o n d i t i o n by, application  f o r example,  o f K u t t a c o n d i t i o n s t o t h e edges o f s l o t s or h o l e s .  Then by examining t h e f l o w t h r o u g h the w a l l from s l o t (hole) the  by  widths  away  a  point  many  from the w a l l s t h e f l o w d e t a i l s due t o  w a l l geometry a r e not f e l t ,  field  the  but o n l y  some  "averaged"  flow  i s d e t e c t e d . The exact b o u n d a r y c o n d i t i o n i s thus r e p l a c e d  a  l i n e a r i z e d , averaged boundary c o n d i t i o n . I n a p p l y i n g  boundary c o n d i t i o n , t h e w a l l i s r e g a r d e d as b e i n g homogeneous.  The  advantage  this  geometrically  i s t h a t a s i n g l e averaged boundary  c o n d i t i o n can be a p p l i e d u n i f o r m l y o v e r . t h e  p l a n e o f t h e w a l l so  t h a t i t i s not n e c e s s a r y  boundary  applied  in  slots  t o have  separate  conditions  (holes) and on s o l i d s e c t i o n s . T h i s  e f f e c t e x p l a i n s why t h e  wall  boundary  condition  averaging  for  porous,  p e r f o r a t e d and t r a n s v e r s e s l o t t e d w a l l s a r e s i m i l a r . F o r d e t a i l s see  Maeder and Wood [ 10 ] . The  same  "averaging"  effect,  i f applied to l o n g i t u d i n a l  s l o t s , l e a d s t o e r r o n e o u s p r e d i c t i o n s . In f a c t , t h e l o n g i t u d i n a l s l o t s render,the variations  flow  three-dimensional  by  imposing  spanwise  on t h e b a s i c t w o - d i m e n s i o n a l f l o w c o n d i t i o n s . On t h e  assumption t h a t , the f l o w i s q u a s i - p l a n e , only  a  small  that  perturbation  i s the  spanwise  variations  are  o f t h e b a s i c two-  dimensional  f l o w , an averaged.boundary c o n d i t i o n f o r t h e  basic  t w o - d i m e n s i o n a l f l o w can be deduced. F o r d e t a i l s see Woods [ 4 ] . In  the  use  of  such  geometrically  homogeneous  linear  9  boundary c o n d i t i o n s , are  lost, in particular,  transverse). porosity the  a l l d e t a i l s of s l o t o r p e r f o r a t i o n  Only  the  their effects  or-CAE a r e r e t a i n e d .  (longitudinal  of bulk p r o p e r t i e s  or  such as -the  Wood [ 1 1 ] shows, f o r example  that  OAR f o r l o n g i t u d i n a l s l o t s would need t o be l e s s than 1%, t o  a c h i e v e a boundary c o n d i t i o n jet  orientation  geometry,  case.  appreciably  d i f f e r e n t from.-;, the  I n p r a c t i c e , a t such a s m a l l OAR, r e a l f l u i d  open.  effects  would be i m p o r t a n t , so a p o t e n t i a l f l o w model f o r t h e c r o s s - f l o w would not be v a l i d . Moreover, Wood's a n a l y s i s o f condition  i n d i c a t e s that only cross-flow  this  boundary  v e l o c i t i e s of l e s s than  0.5%  o f t h e mean f l o w would be i n k e e p i n g w i t h t h e arguments f o r  the  linearization  boundary  of  i n the d e r i v a t i o n of t h i s  condition.  Investigations have  terms i n v o l v e d  found  empirical empirically  by P a r k i n s o n and Lim [ 1 2 ] ,  that the "porosity function  of  wall  f o r each a i r f o i l  parameter" P(s) OAR,  but  and  then t o t r y t o use t h i s same v a l u e of P(s)  determined  data  taken  p a r t i c u l a r i n c i d e n c e and f o r a p a r t i c u l a r s i z e o f a i r f o i l to calculate  the  e f f e c t a t o t h e r i n c i d e n c e s and f o r o t h e r s i z e s o f a i r f o i l .  Generally,  the r e s u l t s a r e t h a t P(s)  particular a i r f o i l practical  conditions. two  be  or pressure  a  the  [13]  i s not s i m p l y an  must  at  the  Mokry  under t e s t . The u s u a l procedure i s  t o choose a v a l u e of P(s) t o match l i f t  wall  and  use  under t e s t , an i m p o s s i b l e of  such  linear  porous  F i g u r e 2.1 from [ 1 2 ] , f o r example,  different a i r f o i l  different variations neither  depends on t h e w a l l OAR and  agrees w i t h  situation for wall shows  boundary that  for  p r o f i l e s t e s t e d , t h e r e a r e two c o m p l e t e l y of  "porosity  parameter"  with  OAR, and  the t h e o r e t i c a l v a r i a t i o n of r e l a t i o n  (2-5).  10  Other  by  results  P a r k i n s o n . a n d Lim  [ 1 2 ] , Parker  [14],  Tsen [ 1 5 ] , have shown t h a t the theory  f o r the l o n g i t u d i n a l  slot  F i g u r e 2.2,  parameter  K(s)  i s not  useable.  and wall  from [ 1 2 ] , f o r  example, shows t h a t f o r f o u r a i r f o i l s of d i f f e r e n t s i z e , of same  profile,  the  theoretical  c o r r e s p o n d i n g to the v a l u e s of grouped  as  though  all  wall OAR  wall  of  the  interference tested,  wall  are  the  curves closely  configurations  were  e f f e c t i v e l y open. C a t h e r a l l [ 1 6 ] s t a t e s t h a t "the limited  by  boundary)  the  doubts  this  great  nonlinear  wall  disadvantage boundary  of  being  conditions  linear  theory.  Wood [ 1 1 ] has  for  two-dimensional  a  as,  wall  appropriate  t h a t the  mathematical  is  depend  on  such  boundary  f o r example, i n complex v a r i a b l e boundary  condition  H e l m h o l t z j e t i s s u i n g from l o n g i t u d i n a l  "porosity" i s  velocity.  analysis  His  of  physically  developed a n o n l i n e a r  s l o t s , where the  extension  homogenous  themselves.  more  s o l u t i o n s f o r most boundary problems conditions  method i s  where the p r e d i c t e d w a l l c o r r e c t i o n s are  same order as the measured v a l u e s The  (linear  of the  c o n d i t i o n " . T h i s i s c l e a r l y the case f o r measurements  on h i g h l i f t d e v i c e s the  about  usefulness  is  a  function  for  t o the case of a l i f t i n g  a  of  nonlifting  airfoil  does  the  cross-flow  airfoil; not  appear  the to  have been made. Sears  [17]  comments  that:  where the flow p e r t u r b a t i o n s estimated,  there  is  a  due  "Even i n t h o s e to  tunnel  (flow)  boundaries  regimes can  be  b a s i c f l a w i n the i d e a of " c o r r e c t i n g "  11  measured aerodynamic d a t a , because such c o r r e c t i o n r e q u i r e s t h a t the e f f e c t s of such p e r t u r b a t i o n s be extraneous  velocities  i s other  i n c i d e n c e , than i n some of t h e most  known. than  a  If  the  uniform  important  field  of  change  of  technical  cases  these e f f e c t s a r e not known and cannot .be c a l c u l a t e d " . F i g u r e 2.2, from [ 1 2 ] , a l s o shows t h a t t h e t h e o r y f o r s o l i d walls absence  g i v e s e x c e l l e n t agreement w i t h t h e d a t a . T h e r e f o r e of  improvements  perforated-wall  corrections,  low-speed t w o - d i m e n s i o n a l developing  high  with s o l i d  walls.  to  lift  the  theory  i t seems  for  i n the  slotted-  or  a d v i s a b l e t o c a r r y out  a i r f o i l t e s t s , even f o r l a r g e  c o e f f i c i e n t s , i n conventional  models  windtunnels  12  2^.3  Low  Correction Test  An  a l t e r n a t i v e approach  windtunnel free-air  to  One  provide  would  an  surface"  layers)  measure,  calculation  the until the  say,  with  f o r an  of  computing  If are  required  to  on  a  wall',  convenient  speed  and  inclination  i f these  wall  inviscid  could  made,  be  of  A  values the  field  same about  iteratively, through  w a l l s e c t i o n s of  variable  the  cost  of  large  number  and  process  extract  there.  achieved  are the  flow  are  boundary  measured  calculated values  and/or  disadvantages plus  a  windtunnel  i n the  This  walls,  the  the  not  adjustments  met.  Thus  (located  infinite  not,  of  possible to  "self-correcting"  to determine  facilities  transducers  flow  as  but  previously  flexible  Obvious  the  tunnel  the  close  walls  small.  sensors  imaginary  airfoil.  porosity.  of  as  the  environment.  of  i n s i d e the  such c o n d i t i o n s use  that  i s performed  compatible  test  i s  modify  conditions  test  array  "control  i s ' to  a u t o m a t i c a l l y be  approach  [17 ] whereby  variables  flow  (unconfined)  corrections  are  Configurations.  "on-line"  of  pressure the  flow  measurements.  Whatever variation order  type  of  must p r o d u c e  to simulate  accomplished  testsection results  correctly  mechanically  the  i s up  like  i s  those  free-air to the  chosen,  flow  of  the  Figure  field.  windtunnel  "porosity"  How  2.3, this  designer.  in i s  13  Is. 3.1  A  Physical Basis  One and  h  reason  flows  i n the  accounted  f o r i n the  nonlinearities  slots,  success  is  the and  of  the  to  the  holes.  theories. the  longitudinal-slot  occurrence  In  main  experimentally  Such  t h e o r i e s , p r i m a r i l y as  s e r i o u s l y degrade  approach with  unconfined within wall  near  flow  their  will  flows  they  add  are  not  these  i n the  of  undesirable  addition,  flow  be  be  test  seen  free  ratio  of  flow  vicinity  of  0.72,  that  almost  of  the  The  effect  and  large  on  the  even  the  the  One  wall  the  to  in this  the  flow  for a will  nearly operate  that flows  negative  reason  a  14%  near  the  on  * a*  side  choice  can  the  between  airfoil.  testsection size  i s extreme at  upper  surface  i s  so  case.  20  the of  small  The  :  *H -is ,  degrees,  distributions  i s to increase  extreme  pressure  s e c t i o n 3.2)  These p r e s s u r e  pressure  rather  pressure  for this  Clark-Y  windtunnel  wall effect  underside  so  methods of  incidence  pressure  slats  wall  The  compares t h e o r e t i c a l  a i r , for 'c'  slats.  s m a l l even  range,  which  wall effect.  a l l of  negative  negligible,  chord  transverse  wakes.  i s slotted. 3.2  uses  solid  be  wall opposite  in free  airfoil  3.1),  Hence a l l t h e  ( c a l c u l a t e d by  w a l l s and  a  will  separated  Figure  solid  create  wall  the  airfoil  from  at  of  only  distributions  large  Figure  airfoil-shaped  the  field.  (see  unstalled incidence  Moreover, the  here  symmetrical  inclinations  to  Theory.  of  slots  Theory.  walls.  The  of  New  lack  theories  separated  the  Soltted-Sall  f o r the  f o r the  porous-wall  separations  New  show  magnitude  the as  a i r f o i l . to  be  14  Another simplify  reason  the  airfoil.  flow  A slotted  energy  a i r from  wall)  to  inflow  would  in  the  field lower  the  mixing  vicinity  downstraam  the  pressure outflow formed the  from and  with  this  shear  their  any  representation on  the  testsection  for  test  any  allow  layer  lower  the  test  test  airfoil  inflow  of  low  and  the  slotted  airfoil.  its  This  associated  of the, main  s u r f a c e . There s e c t i o n back  hand,  on  the  of the  test  wall  will  into  the  by  the  opposite the  airfoil,  into the  presence  there  p l e n u m . The will  of the  were  idealized  as  flow, be  a  plenum  this  streamline  airfoil.  by  of the  The test  entering  the  will shear  be  be layer  airfoil-shaped on  plenum  the  airfoil; will  testsection  be  so  will  wall  flow.  If free  i n such  have o n l y  air  an  s h i e l d e d from  or l o c a t i o n  should  entering there  negative  a constant-pressure  incorrectness in pressure of  test  the  quality  to  airfoil.  downstream  air  of  is the  (surrounding  upstream  shear  wall  s i d e of  an  boundary c o n d i t i o n s i m p r e s s e d  layer  streamline,  the  pressure  degrade the  testsection  airfoil  slats  on  from  upstream  the  a  slotted  i t s a s s o c i a t e d t u r b u l e n t mixing  test  effects  would  airfoil  test  other  side,  wall  would  of the  outflow  the  one  opposite the  of  and  corresponding  On  only  testsection  consist  of  using  w i t h i n a plenum chamber  enter  turbulent  for  a  secondary enter  the  however, i t s e f f e c t much s m a l l e r  upstream  of  the  than test  airfoil.  As  with  irrotationa 1 for  most flow  windtunnel i s assumed, and,  low-speed h i g h - l i f t  testing,  wall  correction  s i n c e the an  theories,  method i s  incompressible  designed, potential  15  flow  method  flaps)  can  be  u s e d . The t e s t a i r f o i l  and t h e a i r f o i l - s h a p e d  lifting  airfoils.  velocity  and t r a i l i n g - e d g e  solid  wall  condition.  Hence  sections  wall  slats  the flow Kutta  satisfies  are  satisfies  conditions. the  (and i t s c o m p o n e n t a l l treated the usual The f l o w  tangent-velocity  as  tangent-  ' past  the  boundary  16  3.2.2  Formulation  The  b a s e d , on Smith  In  and  this  i t s  source will or  non-zero  which  surface  on  solid  either  are  a  velocities  such  sources  flow  boundary solid  condition test  is  and  at  elements both  its  any  are  only  on  of  airfoil  over  any  of  surfaces,  blowing  there.  surface  specified. about  distributed vortex  of  airfoil-shaped  on  Vortex a  closed  over  condition  and  the  the  condition  solid  source  point  in  are  of~ z e r o  the  flow  calculated  normal  over  wall  any is  elements  the  trailing  edges  due  directly.  the are  slats  is  to The  of  the  a l l usual  applied  finite-velocity  flaps.  disturbance  field  velocity  applied  the  boundary  distributed  are  a  is  walls, test  boundary  addition,  <j>  method  distribution  circulation  In  including  a  is  surfaces. at  flow  flaps.  vortices  condition  airfoil,  Again  net  the  or  condition  elements  surfaces  and  by  suction  tangent-velocity  the  and  distributed  the  vortex  while  airfoil  The  set  source  and  wall,  velocity,  for  boundary  Therefore  sections,  test  normal  therefore  to  solid  normal-velocity  velocity,  over  all  zero  distribution  the  represented  distributed the  of  are  Hence  wall  surfaces  A  potential,  [18].  flaps,  which  specified.  singularity  slotted  used  body.  two-dimensional  the  normal  are  a  Theory...  in  normal-velocity  elements lifting  the  elements.  elements a  is  colleagues  slats  vortex  Numerical  surface  method,  prescribe  Source  the  component  and  Exact  here  his  airfoil-shaped with  an  formulation  theory, A.H.0.  of  wall  at Kutta  slats  and  •  velocity  potential,  which  17  .es  Laplace's  equation,  vanishes  at  infinity,  and.  s a t i s f i e s the above boundary c o n d i t i o n s . The  potential  at  a  dimensional, point source  where and  m  point  P  due  a  single  e m i t t e d .by the  i s the d i s t a n c e between the p o i n t s P and  potential  due  to  three-  s i n g u l a r i t y at a point Q i s  i s the volume f l o w r a t e of f l u i d  r (PQ)  to  a l l such  sources  Q.  source  The  total  d i s t r i b u t e d over a s i n g l e  surface S i s  4>(P)  0  s  where  o(Q)  i s the source  1/4IT , of the s o u r c e  Q  )  dS,  (3.2)  s t r e n g t h d e n s i t y , i n c l u d i n g the f a c t o r  element a t  S i n c e the d i s t u r b a n c e velocity  (  r(PQ)  Q.  velocity  is  the  of  as  8©  9n  •>  ->  ~ V^-n  =  +  F,  (3.3)  where n i s the outward s u r f a c e normal,, and a t upstream i n f i n i t y .  The  oo  »  the  undisturbed  f u n c t i o n F denotes the v a l u e  normal v e l o c i t y must t a k e a t the a i r f o i l s u r f a c e . ? i s z e r o a  solid  blowing  the  p o t e n t i a l , the n o r m a l - v e l o c i t y boundary c o n d i t i o n a t a  s u r f a c e can be e x p r e s s e d  flow  gradient  (impermeable) there.  surface,  but  non-zero  the for  f o r s u c t i o n or  1 8  Analysis velocity  at  density  P.  and  a point  distribution  "local" at  (Hess  "farfield"  a (Q)  o n S,  to  the  elements this  boundary  2ira(P)  at  of  points  Q  -  9h i r ( P Q ) J  equation f o r such  of  trailing-edge al  [19  element  The a  (P)  a l l other  source  resulting expression  of  dS  = -V  f o r the  oo  -n  For  a  source  Existence  must  i s a and  be  discussion  or  corners,  as  o r an u n f a i r e d  wing-body  such  a  of in  edges  strength Fredholm  uniqueness S may  The s u r f a c e  singularities  at  functional  vector  (3.5)  equation  a r e w e l l known.  normal  F  unknown  c(Q). This kind.  +  be  continuous difficulties  such at  j u n c t i o n , see  an  source airfoil  Craggs  et  ].  In shaped  equations  position.  with  distributions  parts.  (3.4)  of  on S. T h e  of the second  but the outward  associated  strength  dS  effects  a(Q)  function  distribution  function  two  normal  condition  density  disjoint,  of  due t o t h e s o u r c e  the  equation  theorems  the  S, d u e t o a s o u r c e  ) _d_ 1 a(Q) dn | r ( P Q ) J  p r o d u c e s an i n t e g r a l  integral  that  contribution i s  summation-  a(Q)  shows  consists  i s 2fro"(p)  f  due  [18])  P on a s u r f a c e  contribution  The  Smith  practice, slats,  polygonal  the surfaces  of the solid  and the t e s t a i r f o i l  elements.  The  and f l a p s ,  continuous  wall, are  distribution  the  airfoil-  replaced of  by  sources  19  thereby  becomes  elements.  In  a  succession  the  of  finite  d.istr i b u t e d - s o u r c e  o r i g i n a l method of Smith and h i s c o l l e a g u e s .  each of these f i n i t e  elements.was f l a t and  of. c o n s t a n t  uniform  s t r e n g t h . S u c c e s s f u l r e f i n e m e n t s of the method have used h i g h e r order with  polynomial the  source  parabolic  way  curves f i t t e d strength  along  t o s e c t i o n s of the body s u r f a c e  density  these  curved  Henshaw [ 2 0 , 2 1 ] , or Hess [ 2 2 ] . The are for  varying  in  elements.  a  linear  or  For examples see  higher-order  element  shapes  needed f o r i n t e r n a l f l o w c a l c u l a t i o n s such as i n d u c t s , but e x t e r n a l flow  results  provided  problems  the  flat  elements  give  accurate  a l a r g e enough number of elements, i s used  and  t h e i r d i s p o s i t i o n on the body shape i s chosen c a r e f u l l y . Each v e l o c i t y boundary c o n d i t i o n i s "control  applied  at  a  p o i n t " on each element. For f l a t elements a  single  convenient  c h o i c e i s the c e n t e r of each element.. .Thus  the  distribution  exact  integral  f u n c t i o n may  equation  be reduced  for  a  continuous  to a s e t of N  simultaneous  l i n e a r a l g e b r a i c e q u a t i o n s whose N unknowns a r e the s t r e n g t h s of the f i n i t e surface  elements.  The above a p p r o x i m a t i o n s N -»- °°. The t h a t any  become  exact  in  the  limit  as  method i s d e s c r i b e d as n u m e r i c a l l y e x a c t i n the sense  degree of. a c c u r a c y  By d e f i n i n g the l i n e a r  J  J  S. 3  may  be o b t a i n e d .  operator _9_• dn  3  ^  X  as., J  J  J  (3.6)  20  the  boundary  condition  (3.5) a p p l i e d at  t h e . i - t h c o n t r o l point  becomes N I A . . a . = - V • n . +.F . . j i 1 j =l  (3.7)  0 0  This i n d i c a t e s control  point  another p o i n t 2TT  that  A_^is  velocity  ' i ' by a u n i t s t r e n g t h  at  a  source element . l o c a t e d  at  * j ' . Hence the " l o c a l " normal  the  purposes  of  p o i n t source of equation two-dimensional  induced  velocity,  per  unit the  arc  length  With r e s p e c t  ^ ±' ±  ^  s  three-dimensional  the  a  flat  f o r the d e t a i l s  a (Q) are t h e r e f o r e : along  into  contour  volume per  flow  unit  see rate  length  direction. to C a r t e s i a n axes x  (Figure 3 . 3 ) ,  = log  and y  j  j  f i x e d to t h e  j-th  the v e l o c i t y components induced a t a p o i n t  by a source element  V  problem, the  d i s t r i b u t e d - s b u r c e element;  spanwise  element  this  (3.1) must be i n t e g r a t e d  Appendix 1.. The u n i t s of  'i'  normal  f o r a i l i= 1, 2, 3 , . . . N. . For  in  the  at p o i n t  K y % )  ' j*  + -yj)  2  are  =  2  log R  (3.8)  2  and (  2jtan  V  where  x.  element;  J  and  y. J  -1  J  are  the j - t h element  2"  -  tan  As,  -1  the d i s t a n c e s has length  from the As . j  (3.9)  = 2£2,  1 y.  The  j - t h t o the velocity  i-th  fields  about  a  single  d i r e c t i o n s of V  v  source  element  and 7„  at 'i  x.  direction The range  are  shown i n F i g u r e  a r e p a r a l l e l and n o r m a l t o  1  3  3  inverse  tangents  respectively.  i n (3.9) a r e t o be e v a l u a t e d  f - / 2 + /2) . The two i n v e r s e , t a n g e n t s may. be Tr  /  means o f t h e t a n g e n t l a w i n t o t h e a l t e r n a t i v e  V  tan  =2  Y.  3  this  range  (- , + ) 1T  _J  1  Y  where  l x j  single  inverse  combined  As. i .  tangent  by  (3.10)  y ? - ( ^ j )  +  i n the  expression  J  2  i s t o be e v a l u a t e d  by f a k i n g i n t o a c c o u n t t h e i n d i v i d u a l  Tr  the  y.  of the element at ' j ' ,  T r  3.'4. The  i n the  signs of the  n u m e r a t o r and d e n o m i n a t o r o f i t s a r g u m e n t . When c a l c u l a t i n g f l o w quantities at off-surface points of t h e element than With r e s p e c t  which a r e c l o s e r t o t h e  As/2, the f i r s t  to Cartesian  expression  must be u s e d .  " w i n d a x e s " X and Y,  wind d i r e c t i o n ) , t h e g-th s o u r c e element i s i n c l i n e d 8.  to the X-axis.  A.. = V ii  origin  (X i s i n t h e a t an a n g l e  Thus,  y .  c o s 9.-8. i  j  - V  x .  sin(8.-8. i  (3.11  t  v  '  and  B  > ; L 3  are  = V  c o s (6.-9.) + V 3  1  2  s i n (9.-8.)  3  1  Y  .-  (3.12)  :1  t h e n o r m a l and t a n g e n t i a l v e l o c i t i e s r e s p e c t i v e l y i n d u c e d a t  element point  'i'  due t o a u n i t s t r e n g t h  ' j ' . -The " l o c a l "  d e n s i t y source element a t a  normal v e l o c i t y  JL^ i s 2TT ;  the  "local"  22  tangential  The  velocity  directions and  outward) direction about are  8^  a  is  the  element  for  at  ' i»  (positive clockwise)  single closed  labelled  and  of  parallel  of  zero.  at  contour,  are  normal  respectively  ' i ' . Hence f o r t h e the  computations  source in a  (positive  and  to  the  exterior  flow  vortex  clockwise  elements  order  about  the  contour. In  order  usual  by  surface,  condition  surfaces,  at  and  Since  density  velocity  velocities  convenient and  to  induced  since  The  can  to  be  ' j * the at  The  use  on  a  edge, the  ' i ' are and  size  the  velocities  vortex A ..  of  that  and  be  B..  as  to  a  circulation fo  those  tangential A.,  and  elements  density. source to  of  strength  -B-.  located computed  due  unit  vortex  vortex  in  velocity  of  of  to  lower  equal  the  same s t r e n g t h  elements  and  n o r m a l and  found  Kutta  edge.  for  element  the  velocities  be.  i s simply  elements  i s  to  The  upper  trailing  vortex  same number o f  and  elements  must  the  This  density.  the  corresponding  a l l have  the  applied.  written, corresponding  element number  they  have s o u r c e  exactly.  toward  body,  tangential  degrees,expressions  Y(Q)  point  to  points  a d i s t r i b u t e d vortex  respectively. arbitrary,  90  i s  the  trailing  due  lifting  strength  distributed-vortex  density  at  that  both . d i r e c t e d  for  8, 3. 9) . F o r  same v o r t e x  the  a  d i s t r i b u t e d vortex  control  to  rotated  components strength  the  the  but  the  about  condition  finite  implies  adjacent  magnitude,  source,  a l l of  circulation Kutta  adding  then  established  (3.  f i x the  equal-velocity  accomplished body  to  I t  i s i s  elements coincide  f o r the  source  23  elements a r e then  immediately  useable  Hence t h e n o r m a l and t a n g e n t i a l control source  point  on  and v o r t e x  approach  flow  U  element  'i'  elements  (parallel  ni  an  at  the  of H c o i n c i d e n t  infinite  uniform  N IN  k=l  J  J  induced  due t o a s y s t e m in  elements.  to the X - d i r e c t i o n ) , are  .  3=1  velocities  immersed  N«  V  f o r the vortex  and N V  Since  t .x  B  V  a l l the  body a r e o f e q u a l about  an  N  Jj = jl i j  =  a  J  vortex  strength  N-sided  normal-flow  boundary  N  2  +  U  c  o  V  s  (3.14)  e l e m e n t s on a s i n g l e  Y , the d e s c r i p t i o n  polygonal  O i , o , . . . ,o  Y  n  J  quantities  provides  Jk = ki k l A  +  ,  body  and Y  conditions  is are at  closed  lifting  of the flow  complete  when  field  the  N+1  known. F o r z e r o F ^ , t h e  each  of  the  N  elements  equations  V  = 0.  n  (3.15)  i  while  the . f i n i t e - v e l o c i t y  points  adjacent  Kutta  c o n d i t i o n a t t h e two  control  t o t h e t r a i l i n g edge p r o v i d e s t h e s i n g l e  (N+1)st  equation  (3. 16)  24  For  the c o n f i g u r a t i o n  w a l l s and  test a i r f o i l  N s o u r c e elements and vortex  strength  a lifting  M  I y k  =  to  applied  f l a p s , s o l i d w a l l s and N  K  1  m = 1  on  R(k) I B nu  •V 3Ur j-1 J  JL J  ( mu A  >°l\lv*l  r .  N s o u r c e and  on  to each of the  M  zero normalthe  airfoil,  N equations  = Usin9., i=l,2,...N. i  applied  of  (3.17)  M bodies  (airfoil,  M equations  R(k)  M + B  element  solid  with a t o t a l  be d e t e r m i n e d . The each  f l a p s , w a l l s l a t s ) y i e l d s the  B  slats,  b o d i e s , t h e r e are  w a l l s l a t s , y i e l d s the  A Kutta condition  N  wall  p l u s f l a p s of F i g u r e 3.1,  densities  velocity condition  I Ad. j=l 3i 3  of a i r f o i l - s h a p e d  k=l  m=l  +A  r  mL  r  >  =  "U  (cos9  D  +cos6  r  L  ),(3.18)  r  r=l,2,...M.  The  subscripts  U and  L i n d i c a t e the  the t r a i l i n g edge"on the of  the  r-th  lifting  v o r t e x ) elements on  body;  this  In summary t h e r e  upper and R(k)  lower i s the  surfaces  to  respectively  number of s o u r c e  (and  body.  are:  - a t o t a l of N s o u r c e elements a i r f o i l , i t s f l a p s , the wall  control points adjacent  distributed  airfoil-shaped  over  the  test  w a l l s l a t s , and  the  solid  sections. - a t o t a l of M b o d i e s r e q u i r i n g  Kutta  conditions.  M  a lifting  total  of  J R (k) v o r t e x elements d i s t r i b u t e d o v e r k=l  b o d i e s ; t h e r e are R (k) s o u r c e elements and  R (k)  s t r e n g t h d e n s i t y v o r t e x elements d i s t r i b u t e d o v e r the - 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 a j  k-th  the  equalbody.  25  - M unknown v o r t e x s t r e n g t h d e n s i t i e s - M+N equations Y  M"  i n the M+N unknowns a i , a , . . . a 2  , yi / y , 2  2 6  3^3 O t h e r  Airfoil-Wall  Obvious (.3.17,3.18) (free  air),  (ground  simplifications  of  are  airfoil  for  (c) between  solid  lower boundary  sided  transverse  are  each  slats,  slats  general  over  (a) i n an unbounded  two s o l i d  solid  walls,  boundary  and t h e s o l i d  airfoil,  wall  density  surface a  of s i n g l e -  applied.  of N source  test  elements  the transverse  In a d d i t i o n  on t h e  stream  a n d . (d) between  i t sflaps,  sections.  lower  consisting  (a) - (d) a t o t a l  the t e s t  equations  t h e r e i s an  airfoil  and  on  of i t s flaps.  For air,  above  w i t h no K u t t a c o n d i t i o n s  unknown v o r t e x s t r e n g t h each  the  and an upper  o f -the c a s e s  distributed  wall  a test  Examined.- . '  (b) i n t h e p r o x i m i t y o f a s i n g l e  effect),  In  Configurations  example, f o r a s i n g l e  equations  test  (3.17,3.18) r e d u c e  airfoil  1  3  K  l  i n free  to, respectively:  N N •I A . . a . - y I B. . = U s i n 8 . , j=l k=l 3  (no f l a p s ) ,  i=l,2,...N,  (3.19)  1  and  Jj = l < V V J +  For equations  A  +  a single  Y  J  ( A 1  kU  + A  k=l  test  kL  3  =  " U(cose cose ) u +  airfoil  (3. 17,3. 13) r e d u c e  N I A j=l  )  i n (b) , (c) ,  K  1  or  (d)  (3.20)  above,  the  to, respectively:  NA . a . - y I B, . = U s i n G k=l 3  .  L  1  ,  i=l,2,...N  (3.21)  27  and N  NA  * j=l Here  ( B  DU  jL  + B  J  there  airfoil  ) a  J  (  J  are  and  \^ kU k=l  +  NA  A  +  A  kL  source  (N-NA) s o u r c e  s u b s c r i p t s 0 and  adjacent  to  the  different Jacob to  of the s i n g l e  here  and  use  trailing  representation than  second  lower s t r a i g h t  L  on  test  the a p p r o p r i a t e s o l i d  wall  vortex  is  an  airfoil  in  be  Mavriplis i n the  unbounded  exactly.  distributed  solved f o r  the  stream  over  t o be  boundary inherent case  is  same  (free  N source  surfaces  effect  "image" p o s i t i o n  is  lower  the  The  plane".  and  the the  solid and  N  i t s flaps,  twice the  configuration  number in  an  t h e r e i s a s a v i n g by  the  boundary can shape  present  t h a t t h e b o u n d a r y be  for a "reflection  Hence  straight  the t e s t a i r f o i l  a i r ) . Hence  in  plane".  this  is  so t h a t  I f there are N source  airfoil/flap  elements.  arbitrary  requirement  lower  points  [ 2 4 ] . Their approach  solved i s exactly  p r e s e n t method where t h e s o l i d by l e s s t h a n  control  ground  boundary i s a " r e f l e c t i o n  t h e number o f e q u a t i o n s to  on  i n t h e method u s e d by. o t h e r s , f o r e x a m p l e  represented  elements  the  t h e u p p e r and  boundary c o n d i t i o n of z e r o f l o w normal to boundary  elements  indicate  airfoil.  of  (3.22)  L  the  test  airfoil  solid  vortex)  edge on  S t e i n b a c h [ 2 3 ] and a  u +  elements  The  The  " U(cos0 cos0 ).  =  (and  sections.  respectively  }  of  be  represented  the  solid  lower  method;  there  is  straight  as  is  no the  28  iii Assembling  the  A computer B.  This  system.  program The  its  the  flaps,  sections.  coefficients Typically  the  thus  source  of  to  c o n s t r u c t the  are  and  the  vortex  matrices  more t h a n  A and  wall  B are  about  400,  system  B of  are  used  slats,  then N+M  matrices  t o be  IBM  and  370/168  the  to  matrices  a  airfoil, wall  assemble  the  (3.17,3.18). A  and  entries.  third  and  solid  equations  solved  (or p a r t s t h e r e o f )  the  and  used  hence the  to assemble  equations  A  coordinates, lengths  150,000 n o n - z e r o n o n s y m m e t r i c  A and  OBC  e l e m e n t s on  unknowns i n t h e  i s  a l l three  inputs  matrices  N+M  the  i s used  airfoil-shaped  c ^  matrices  that  the  These  Solution  i s w r i t t e n i n FORTRAN f o r t h e  program of  Numerical  Jguations..  program  orientations  contain  H§thods o f  matrix  B  each  In  fact  C  such  i s w r i t t e n C(a,y)=d; must  reside in  memory  simultaneously.  The  actual computational  250,000 e n t r i e s , for  computation  devices the is, be for  such  their  in  as  relative  such  relative  similar Mavr.iplis  matrices  source  position  and  orientation.  in  uses  f o r each  in airfoil  method  of  present  [24], Labrujere  (3.17,3.18)  The  and  change  be  storage  the  original  methods  temporary  memory)  must  of  example, a change  the  and  magnetic discs. geometry  (useable  large matrices  blocks  recalculated completely  In  of  hence  capacity  A  vortex  use  and  (Jacob  B  describe  elements,  in relative or  Smith  that  and  superposition  wall  OAR.  and  Steinbach  of  the  must  geometry,  [18]  [ 2 5 ] , Henshaw [ 2 0 , 2 1 ] ) ,  numerical  peripheral  These m a t r i c e s  incidence, size  Hess  about  partitioned on  and  i s  in  [23],  solution  three  "basic  29  flow"  solutions. This  there  is  no  incidence flaps  change  long as  a  as  possible  in  i s changed.  as  attitude  i s  relative  This the  whole,  for  i s also  isolated  geometry possible  airfoil-flaps  that  i s ,  airfoils,  when  the  an  airfoil  for  combination  without  a  change  as  airfoil with  changes i t s in  relative  geometry.  The is  due  three  to  chord.  a  uniform  The  circulatory the  flap), flow.  give  the  to a  "basic  satisfy  prescribed  third  condition  solid  surfaces. solid OAR,  of  the  with  a  they  For  matrices  at  i s  not  uniform The  one one  for  and  a change  present  airfoil  l i f t  are  the  fourth  onset  linearly edge  and  with are  elements  an  not  second a  edge.  solutions yield  walls  be  (i.e. a  combines the  trailing  possible  in a i r f o i l  pure  coefficient.  circulatory  B must  then  trailing  condition automatically  i s no  a  independent c i r c u l a t o r y each  flow  circulation  f o u r t h s o l u t i o n whereby  vortex  airfoil  onset to  pure  i s  first  the  i s due to  airfoil  airfoil  solid  have  A and  a  -to stream  third  condition at or  f o l l o w s . The  parallel  corresponds  the  not  the  Hence t h e r e  walls. the  do  yield  Kutta  procedure  as  f o r example, l i n e a r l y  first  walls. Since  bodies,  Kutta  incidence  is satisfied  incidence  This  flows"  solutions to  combinations any  than  the  a  chord.  that  more  flow  to  airfoil  Henshaw [ 2 0 , 2 1 ] , and  onset  I f more t h a n  i s  three  s o l u t i o n s are  due  flow  airfoil.  The  i s  onset  there  combined to  to  flow"  stream  second  perpendicular  about  "basic  the  Linear flow  airfoil  between  closed  lifting  associated  incidence,  calculated  at  satisfied.  d i s t r i b u t e d over  flow  Kutta  size  afresh.  their  with or  the wall.  30  Referring to  be  solved  matrix are  storage  With  of  a  This  FORTRAN  a matrix  element  1,  of matrix  compilers,  are assigned  (1,1)  first  reside  and  access  of  out  which  storage  of real  must  elements  reversing example,  which  ( 4 . 1) .  That  the  of  under  beginning  they  of elements  elements  i n  real  memory  system f o r  Moler  [26],  i n a row r e q u i r e s a t r a n s f e r i n and of each  i s completed. row  matrices  i s due t o t h e IBM v i r t u a l  see  row-  accessing  reside  F o r more d e t a i l s , ,  by  may a l l  situation,  entails  will  (left-  subscript  However, i f t h e  matrix  with  of the  matrices,  matrix  elements  i s , the elements  e m p l o y s some k i n d o f p a g i n g  allocation.  stored  a  by  matrix,  This  of  row.  This  the  block  by  i s not e f f i c i e n t .  the s u b s c r i p t s of the elements the  form,  memory. I n t h i s  of  This  ( 1 , 1 ) , ( 2 , 1 ) , ( 3 , 1 ) , . . . (N,1). eguations  of the  possible  a l l values  i n memory.  memory o f b l o c k s  be  through  a p o r t i o n o f each  the accessing  matrix  •j , i '  efficiency  i s then  by c o l u m n .  real  are already  system  accessing  until  elements  eguations  notation, but  i n c r e a s i n g t h e second  i n  memory a t a n y o n e i n s t a n t .  Thus  increased  I f A, B a n d C a r e s m a l l  the  large, only  dynamic  as  sequentially t o storage  s u b s c r i p t , then  operating  of  The s u b s c r i p t s  i n conventional  proceeding  simultaneously  elements are  i s deliberate,  a r e s t o r e d column  and r e p e a t i n g .  wise  below.  the system  compilers.  matrix  most)  written  and summation  FORTRAN  of  i s  (3.17,3.18),  e l e m e n t s o f A, B , a n d C a r e n o t o f s t a n d a r d  reversed.  the  to equations  This of  block, Hence  i s accomplished a  first  subscripting  matrix row i s  so  are evident  a by  f o r then i n  31  The  P.L/1 c o m p i l e r s  problem of in  does  a  not arise  matrix,  t h e same  The  store  when a c c e s s i n g  b u twould  system  o fe q u a t i o n s t obe s o l v e d  a  i  +  C  2,1  Q 2  +  -'-  N,l N  +  C  l , 2  a  i  +  C  2,2  a 2  +  - - ' N,2 N  +  l,N  a i  C  C  C  2,N  1,N+I  a i + C  1,N+M  0 1 + C  a 2  +  - -  2,N+I  2,N+M  +C  a  +C  a 2 +  a 2+  + C  a  N,N N a  -*•  + C  +C  N+l,2  Y l  l,N  C N  a  +  + C  Y  l  +  i n t h e same r o w elements  +  +  +C  (3.17,3-13),  j : L  R(k) I " I B .  from  i s written:  ' - - N+M,1 M +C  " -  "-  +  N+l,N+l  a n d t h e column  equations  m=l  Y l  a  a r e assembled  j i  C  N+l,l  N,N+i N  C(0,Y)-d  A  +  C  ' '* N,N+M N N+l,N+M  where t h e m a t r i x C  C  elements  column.  l , i  +  r o w by r o w , s o t h i s  i f i t were n e c e s s a r y t oa c c e s s  C  C  matrices  Yl +  Y  C +  C  M,2 M Y  N +  =  =  d  d  l  z  N+M,N M = N ^ - > Y  Y l +  -'•  **  -+C  vector  d  + C  1  M,N+l M Y  N +  = d N  +l  N+M,N+M M N+M' Y  d  _d  i n t h e system  t h e m a t r i c e s A a n d B b y means o f  that i s ,  j = 1, 2 , . . . N;  k=l,2,...M;  i=l,2,...N-  j=N+k; i = l , 2 , ' . . . N  ,  32  j = l , 2 t ' • •N ; r = l , 2 , . . . M ; i=N+r (4.2)  B. c.  R(k) I' (A_ +A m= 1 r  )  k=l,2,...M;j=N+k;r=l,2,...M;i=N+r  r  and  Usin6.  i=l,2  [ -ucose  t ' ''  N  r = l , 2 , . . . M;  -ucose  (4.3)  i=N+r  r  Thus the computation both  matrices  assembled Appendix  peripheral allocate  For d e t a i l s  see  the  de-allocate  t h e memory a s s i g n e d  B  into  parts  of C that and  memory  present  from  involve appears  summations o f m a t r i x equations  access  are large, FORTRAN  the  memory  (4.2);  this  paging  allocate  peripheral  B. in  t o A,  This  alternative  Appendix  elements  storage  i n the  10. same  The row  program  i n  B into to  B,  involve  A,  memory  for  B,  and c a l c u l a t e a l l is  simpler  to  large  number  of  i s  evident  i n  i s t h e main s o u r c e o f i n e f f i c i e n c i e s  systeo.  to  C must be  assigned  memory f o r C, c a l c u l a t e a l l p a r t s o f C t h a t  read  the  requires  The a l t e r n a t i v e i s : c a l c u l a t e A and. B, w r i t e storage,  de-allocate  program  C  A and B. When t h e s e m a t r i c e s  i n blocks. 2.  of the matrix  under  33  4_-_2  S o l v i n g the  Usually equations  Eguations-_  the  solution  i s obtained  directly  a FORTRAN G a u s s - e l i m i n a t i o n above mentioned  paging  Another d i r e c t s u c c e s s i v e row In  this  that  a series  augmented is  used  an  of  the  by  system,  see  takes  Moler [26]  by  Hess and  matrix  i s constructed.  to c o n s t r u c t  that  or  a set of  The  N+M  of  Appendix  3.  of  row  right-hand  vectors  in  N+M  account  For the  is  Purcell  i s t r e a t e d row  t o each  of  methods.  Smith [18]  o r t h o g o h a l i z a t i o n process augmented  system  Gauss-elimination  vectors orthogonal  matrix  complete  subroutine  method, u s e d  vector  method  of  the [27].  by  row  such  vector  of  the  side vector  'd*  (N+M + 1 ) - d i m e n s i o n a l  space,  (  G  l i '  The the  C  2 i '  3 i  solution  N + M , i '  C  vector  -  d  i  ( a , y ) °f  .  }  i=l,2,...N+M.  equations  (4. 1)  is  such  that  vector  Cai,  is  C  a , 2  orthogonal  solving  as Each  stage  of  3 f  to  equations  dimensional unity  a  ...,  o^,  a l l (4.1)  y i , Y2,..-, y ,  the  row the  of  the  process,  vectors  i s equivalent  vector orthogonal  i t s (N+M+1)-th  M  to the  of  1)  (**-5)  ( 4 . 4 ) . The  to determining N+M  v e c t o r s of  process an  of  (N + M+1)-  (4.4)  with  component. coefficient  and  i s not  matrix  needed  C i s used at o n l y  before  or  after  one that  34  stage. at  Thus  a time,  that  of  storage  C i s t r a n s f e r r e d from  with the  each  row o c c u p y i n g  previous  row.  virtual  an  of a s i n g l e  row.  vectors w i l l  memory,  repetitively.  they  are  number o f components finished;  the  approximately be is  solved about  total  by t h i s  number  process  that  Indirect  based  of  process  memory  iterative  procedure,  methods  see  diagonal  Hess  and  elements,  However, by e x a m i n i n g over  entries  off-diagonal  entries.  non-singular  except  trailing surfaces linearly  which  almost  capacity) For  a  3.  successive-over-  discussion  of  such  a t t e m p t s we're  C i s d i a g o n a l l y dominant,  2TT, a r e  the  largest  provide  In essentially very  thin  dependent)- can creep  i n the matrix.  equal  bodies,  of  the  a l l the  such  matrix as  e l e m e n t s on  so matrix a  the  t o t h e sum o f a l l t h e  a l l cases,,  i n . For  that  l a r g e elements i n t h e  t h e , sum  and v o r t e x  coincident  that i s ,  (4.2), i t i s seen  approximately  e d g e s . Here t h e s o u r c e are  can  was a b a n d o n e d .  the relations  for  half-  process.  Smith [ 1 8 ] . U n s u c c e s s f u l  B .. , A „ a n d A _ mi mu mL  i s  as  a  (N+M)-th c o l u m n o f C. I n g e n e r a l ,  diagonal  about  see Appendix  such  (SOB) a r e a l s o p o s s i b l e . F o r  summations  is  total  locations required i s  Gauss-elimination  on t h i s  In g e n e r a l , the matrix  (last)  the  The maximum  (for a given computational  for a  made t o u s e SOR; t h e method  the  when  the  t o be i n r e a l  2  FORTRAN s u b r o u t i n e  methods,  occurs  However,  tend  as  amount o f  (N + M ) / 4 . Thus t h e number o f e q u a t i o n s  twice  relaxation  used  location  insignificant  components o f a l l t h e o r t h o g o n a l since  memory a row  t h e same s t o r a g e  Thus  i s required f o r storage  to r e a l  cusped  the  singularities discussion  i s  of  two (rows such  35  problems,  The  see  Hess and  Smith  [18].  summations  v  I  n.  R(k)  M  N  A. . a . -  Y  y.  I  k=l  B  m=l  (4.6)  UsinS. l  mi  and  R(k) , i i / " • + Ucos6 i . i i - , k_J:-, 1 k=l m=l m i M  N  y B . .a  j=l provide points  the »i'  net  due  V  and  of eguations ,V  n  i or  normal  and  to a l l source  respectively, set  D 1 1I D  the  tangential velocities  and  uniform  (4.1)  vortex  onset  i s checked  elements  flow  by  U.  The  computing  at each p o i n t of a p p l i c a t i o n of the z e r o i Kutta boundary c o n d i t i o n s . At a l l c o n t r o l t  surfaces,  V  calculated  i s zero, 1 from  and  (4.7)  the  local  pressure  at  control  • j ' and solution the  ' m'  to  the  velocities  normal-velocity p o i n t s on  coefficient  solid C  is  :  V.  The  resulting  trapezoidal  rule  test  and  airfoil  values or  flap  nose-up  midchord  from  expressions  the  and  =  1  of  C  the  -  p  are i n t e g r a t e d numerically  fitting  contours  dx. P  J  3  of  c u b i c s p l i n e s ) around  to determine the  guarterchord  1 c|>C c  (4.8)  U  pitching  1 4>C c  lift,  moment  dy.  drag  (by the and  coefficients,  36  = 772 <|>C  C .  'Mo M  where and  dx = d s . c o s 9 . 3  (x.dx.+y.dy.)  C  T  3  and  3  ,  dy = ds s i n 6 3  3  i n t e g r a t i o n s are performed c l o c k w i s e  (4.9)  (4.10)  3  around  the  polygonal  contours. From  a  calculation  body, r e p r e s e n t e d  of the net c i r c u l a t i o n  by NA s o u r c e  f.  r  lifting  and v o r t e x e l e m e n t s .  t  ^  y  about a  NA  = Av-<U = ov. ds. = J J t . i  I V, As., . ^, t . I 1= 1 X  X  (4.11) I  and  since the l i f t  circulation  C  coefficient  C  is  related  to  the  total  r by  i s g i v e n by  ~  °L  By s u b s t i t u t i o n  y  i s  = Uc"  ^ t / i 1=1 l V  S  *  ^-  o f ( 4 . 7 ) , i t c a n be shown t h a t  C  where  NA  Ii  the  = ^JH-(perimeter  1 3  >  (4.13) r e d u c e s t o  o f t h e body)  ,  (4.14)  U C -  vortex  strength  d e n s i t y f o r t h e body  under  consideration. The equivalent  C  values only  calculated  f o r an i s o l a t e d  from  (4.9)  airfoil.  and  (4.13)  are  I f a s e c o n d body o r a  37  boundary  is  present  calculation d e p e n d s on  of the  reduces  shrinks  to  In from  the  the size  (4.11)  the  of  must be  flow  field  contour  of  surface  any  3.4  not  equal.  a particular  only  as  the  The  body  i n t e g r a t i o n . The  discontinuity  as  element  then  integral  contour  size  zero,  flow  field  streamline  can  Alternatively 1  a given  streamline,  be the  can  with  are  position, normal  non-zero a t a l l At t h e  edges  approaches (3.8)  expression  solved  field  The  The  expression  x-coordinate, that  control point.  point  perpendicular  stepping  tracked.  be  vary  of  infinity  and/or  s o l u t i o n i s uniformly  and  computed. by  velocities  velocities  for  the  slope.  .parallel  and,  surface  i t i s i n general  i n the  components  calculated  elements,  at the c o n t r o l p o i n t s ;  tangential velocity  p o i n t s . Hence a t a  be  surface  s u r f a c e element, i f the  except the  off-surface  can  quoted a r e c a l c u l a t e d  body  calculated  singularity  the  the  only  a given  in surface  However,  values  on  shows how  elements the the  that  solution i s valid  e l e m e n t . On  the  because of  at  of  c o r r e c t value  emphasized  i s prescribed  Appendix  about  pages, a l l C  points  a surface  direction  are  (4.9).  meaningless. Figure  the  the  the  following  It  points  values  circulation  to  expression  velocity  two  zero.  "off-center"  on  the  local from  to  the the  algorithm  to  streamwise  point, a  i s given  stream  function  f o r say  the  to  points  on  a  (x,y)  can  be  i s , a locus of  points  can  be  particular  i n Appendix  iteratively locate  at a l l  velocity  flow . d i r e c t i o n  point  f o r the  'i '  valid  given  4. in  y-coordinate, particular found  along  33  which  the computed  value  of  the  stream  function  i s a  constant.  39  5.. R e s u l t s o f t h e New The  use o f t h i s  distribution isolated  method,  bodies,  pressure  type  of two-dimensional  to  i s  calculate  well  distributions  obtained  For lift  purposes C  L  experiments  of the present  were c a l c u l a t e d  control  points  to  f o r the  represent  c o o r d i n a t e s of t h e 50 c o n t r o l values  a  0,2,3,5,8, zero +3  of  the l i f t  polynomial  comparison  of  pressure  two-dimensional  degrees  was f o u n d  i s 0.1229.  incidence  NACA-0015  a r e given  5  incidence,  airfoil  theory  slope m f o rsymmetrical  through the  using  By  Table  exact  the  6  50  1.  curve-  points at  lift-curve  slope  The c o r r e s p o n d i n g  at  value a t  higher i f a  used.  ( s e e Pope  airfoils  free-air  in  T h e s e v a l u e s would be s l i g h t l y p o i n t s was  be  so o b t a i n e d a r e l i s t e d i n  incidence a .  order  will  method,  airfoil,  of  p r o f i l e o f t h e a i r f o i l . The  p o i n t s used  t o be 0.1193  number o f c o n t r o l  From t h i n  the slope o f the curve  theoretical  the  the  of  and 10 d e g r e e s  degrees  larger  a  and o t h e r  coefficients  3, as. a f u n c t i o n o f  fitting  d i s t r i b u t i o n s on  method, w i t h  as a f u n c t i o n o f a i r f o i l  As an example  Table  this  o f comparison here,  loadings  The  from  For  singularity  f l o w t h e o r i e s , s e e Hess and S m i t h [ 1 8 ] .  coefficient  used.  surface  pressure  established.  d i s t r i b u t i o n s d e r i v e d from potential  Theory.  [28])  this  lift-curve  i s g i v e n by ( p e r r a d i a n )  1 + .773m =  2TT  U  +  c (»773|) J  where t / c i s t h e m a x i m u m - t h i c k n e s s t o c h o r d 0015  airfoil,  (5.1)  2  ratio.  F o r t h e NACA-  t h e v a l u e o f is i s 6.919 p e r r a d i a n o r 0.1208 p e r  40  d e g r e e . The the at  agreement  expression zero  (5.1) g i v e s  degrees,  independent  The extent,  of  s i n c e an  aligned  the  theory  a set  with  of  the  direction  test  airfoils  presence  of  the  lower  w a l l , with  will  be  results  70%OAR  of the  assumption  is  a uniform  flow  developing  v a r i o u s upper  i n the  to  by  undisturbed  flow.  were  high  the  Figure as  a  very  small  as  lift  C  lift  5.1  with  function  of  c o n t r o l points)  is a  14%  OAEs.  This  l o w e r . By  this for  in  the  configuration TSOSL.,  comparing  meaning the  lift  T  T  -C  consequence  an  NACA-23012 at length)  at  zero would  f o r the  4.6% and be  present  8 degrees  slotted  T  coefficients,  size,  thick,  actual a i r f o i l  no  chord  Sith  ) f o r a l l the F  L  and  up  to  airfoil  unity.  airfoil  by  the  surfaces  a transversely-slotted  shows t h e c a l c u l a t e d r a t i o  airfoil  that  infinite  coefficients  c o r r e c t i o n s (C  at a l l l i f t  l a r g e as  first 50  is  calculated  abbreviation  upper, s o l i d  considered,  c/H,  m  wall configuration. I t consists  wall  windtunnel,  of  flat  L  sizes,  that  only  , to the f r e e - a i r value, C , T F i n d i c a t e d t h a t a t r a n s v e r s e l y - s l o t t e d w a l l of about  gave  airfoils  Strictly  slope  and  loadings  wall i n conjunction  transversely-slotted  the  lift-cnrve  airfoils,  above mentioned  referred  coefficient  theoretical  multiple  different  upper  t h e o r i e s i s good.  represents  (Figure 3.1),  a solid  two  underlying  configuration  of  the  incidence.  present past  between  flap  c/H,  of l i f t  coefficients  f o r three  caaber  Clark-Y  airfoils.  (represented  20 d e g r e e s i n c i d e n c e stalled  at  purpose).  incidence, d e f l e c t e d 20  this The  with  degrees  (the  fact  incidence i s of  second a  The  25.6%  airfoil  is  (overall  (represented  by  41 46  control  third  points  airfoil This  i s an NACA-0015  ratio  of l i f t  configurations. 70%OAH lift be for  One  TSUSL  thau  l i f t  corrections  1.0.  Indeed,  shift than  wall see  the details points  configuration Table  aerodynamic pressure  used,  slat  tested  pressure  n e a r .701 size  lower l i f t  walls,  could  the present  for airfoil  and g i v e  whereas t h e  c/H  value  theory predicts  less  o f OAS  corrections  than would  of  less  0.8.  size  of  wall  slats,  and s p a c i n g s ,  number  f o r any  t h e o r e t i c a l l y , and r e f e r r e d  f o r two-dimensional  Figure  5.2  airfoil  by  a t 20  wall  a  of  airfoil-  to  herein,  dimensional  pressure  distortion  i s plotted  p  TSUSL  wall  of  the  theoretical i n  the air.  i s zero  l i f t -  usual  non-  the  The f i g u r e  shows t h a t  on t h e . a i r f o i l  configuration  of  and i n f r e e  i s , there  i n terms  subsequent  incidence  configuration,  coefficient C .  i n  comparison  degrees  of the p r e s s u r e . d i s t r i b u t i o n of t h i s  use  testing . i s  .the present  c o e f f i c i e n t s a r e t h e same, t h a t The f i g u r e  airfoil  for  shows  calculated  o f a 70%OA.R T S U S L  correction.  presence  i s a  o f OAS  distributions  analysis.  on t h e C l a r k - Y  lift  and t h e second  value,  a slightly  than  wall  between s o l i d  wall  o f t h e number  distributions  presence The  1%,  - f o r two.  3.  obtain  method  tested  up s l i g h t l y ,  A p r i n c i p a l reason to  that  walls,  free-air  than  <A% f o r c / f l l e s s  about  control  of less  shown  I t i s seen that  a n d one s l o t t e d  i t appears  the curves  For  configuration.  The  incidence-  i s  two s o l i d  f o r an a i r f o i l  wall  a n d 35 o n t h e f l a p ) .  at 3 degrees  50% of the true,  one s o l i d  airfoil  coefficients  i s with  wall  correction  mora  on t h e main  i s small,  even  i n  the the  at the  42  high  C  o f 3 . 0 9 . The p r e s s u r e L  larger  F  negative  airfoil than  and a l o w e r in  the  configuration. coefficient than  pressures  the  negative  free  configuration.  The  air  wall  configuration  correction A  over  ratio  might  of  field  of  computation  fields  and  was p e r f o r m e d with  The r e s u l t s  of C are  along similar  p r e s s u r e s and momentum about  section,  TSUS.L  pitching  wall moment  for  the  that  larger  TSOSL  pitching  wall moment  value  C  ,  i s  Mo p  T  this find  zero-lift-correction use  as  a  low  TSU5L  moment-  o f a TSUSL w a l l c o n f i g u r a t i o n and  the.  coefficient  test within  slightly  inside  f o r the f r e e - a i r  a r e shown  and d i r e c t i o n to the plane the walls. A  case,  positions  for  above,  these  surfaces.  Qualitatively  that  a  detailed  in  similar  two  flat to the  the value  same  terms o f t h e the  flow  calculation  fluxes f o r a rectangular confirms  of the  i n F i g u r e 5.3, f o r t h e  described  in  were  relative  case  airfoil 5%.  speed  surfaces parallel  t h e same c o r r e s p o n d i n g  lift-correction  variation  the  , to the f r e e - a i r  comparison  wails,  airfoil.  zero  of the  after  for  midchord  also  a t p o i n t s on f l a t  testsection  test  the  nose-up  0.573  f r e e a i r was made a s f o l l o w s . The f l o w  surfaces  section  shows  test configuration.  flow  computed  air  i s 0.603, w h i c h i s t h e r e f o r e  value  which s u g g e s t s  free  forward  midchord  Mo here,  for  distribution  i n the t u n n e l , C  0.95  the  pressure  The r e s u l t i n g  corresponding  coefficients  over  pressure  for  distribution  control  using volume  of the a i r f o i l  lift  43  Experiments  t o Ver ify_  t h e New T h e o r y .  J 2 i l T e s t s e c t i o n DesJ.g_ru  The solid  success  lower  o f ,the p r o p o s e d  wall  (TSOSL)  experimental  verification  Initial  experiments  testsection (Figure by  test  of t h e present were  an e x i s t i n g  deep,  flow,  windspeed  over  with  range  a  length  a turbulence of  zero  for  boundary  growth),  area  i s 0.582m .  these  testsection insert.  initial  the  octagonal  closed-circuit  windtunnel 915mm  wide  o f 2.59m, a n d p r o d u c e s a v e r y  level  less  tapered  than  0.1%,  downstream  over  a  h a s 152 by  to  compensate  so that the octagonal c r o s s s e c t i o n  the  was  surrounded  testsection,  with  frame.  a two-dimensional  a r e mounted v e r t i c a l l y windtunnel  balance,  testsection  on t h e yaw-  at the midpoint  and s p a n n e d t h e 388mm d e p t h .  airfoil-shaped  open  were f i t t e d  the existing  One s i d e - w a l l  by a 0.39 by 0.30 by 2.44m plenum, a n d  and c h o r d s  wooden  airfoils  of a six-component  of  wall  were e n c o u r a g i n g ,  i s 915mm wide by 388mm deep i n c r o s s s e c t i o n ,  2.59m l o n g . T e s t  fitted  experiments  was m o d i f i e d t o a c c e p t  This insert  turntable  of  in  2  As  2,3)  and  d e p e n d s on t h e  t o 50m/s. The t e s t s e c t i o n  (actually  layer  upper  theory.  performed  low-speed  152mm c o r n e r f i l l e t s  and  configuration  6.1, P l a t e 1 ) . T h i s t u n n e l h a s a t e s t s e c t i o n  686mm  uniform  of  transversely-slotted  with  spacers  slats  o f NACA-0015 s e c t i o n  o f 46 o r 92mra, a t z e r o area metal  ratios  which  full  be  (Plates range  be t e s t e d , a s t h e s l a t s  i n t u r n were  i n an aluminum c h a n n e l .  incidence. A  (OAR) c o u l d  sliders  could  recessed  separated  by  i n theside-wall  Modifications inserted to  the  nozzle  388  by  to  the  and  e x i s t i n g windtunnel  diffuser section  915mm t e s t s e c t i o n . The  shape of c o n v e r g i n g  sections  spatially  flow  for  uniform  example,.see  use  the  "same  direction)  see  The no  and  circular Appendix  floor  wall  to  two  within  boundary  pitotstatic  solid the  0.3%  layers.  the  order  of  Reynolds  Figure  of  6.3  windspeed  the  12mm,  produce  established; here  (in the  was  to  streamwise as  details  ceiling  are  walls)  parallel  for  the  of  the  for  solid;  boundary  layer  tube t r a v e r s e s  windspeed  c e n t r a l "core" shows  traverse.  a  numbers a  required typical  is  flow,  typical  Boundary  various  boundary  "core"  be  the flow  pitotstatic  where t h e  test  thickness  test  layer  would  outside  windspeeds c o v e r i n g  for  empty  spatially  layer  empty t e s t s e c t i o n , a g a i n  over a range of  i n the  where a t e s t a i r f o i l  testsection  6.2  and  mounted, i n d i c a t e a d i s p l a c e m e n t  shows  the  of  range  airfoils.  pitotstatic  tube  traverse.  Initial trends  be  For  addition  to  crosssection  compensation  Figure  tube windspeed  would  and  i n the  t u b e measurements i n t h e airfoil  approach  an  theoretical  i s well  variation  Pitotstatic  mounted, i n d i c a t e t h a t uniform  exit  rectangular  mechanical  (with  the  of  5.  attempted.  testsection  of  in  crosssection  The  crosssection.  testsection  is  at  Wang [ 2 9 ] .  equivalent  s t r u c t u r a l or  growth  conditions  length)  design  circular  c r o s s s e c t i o n a l area  f o r an  theoretical design,  Smith  of  (2.6m  consisted  in  slotted-wall  experiments without the of  data OAR  taken  10%  or  as  the  plenum  functions  greater.  The  gave  of the flow  inconsistent  wall  exiting  OAR,  with  from  a  the  U5  testsection model,  through  at  large  testsection airfoil.  the  OAR  through  was the  Conservation of a i r into  between  the testsection was  cured  transversely-slotted functions whatsoever.  of  wall  not  wall,  slotted  the diffuser  by  exit  wall  are  was  section diffuser  using  wall. OAR  and  upstream  constrained  o f mass f l o w  influx  problem  slotted  a  now  test  re-enter  the  downstream preserved through section  plenum  Consistent  to  of the  trends  achievable  of the by  a  test large  a breather  slot  entrance.  This  surrounding i n data with  taken any  the as OAR  46  6.i2  Airfoil  Models  Altogether drag  and  Tested.  nine  pitching  addition,  different  moment  airfoils  data  surface pressure  were t e s t e d , w i t h  taken  f o r each  lift,  airfoil.  In  m e a s u r e m e n t s were made on two o f t h e  airfoils.  Four 307, and  airfoils  462, 0.67  billets  616mm c h o r d  respectively) ,  Each  airfoil  through  the  were  and 153,  c/H o f 0.17, 0.34, 0.51  machined  mounted  hole  on  from  .. s o l i d  a circular  i n the testsection  between t h e c i r c u l a r  floor  or c e i l i n g  Four  laminated  section,  were l e s s  aluminum  were a l s o  tested. Since  circular  c/H  they  these  h o l e and t h e  were  fitted 3mm  with thick floor  r e c e s s e s , o f 724mm d i a m e t e r  airfoil  i n c i d e n c e c o u l d be v a r i e d combination  A single  3mm  mounting  a i r f o i l and  2.5mm on a l l t e s t s .  airfoils  with t h e t e s t s e c t i o n  endplates  with  o f 14% t h i c k n e s s , 4.6% camber  stepped  endplate  which  608mm  chord  o f 0-25, 0.39, 0.53 a n d 0.66 r e s p e c t i v e l y )  aluminum e n d p l a t e s  flush  spar  floor  683mm s p a n a n d 2 2 7 , 3 5 4 , 481 . a n d  sizes  testsection,  than  wood a i r f o i l s  (airfoil  the  span  ( P l a t e 4 ) - The g a p s between t h e t i p o f t h e t e s t  Clark-Y  fit  (model s i z e  was  a circular  c l e a r a n c e a l l around spar  383mm  t o c l o s e t o l e r a n c e s , on a n u m e r i c a l l y - c o n t r o l l e d m i l l i n g  machine. passed  and  o f NACA-0015 s e c t i o n ,  t o touch  laminated  about  large  outside  (716mm  (Plate 5)and c e i l i n g  The  the  diameter) endplates  into  circular  and 6ram d e p t h . . T h u s t h e t e s t  by r o t a t i n g  the v e r t i c a l  the t e s t s e c t i o n  wood  extended  airfoil  axis,  floor  of  the test without  airfoilallowing  or c e i l i n g .  11%  thickness,  2-3%  47  camber size  Joukowsky  section,  c/H o f 0.34) was a l s o  circular  endplates.  pressure  contains  theoretical  this  the  the  necessary While  five  to  to  with  e d g e r e g i o n was t h i c k e n e d t o be  located  and m o d i f i e d  be  airfoils  depending  detected  out  between t h e c i r c u l a r testsection  floor  loadings  particularly loading  to  streamwise  The  there.  profile  Table  1  coordinates f o r  of  the  on  the  the  test  and t h e c i r c u l a r  test  themselves.  through  recesses i n the  airfoil-endplate  Thus t h e r e  drag  affect  combination,  through  skin-friction  flow  t h e gap  would c e r t a i n l y  due t o t h e o u t f l o w  expected  i t was  incidence,  section  l o a d i n g on an e n d p l a t e .  the  the  was t h e n the  endplate  endplate with  endplate  was  was  would be a  the  gap i n  force  vertically  flow  field  adjusted  therefore  the windtunnel  the testsection  suspended  correct  on  i n c i d e n c e s . T h e gap between  the  airfoil  and c e i l i n g - . T h i s flow  upper s u r f a c e f l u s h  airfoil  endplates,  i n the  direction.  mounting  establish  with  on t h e t e s t  l o a d i n g on a s i n g l e  airfoils  fitted  endplates  on an e n d p l a t e  addition  and  taps  fitted  measure t h e l o a d i n g s on t h e e n d p l a t e s  under t e s t ,  could  by  trailing  (airfoil  airfoil.  For  the  t e s t e d . I t was s i m i l a r l y  The  3.8mm t o a l l o w  683mm s p a n and 307mm c h o r d  floor. above  over  balance  with t h e  Each.of  the five  the  the  the t i p of the test by  raising  or  Measurements o f t h e l o a d i n g on t h e e n d p l a t e  for  each  airfoil  and  f o r v a r y i n g t i p - e n d p l a t e g a p s . The r e s u l t s  a complete  endplate  endplate  airfoil.  over  determined  range o f t e s t  to  at a l l airfoil  lowering the  airfoil  were  made  incidence  were e x t r a p o l a t e d  48  to  zero  gap t o d e d u c e  airfoil/endplate The  net effect  was  typically  large,  combination  on t e s t less  endplate  loadings  corrected  airfoil  fact  markedly chord on the  30%. Table and  that  minimum  the  pressure aluminum  metal.  than  test  that  airfoil  provided  0.67-NACA-0015 taps  (Plate  pitching  airfoil  Figure  moment  drag  6.4 show  was  typical  uncorrected  and  airfoil.  on t h e e n d p l a t e corresponding  d i d not vary  to  a  gap t o  the claim that theloading  s h o u l d n o t be t o o s e n s i t i v e t o  t h e gap  i s less  airfoil  center-span  Plastic  1.0m  section  tubes  length  were measured  pressure  transducer. transducer  balance  the p l a s t i c  a  housing  than  with  37  a  certain  external  has a  taps. A l l orifices  and  approxim-  pressures to  the  wide  via  the  testsection.  u s i n g a 48 p o r t " S c a n i v a l v e " m a n u a l - s c a n The  and t i e d  tubes  were  disconnected  t o the windtunnel  from  the  balance "turntable  measurements t o e l i m i n a t e a n y e f f e c t tubes.  75mm  diameter  diameter  the s u r f a c e  location  65 c e n t e r - s p a n  pressure  a n d h a v e 0.5mm  o f 1.6am i n s i d e  transmit  Pressures  pressure  to  was f i t t e d  6 ) . The Joukowsky a i r f o i l  spar  in  and  0.005, s u p p o r t s  mounting  during  on t e s t  corresponding  pressure taps are surface f l u s h  ately  and m i d c h o r d  value.  The  in  4  the loading  o f about  gap,  lift  l o a d i n g s f o r t h e Joukowsky  a two-dimensional tip-wall  was due t o t h e two e n d p l a t e s .  2%; t h e e f f e c t  f o r a gap l e s s  ratio  which  airfoil  than  typically  The  t h a t p o r t i o n o f t h e l o a d i n g s on t h e t e s t  of  tension  49  6 ._3 T e s t  Procedures.  The tube  testsection  mounted on  the  between  the  Located  thus,  to  relatively  be  Also in  the  settling  nozzle  sufficiently  chamber  t u b e would the  tube  accurate  the  flow  reading,  l o c a t e d . . The pressure  transducers.  monitored  on  pressure  the t o t a l  head  the for  tests,  same the  pitotstatic  "Scanivalve"  a  t u b e ) , and  pressure  Since is  the  a  head  airfoil  the t o t a l  airfoil,  deduced  pressure  total  test  same a s test  and  from  the the  downstream  produce  numerically  water).  a "Betz"  walled  test  airfoils  connected  would  to  pitot  be  "Barocel"  reading  is  During  measured  as a c a l i b r a t i o n  static  pressure  are  (with  pressure  testsection  windspeed  pitotstatic  and  used t o  determined  i n place, at incidence)  nozzle  testsection,  micromanometer.  at the n o z z l e p i t o t s t a t i c  reference  a-second  transducer.  measurements  head i n t h e  obtained.  against  nozzle i s  a  micromanometer..  be  the nozzle  used  Thus  solid  "Betz"  effects^  far  of  were  i n the  The. r e f e r e n c e w i n d s p e e d and balance  the  tests,  simultaneously  upstream  calibrated  empty  tubes  During  f a r enough  would  on  midway  entrance.  be  (mm  was  nozzle  testsection  voltage, could  where  pitotstatic  tunnel  pitotstatic  model " b l o c k a g e "  speed  either  tube  a  sufficiently  readings  centerline,  the  test  be  t u b e mounted i n t h e  flow  annd  by  from  i n the  t u b e would  unaffected  that  deduced  exit  pitotstatic  nozzle p i t o t s t a t i c  pitotstatic  was  centerline  or as a " B a r o c e l " o u t p u t  • The  on  the  pitotstatic  scale  flow  pitotstatic  the  large  windspeed  tube  as  reduce follows.  (when  there  i s essentially  the  i n the  vicinity  of  static  pressure  can  t u b e measurements  of  the be  total  50  head  and  only  be  test  dynamic  verified  airfoil  testsection part  in  200.  Let  H,  pressure i n the  in  total  head  p and  q be  pressure  refer  the n o z z l e  testsection  (solid  ratio  k  is  dynamic  to  incidence, equivalent that  the  the  reference  past  the  static  Figure  6.5  that  the  lower  than  total  +  q  = H  the  taken  occur  airfoil  pressure shows  curve  a using  to the  + qz  2  section  empty  for  i n the  nor  i s the  i n the  walls)  gj are  kq  two  x  test  (solid  and  i s not  empty  (solid  walls)  and  x  (H.j-kqi)  static  pressure  airfoil, OAB,  at  any  are  the  stream  conditions  walls)  testsection  x  the  calibrated while  q . i n the  nozzle.  a c t u a l flow  reference s t a t i c while  tubes.  Hence speed  pressure under  nozzle-testsection pitotstatic  2  measured  slotted-wall  testsection,  typical the  H  and  (6.1)  uniform  empty  the  -  are  any any  measured  In  testsection  pressure  of  pressure';,  above,  (solid  and  head by .1  s u b s c r i p t s 1 and  r e f e r e n c e w i n d s p e e d and  presence  that  static  l e t the  s i n c e only Hi  static  total  respectively.  = p  2  nozzle  head,  according  i  (solid  the  and  w i n d s p e e d "so "deduced  test  actual  calibration  to  measurements i n d i c a t e d  i n the u n d i s t u r b e d  would a c t u a l l y  corresponding  such  empty t e s t  data  values  no  equivalent  Thus  in  l  and  reduce  with  testsection  then  the  respectively.. used  P  the  pressure  walls)  the  and  =  2  test,  was  walls),  o f q! t o q ,  under  can  respectively,  Hi  If  heads  empty t e s t s e c t i o n  place;  dynamic to  (q). This e q u a l i t y of t o t a l  The  the  test.  windspeed ratio  k  51  is  determined  through  the  The  and  a straight  strain  and  windtunnel  gauge  simultaneously.  amplifiers,  nulling  they  switched  left zero for  windspeed  every  20 t o 3 0  linear  These  i n time  and  m i n u t e s and zero  and  time-ayeraged  significant, digit  The  transducer  relative  measurement possible  errors  degrees (Re)  are,  as  strain  are  on  a  units.  type,  the  drift  standard off  the  r e c o r d the  in  the  a p p l i e d to  each  drifts  then  these  strain  zero  single  endplate  drifts  i s the  gauge  voltmeter  follows. Typical  (including  available  The  are  same.  voltages.are steady  3 or  4  displays.  windtunnel v a l u e s and  f o r e x a m p l e , f o r t h e Joukowsky  incidence  gauge  vacuum t u b e  between r e a d i n g s  digital  accuracy  i s indicated  two  t o produce e s s e n t i a l l y  r e a d i n g s on  three  independently  to minimize  are  and  independent  readings i s to shut  drifts  time  read  the  readout  balance  t h a t the  pressure  amplified  number  are of  i n a p r o p o r t i o n , assuming t h a t  All  +3  the  six  units  at a l l times  on  windtunnel  readings.  reading  curve.fitted  that in principle,  only  readout  'ON'  has  measured and  amplifiers  readings  procedure  zero  these  so  practice,  and  As  the  least-squares  balance  cells be  In  simultaneously.  in  load  t h r e e moments can  are  line  origin.  six-component  four-arm forces  from  balance maximum  airfoil  effects),  at  Reynolds  0.5(10 ): 6  C =0.713±0.004 (0.6%) ; C L  Mc_  =-0.0625±0.0005 (0. 8 % ) ;  4 and At t h i s  C = 0 . 0 2 8 6 ± 0 . 0 0 0 4 (1.4%) i n c i d e n c e and  Re  the  net  lift,  (6.2) quarterchord  pitching  moment, d r a g and  0.2%,  determined  and w i n d s p e e d  c a n ba measured, t o w i t h i n 0.2,0.4,1.0  respectively.  Similarly  by i n t e g r a t i o n  of a  a value of l i f t  measured  pressure  coefficient distribution  has C =0. 7 0 9 ± 0 . 006 (0. 9%) . In of  comparing  incidence  degrees  coefficient  there i s a  i n t h e measured  worm g e a r  mechanism.  turntable  in  the  values  maximum  (6.3)  taken  at different  possible  error  i n c i d e n c e due t o s l a c k  Care  same  was  taken  direction  backlash  i n t h e worm g e a r  is  the  largest  the  slope of the C  to  to  errors  ±0.005  i n the t u r n t a b l e  always  minimize  rotate  the  the effect of  mechanism. T h i s p o s s i b l e  of the possible  of  angles  error  which a r i s e  in  a  i n computing  (a) c u r v e s .  F i g u r e 6.6 shows a t y p i c a l  C  (a) c u r v e ,  with  error  bars.  L  The  vertical  error  error  bars  indicate  o b t a i n a b l e on a s i n g l e  horizontal  error  measured  incidence. .  t h e maximum  accuracy  measurement i s good f o r t h e t y p e F i g u r e 6.7 shows  three  C  obtainable  (ct)  consecutive runs The  degree  of  measurements the  type The  under i d e n t i c a l repeatability  taken  obtainable  error on  curves  and  a  i n the single  here.  indicates  the  •  measurements test  taken  in  three  c o n d i t i o n s and p r o c e d u r e s .  obtainable  on a s i n g l e  o f m e a s u r e m e n t s made level  in  possible  o f m e a s u r e m e n t s made  L  repeatability  cumulative  measurement a s d e s c r i b e d a b o v e . T h e  bars i n d i c a t e The  t h e maximum p o s s i b l e  r u n (which  gives  confidence  in  i s not repeated) , f o r  here.  of turbulence i n the two-dimensional  testsection  insert as  was  h i g h as  0.1%.  It  streamwise now  not. m e a s u r e d . in  the  is  probably  is  assumed  unmodified , octagonal less  fluctuations  11.8 v e r s u s  It  7 for  due  the  due to  to the  octagonal  the  that.this  level  crosssection, expected  increased  might  be  that  is  reduction  contraction  crosssection.  in  any  ratio,  54  Zi. E x j B s r i m e n t a l R e s u l t s .  The listed  results  i n Tables  coefficients  of t h e experiments 5, 6  of  and  lift,  7.  drag,  moments, a c c o r d i n g t o t e s t number  (Re), as  lift-curve  line  on  an  of  zero l i f t  calculated lift  angle  For  the  on  increments  with  (-2°,8°).  the  when C  L  a  Joukowsky  a i r f o i l with  zero-lift  pitching  moment c u r v e ,  a i r f o i l incidence from 10  from  a  degrees 2  dc  incidence,f o r  degrees  above t h e z e r o  m i s calculated  0<>,  m  i s  with a z e r o on  angle  the  angle. For  of  airfoils  the  below  lift  angle  a zero-lift  (a) . The  least-square f i t  c a n be d e t e r m i n e d  i s zero. In a similar  as  Reynolds  on '•(-6°,4°). From  angle  tabulated  (c/H) and t e s t  For the Clark-Y  -6.3 d e g r e e s ,  are  pitching  zero-lift  o f about  m i s calculated  line  of  t o 8 degrees  NACA-0015 a i r f o i l s  degrees,  of test  are  airfoils  and q u a r t e r c h o r d  a i r f o i l size  interval  the nine  results  midchord  functions  measurements i n 1 degree  the  The  s l o p e dc / d a i s c a l c u l a t e d  straight  angle  using  (-8°,2°).  o f about  fitted  -3.8  straight  as the a - i n t e r c e p t  manner t h e s l o p e o f  the  / d a , c a n be d e t e r m i n e d .  midchord  The  position  the  airfoil  Mo of  t h e aerodynamic c e n t e r  leading  x a  c  a  _ Xo _  C  c  the  slope, slope,  respect  c  to  /7 1\  _T  m  x , m, c , and x a r e t h e d i s t a n c e f r o m 0  axis  curve  with  c  edge i s x  where  /  the leading  o f measurement o f t h e moment c o e f f i c i e n t  slope, the chord respectively. the z e r o - l i f t  and t h e Table  angle  midchord  C , the Mo  pitching  of  the  lift-  moment-curve  8 contains values of the  and t h e p o s i t i o n  edge t o  lift-curve' aerodynamic  55  center  f o r the  nine  airfoils  and  the  various  wall  variation  of C  configurations  tested.  Figure  7.1  shows  a typical  (a)  with  upper  L  wall  open-area  corresponds slopes  to  i n Figure  resulting  of the  airfoils  quite  sensitive  the  the on  are, to  least-squares  example,  f o r the  value  of  is  7.5  pronounced  [31],  and  will  to the  d e g r e e of  the  the  choice  0. 67-N AC.A-00 15  m computed  on the  Ci,(a)  curve airfoil  value for  curve  on this in  this  separation  the  As  progressively  Figure values the  of  used  less  7.2  of  f o r one  Re  is  any  here, which  done. solid  For  walls,  the  value  i s 0.1138.  This  as  range [30])  there  is a  of  Reynolds  the  formation  of  subsequent  increased,  for  over  is  separation  l e a d i n g edge and  shows  theory, the  by  is  the  the  laminar  reattachment jog  becomes  pronounced.  lift-curve  present  function  the  bubble  fact  required  airfoil  of  downstream.  m,  incidence  (00,12°)  (see T a n i  l a y e r near the  slopes,  between two  jog i s a t t r i b u t e d to  boundary  this  zero-  there.  fitting  numbers. T h i s a laminar  40%;  accuracy  range of  The  (-2°,8°) i s 0.1114, w h i l e  noticeable  jog i n the  lift-curve  of  line  and  plotted lift-curve  [32]).  discussed  of  line  free-air  Riegels  values  straight  particularly  straight  l i e between z e r o  i s 0.1156 and  (0-o,10O)  The  experimental  Sherman  a p p e a r s to  apparent The  and  OAR  (OAR).  established  (Jacobs  correction also  ratio  upper  s e t of  a  comparison  of  two  slope  m,  with  the  for  a  14%  0.53-Clark-Y  w a l l OAR. measured  The  sets of  corresponding  and  the  slopes  airfoil,  large wall s l a t s  values,  measured  small  from as  a  (92mm) were slats  (46am)  56  for  the  the  other  s e t . The  lift-curve  (zero OAB). 0.5(10*), values  ordinates  slope,  m ,  The  test  Re  based  on  the  of l i f t - c u r v e  straight  line  computed  a t -8,  F i g u r e 7.3  and  +2  shows  tested at  slopes  are  shown  slats.  The  theoretical  determined values  of  from lift  a Re  m/m  determined  are  values  1.0 ( 1 0 ) .  values  and  higher  than  lift  The a  walls was  theoretical least-squares  coefficient  f o r the  both t h e of  C>  straight  the  lift-curve  l a r g e and  small  line -2,  show t h a t t h e  0.67-NACA-00 15  measured  lift-curve  , computed a t  7.3  solid  of  measurements  from  The  6  C  value  incidence.  f o r t e s t s with  7.2  of  the  two  chord.  results  a least-squares  incidence. Figures of  m are  of  coefficient  of  airfoil  similar  airfoil  sets  test  degrees  by  presence of  both  f i t through three -3  normalized  i n the  in  slope  are  slope  wall  m  are  f i t through  three  +3  +8  degrees  theoretical  values  experimental  and  values,  for  both  2.8%,  at  s  airfoils, 70%OAR,  and and  present  (per  large  0.67, could which  was  were t h e n published  slats.  for of  and  four  reached  tested at adjusted  by  values  sizes  different  0.34)  for  were  OAR,  run  of 0,  60,  three at  a Re  0 . 3 ( 1 0 * ) . The  correspond f o r the  to  lift-curve  70,  and  80S,  larger a i r f o i l s  a Re  data  extensions  NACA-0015 a i r f o i l  smallest  to  two  of  f o r the of  about  to  the  8.2.  A l l t e s t s f o r the  and  m (Re)  accounted  shows e x p e r i m e n t a l  walls  be  be §§8.1  degree)  0.51, not  in  7.4  presence of the  will  theory  Figure m,  f o r a l l OABs. T h i s d i f f e r e n c e i s  in  the  using (c/H  of  of .0.5(10*). T h i s  Re  airfoil  the  slope,  data  (c/H  of  0.17),  for this  airfoil  0.5(10*)  NACA-0015 a i r f o i l  Re,  using  (see J a c o b s  and  57  Sherman [ 3 1 ] ) . The  adjusted  figure  results  7.4.  ( z e r o c/H)  The  lift-curve  with  [31].  The  upper  w a l l OAB  Figure for  four  different larger of  7.5 sizes  OAR,  0.25)  data  a (Sa)The  (c/H  extrapolation  a  Re  to e s t i m a t e  was  lift-corrections  good  experimental  and  agreement for  with  an the  a  of  for  0.39)  of  the  the  of  m (He)  aspect By  equivalent  a t a Re  using  value  aspect of  ratio  for  ratio,  air  Re  (zero  i n d i c a t e d f o r an  11  14%  of  a  value  with An  w a l l OAR  an  o f ra  relation  thickness,  0.096.  upper  c/fi)  [ 33 ], f o r  estimated  0.5(10*) i s  as  [33]).  theoretical  the  The  section i s scarce.  6 gives  the  Re  (c/H  0.45(10*) -  0.5 (10*)  free  Figure  of  at a  smallest  (Silverstein  of  m  three  0.096, w h i c h a g r e e s f a v o u r a b l y available  of  walls  were r u n  for the  to  values  presence  t h i c k Clark-Y toward  an  are  in  a free a i r  agreement  t e s t e d a t a Re  14%  0.5(10*).  thickness  in  adjusted  of  for  14%  is  infinite  points  lift-corrections  be r e a c h e d  s e c t i o n of  to  the  not  of  curve  correcting  in  i n the  0.53,  were n o t  value  the  70%,  airfoil  0.66,  which  thick Clark-Y at  (5.1)  of  0.093,  l a r g e slats.: A l l t e s t s  of  that  of  corresponding  f o r the  slope  flagged  theory.  could  airfoil  information  0.071  and  show a c o n v e r g e n c e  lift-curve  11.7%  He  information  results  the  the  airfoil,  for this  60  Clark-Y  using  airfoils  value  present  of  the  i n d i c a t e zero  shows t h e  0.5 ( 1 0 * ) . T h i s  of  slope  between  are  show a c o n v e r g e n c e t o w a r d  results  , p r e d i c t i o n s of the  data  then  value  of  Here  zero  less  m  than  60%.  Experimentally, correction  OAR  from  both  Figure  a p p e a r s t o ±>e somewhat  7.1 less  and than  7.5, 60%,  the based  zeroon  a  :  58  free  air lift-curve  0.092 The  were  s l o p e o f 0.096-  chosen,  difference  in  the  test  smallest  airfoil  have been  airfoil  has  relatively  airfoils,  a  which  Figure  configuration,  slotted  o f two s o l i d using  the  the t h e o r e t i c a l  wall  the negative  airfoil,  positive  p r e s s u r e s on t h e u n d e r s u r f a c e ,  measurements w i t h coefficient, conditions. distribution  without  those  obtained  the larger  a t +3 d e g r e e s  incidence  slats.  TSOSL  The  appreciably  The c l o s e a g r e e m e n t  this  pressure  of the  the  upper  modifying  or generally of C  wall  comparison  effect  pressures over  of  distribution.  that  experimental  that the chief  surface  the  the  large  to  m-values.  w a l l s , a n d t h e 70%OAR,  prediction  wall i s t o lower  appears  f o r the higher  airfoil,  due  t e s t s £ 3 6 ] with the  It  of  as  be a b o u t 6 0 % .  s m a l l e r nose r a d i u s than  comparison  such  be l o w e r e d  Re. P r e v i o u s  on t h e Joukowsky  t h e presence  supports  a  value  would  should  unreliable.  might account  7.6 g i v e s  distributions  lower  t h e z e r o - c o r r e c t i o n OAR  m-values f o r t h e s m a l l e s t a i r f o i l  the  in  If a  distorting  v a l u e s from  by i n t e g r a t i o n o f t h e  balance pressure  C , indicates satisfactory two-dimensional P C u b i c - s p l i n e p o l y n o m i a l s a r e used t o c u r v e - f i t of C  the  flow the  f o r integration. P  In solid  Figure  wall-correction  actual correction  These c o r r e c t e d data as the  pressure  data  from  F i g u r e 7.6 f o r two  walls are corrected to equivalent free-air  conventional The  7.7,. t h e  taken.  formulae  a r e seen  wall data.  {Pankhurst  used  a r e compared  The 70%OAR d a t a  corrected solid  theory  and H o l d e r  are recorded  with  t h e 70% OAR  t o agree  conditions  quite  The c o r r e c t e d s o l i d  by  '£34]).  i n Appendix  7.  pressure  data  closely  with  wall value of  59  C  is  0.675, w h i l e  indicates at  an  t h e 70%OAR  that zero l i f t - c o r r e c t i o n  upper  wall  OAR  pressure  distributions  by  thicker  the  replacing Por  completeness, affects  of pitching  Figures coefficient 0.53-Clark-Y  for  7.8 c  D  . mxn  test 60  and  edge  and  trailing  t o show other  how  of  the  edge  of  7.9  occur  70%. The hump i n t h e surface  i s  experimental  caused airfoil  cusp.  aerodynamic  show  data,  a TSUSL some  wall  typical  coefficients  are  shown,  the  pitching  as  midchord  C_  respectively  moment  for  the  D  as  a f u n c t i o n of a i r f o i l  agree  (near  well with  i n c i d e n c e , and  free-air  corresponding  conditions) values  from  0  and G r e e n b e r g  at a given  decrease,  with  increasing  observed  for  the  [ 3 5 ] , and  incidence i n i t i a l l y OAR.  T h i s same  other a i r f o i l s  e n d p l a t e s , and a p p e a r s configuration.  again  will  t h e use: o f s u c h  The v a l u e s a t 70%OAR C_ M  conditions  this  incidence-  Mo airfoil,  and  i s 0.651;  T  t h e r e a r upper  moment and d r a g  F i g u r e 68 o f P i n k e r t o n values  of C  and d r a g , c o e f f i c i e n t  w a l l OAR. C  toward  trailing  f u n c t i o n s of a i r f o i l  upper  between  the t h e o r e t i c a l  configuration curves  value  Lim  i n c r e a s e and  behaviour  t e s t e d , both  t o be a p r o p e r t y o f t h i s  [36].  with  of and  C  D  The then was  without  particular  test  60  Si.  Extensions to  What i s a v a i l a b l e a t a  present  f o r an  predicts  lift-corrections  and  low  incidences.  occurs Thus the  i t were o p e r a t i n g The  result,,  theoretical normalized are  too  theory  of  their  high.  is a  p o t e n t i a l flow  solid  wall  OAB  lower of  is  less  a larger a  than  given  i s that  slope  m or l i f t  theoretical  because  p o t e n t i a l flow  and  70% sizes  which  OAR  zero  predicted  experimentally  OAR,  corresponding is  at  the  behaves  60  airfoils,  OAB  for  configuration  between  the  theory  as  OAB.  lift-curve  This  wall  f o r a v a r i e t y of  slotted wall  at  at  values by  Theory.,  However, e x p e r i m e n t a l l y ,  lift-correction theoretically.  upper  New  is a  t r a n s v e r s e l y - s l o t t e d u p p e r and  (TSOSL), t h a t ,  if  the  the  the  r a t i o s , of  the  coefficient  C ,  free-air  TSOSL w a l l  representation  of  a  L  values,  configuration viscous  flow  field.  For  example,  developing flow  their  theory  180,000  predicted at  respect on  the  v i s c o s i t y , the  c i r c u l a t i o n s predicted are  to  operating  their  slotted  chord  OAR  w i l l be  which i s due  wall s l a t s by  i n a Re  to  much l e s s t h a n t h a t  of  the  circulation  portion  of  the  the  are  not  potential  l e n g t h s ) . , Thus  portion  the  the  range of  wall i s l e s s than  t h e o r e t i c a l l y . Hence t h a t  a given  slats,  full  s i n c e they  (with  circulation  because of  37,500 the  to net  circulation  measured,  lift  on  wall  the  lift  predicted  theoretically.  In that  other  words, i n o r d e r  would a c t u a l l y be  to  predict  t h e o r e t i c a l l y the  measured e x p e r i m e n t a l l y  on  an  airfoil  lift in  61  this to  TSUSL w a l l be  flow  configuration,  accounted  field  for.  about the  field  has  wall  configuration.  and  any  Let from C  of  C  ( ;E)  experiment  (F; )  reasonable equal  to  theory  and  C  respectively.  ratio  C  For  the  each will  be  the  will  same f o r  produces near useful of  C  (OAR; T) ;C  usual  the  the  wall  plenum.  c o e f f i c i e n t s obtained let  OAR  C  and  (OAB;E) :C  (F;T) ,  TSUSL  (OAR; free  )  and  air test  OAS,  i t  is  (OAB;T) w i l l  particularly  in  a  be low-  In  this  case the  same. Thus t h e  free-air  t e s t s or  viscous  ratio  of C  f o r t e s t s at  test conditions. Therefore ratio  of  C  (OAS;E):C  effects  (E)  C  (T)  whatever  OAR  i t  (F;E) , w i t h  to  on  is  still  the  ratio  (F; T) . Li  . What  which  airfoil,  particular C  the  flow  Li  the  free-air  t o compare t h e  L»  will  be  ratio  (F; E) : C  t e s t environment.  ratio  a  viscous  would r e q u i r e  i n the  from  fields  viscous  through  test  denote l i f t  flow  free a i r  a second  flow  the  coefficients  Li  correction  usual  Now  the  viscous  r e s p e c t i v e l y , and  t o assume t h a t  the  of  to  { ;T)  theory  lift  configurations  i s the  l a y e r s which d e v e l o p  and  denote  that  two  Thus a c o m p l e t e s o l u t i o n  viscous shear  first  are  isolated airfoil.  been i n t r o d u c e d ,  application slats  The  there  is  then desired  account f o r are  due  viscous  required.  the  only  viscous  t o the  flow  is a  p o t e n t i a l flow effects  TSUSL w a l l  analysis  f o r the  calculation that  present  configuration. test a i r f o i l  experimentally That will  i s , the still  be  62 8.1  Potential  Flow C o n s i d e r a t i o n s o f V i s c o u s  T h e r e a r e two ways t o e x t e n d a potential effect the  flow  calculation.  of viscosity  formation  shapes.  The  second  The  in  airfoil  is  like  to  only  completely  that  attached  flows  Thus,  surfaces statement  since  adjacent  consecutive  from  an  the flow  lower  surfaces.  edge)  these  through  streamlined  actually  t h e plenum  the  F o r boundary  airfoil  occurs  surrounding  equal  This  leaving  relation  velocities edge  layer  effects,  are  the a i r f o i l  the t r a i l i n g  wake of  not  trailing and  edge, must be e q u a l .  s i n c e the formation of  requires  the  alternate  sign.  shedding  edge v e l o c i t i e s  is  physical  of  For a given  circulation  trailing the  airfoil  and p r e s s u r e s a t t h e upper  vortices  the  test  surface  s h a p e and i n c i d e n c e , t h e r e i s a u n i q u e  set  trailing  to  i n the a i r f o i l  trailing  will  the  on  which  i s t r u e on a t i m e - a v e r a g e ,  street  theoretical  f o r the  a r e c o n s i d e r e d , t h a t i s , boundary  edge i s a t t a c h e d , t h e v e l o c i t i e s  airfoil  account  such  make t h e g e o m e t r y o f t h e f l o w  wall slats.  f l o w s which s e p a r a t e  a vortex  toward  and w a l l s l a t s ,  the t e s t s e c t i o n , with  airfoil-shaped  considered.  This  to  f o l l o w i n g d i s c u s s i o n a p p l i e s t o both  the  lower  theory  wall.  and  layer  way  more  experimentally slotted  is  of v i s c o u s boundary l a y e r s  representation  the  the present  One way  on t h e t e s t  Effects.  that  on t h e u p p e r a n d  reasoning  behind  the  known a s t h e K u t t a c o n d i t i o n , t h a t i s , t h a t  a t t h e u p p e r and l o w e r ' s u r f a c e s a d j a c e n t  ( o f any body  which  possesses  a  sharp  t o the  trailing  must be e q u a l .  Experimentally  the  measured  value  of  lift  (which  i s  63  proportional predicted these  by  lifts  procedure  From  might  be, f o r e x a m p l e , the  usual  uniqueness  shape  and  flow  reduction  comparison  of  distributions.  of  incidence  of is  of  The  theoretical  •  the  pressure  upper s u r f a c e ) measured.:  as  possibility  i s o f t e n used  experimental  pressures  of  lift  pressure at a  (determined  at  measured l i f t  the  by  reduced  a t t h e measured  satisfactory  as  the  p r e s s u r e . peaks i n t h e t h e o r e t i c a l  (near t h e a i r f o i l  the  i nthe  are calculated  distributions)  i s not completely  may n o t e q u a l  Physically  fora  t h e shape o f t h e p r o f i l e - .  and  the  negative  distributions  only  that the t h e o r e t i c a l  same  the  the  i f the Kutta c o n d i t i o n i s r e t a i n e d , i s  incidence.•, T h i s procedure magnitude  of  o f the value of t h e c i r c u l a t i o n  theoretical  the  the r a t i o of  value  k of r .  the  the  lift  theoretical  i n c i d e n c e approach  i n c i d e n c e , such  integration  theories;  the  i s required i s a  t h e i n c i d e n c e , and/or a l t e r  The  than  k. What t h e n  incidence,  the c i r c u l a t i o n ,  pressure  i s always lower  potential  to a fraction  the  reduce  reduced  usual  reduce  r  airfoil  reducing to  the  to  circulation  given  to the c i r c u l a t i o n )  leading  edge  t h e magnitude o f t h e peaks thin  laminar  on  the  actually  boundary l a y e r ,  which  forms beginning  a t t h e forward  of  grows i n t h i c k n e s s as i t r o u n d s t h e l e a d i n g e d g e  the a i r f o i l ,  and  passes  this of are of  thin  through  boundary l a y e r  the a i r f o i l much the thin  the negative  s t a g n a t i o n p o i n t on t h e u n d e r s i d e  pressure  than  boundary  those layer  The ' e f f e c t  of  i s to i n c r e a s e the radius of curvature  l e a d i n g edge s o t h a t t h e  lower  peaks.  flow  velocities  p r e d i c t e d by t h e o r y . over  t h e l e a d i n g edge  there  Thus t h e e f f e c t i s  to  reduce  64  the  magnitude  magnitudes incidence  of  predicted for  t h e same d e g r e e  boundary  condition,  calculation  i f  from  trailing  lift  obtained  can  be  8.1  may  to this  altering  the  lift.  the  airfoil  calculation  will  peaks,  not  results  the  the  i s  possible  physically  profile  shape.  Recall  that  when  for  for  but  for  the  the  incidence.  i s  distribution  measured  two  in  lower  the  i f  solid  the boundary  an a i r f o i l ,  layer  8.  measured at  walls.  10 The  unsatisfactory as  such  large  there.  circulation This  the a l t e r a t i o n s  The  Appendix  airfoil  edge  profile.  lift.  and  method a r e p h y s i c a l l y  not found  pressures  but the t h e o r e t i c a l  0.67-NACA-0015  trailing  theoretical  The v e l o c i t i e s a t  presented  of  otherwise.  the  theoretical  presence  airfoil  to  by t h e K u t t a  theoretical  experimentally  the shape o f t h e a i r f o i l  which o c c u r  The  resulting  by t h i s  of the  fixed  of t h e p r e s s u r e  pressures are i n practice  justifiable those  the  the c i r c u l a t i o n  determined  calculation  in  not  no l o n g e r be e q u a l ,  equal  compares  the v i c i n i t y  i s  a t t h e measured  incidence  theoretical  be  t h e measured  pressure d i s t r i b u t i o n s  It  pressure  below  o c c u r s due t o t h e p r e s e n c e o f t h e  from t h e i n t e g r a t i o n  made  negative  Reducing  negative  to specify  edge w i l l  procedure f o r  in  peaks  theoretical  circulation  i t i s possible  the  degrees  the  of t h e s e  the  then c a l c u l a t e d  Figure  of  as a c t u a l l y  circulation  are  pressure  layer.  However,  The  negative  theoretically.  purposes  reduce t h e magnitude to  the  developed procedure  a r e i n keeping  effectively  the effects  by i s  with  modifies  of viscosity are  65  confined  to the t h i n  surface,  and  considered profile  the  boundary flow  layer  outside  as i r r o t a t i o n a l .  shape,  The  using  the usual  following  sections  to  be  mapped  is  conforraally  Theodorsen  original  the  modified  profile  airfoil is  the  camber t h a n profile  modified  pressure  direction  effectively  which r e s u l t s  the o r i g i n a l by  thickness)  m i g h t be c o n s i d e r e d , would  have  thickness than  the  thickness  symmetrical  thickness airfoil  zero  lift.  taken  (vertically)  8.3. w i t h modified  Therefore  i f  of  at  the t r a i l i n g  be s i m i l a r  were f a i r e d  when  at  that  surface,  and  point,  profile  less  calculation edge  and  p o i n t . The i s  greater  except  two s o l i d  i s  layer  for  walls,  edge was i m a g i n e d  to that  to  (original  boundary  trailing  o f t h e wake,  to the e f f e c t i v e  the  respect  profile the  the  wake  lower  as the midpoint  by  profile  which a p o t e n t i a l f l o w  the  shows  and h a s s l i g h t l y  The a c t u a l  i n f r e e a i r o r between  of the p r o f i l e  s h a p e would  on  circle  of the  the  the  airfoil  i n the c a l c u l a t i o n  addition  of  of  walls.  incidence  would be b l u n t  that  procedure  o f t h e b o u n d a r y l a y e r on t h e u p p e r s u r f a c e  the  portion  about  condition-  between two s o l i d  i n Figure  profile.  8.2  this  and t h e r e s u l t i n g  the present  flow,  lower  the  about  extends  airfoil  used  distribution  at a s l i g h t l y  swollen  displacement  profile  of the approach  and  method. F i g u r e  from  be  layer modifies the  onto a m o d i f i e d  NACA-0015  can  two-dimensional  i s a t 10 d e g r e e s i n c i d e n c e  the  theoretical  of  for  airfoil  layer  calculated  follows  means o f t h e c l a s s i c a l profile  i s  the  e q u a l - v e l o c i t y Kutta  procedure  [ 3 7 ] whose a n a l y s i s  to  boundary  Thus t h e b o u n d a r y  Pinkerton  This  the  s h a p e and t h e p o t e n t i a l f l o w  modified  this  adjacent  the  a at  t o be after  the r e s u l t i n g  of Figure  8.2-  66  The the  airfoil  i s supposed  p r o f i l e are  to  be  proportion  to  their  direction  of  rotation  of  the  p r o f i l e - As  amount,  this  leading  the  as  all  not  a  the  resulting  in  of  rigid  -  Points  p r o f i l e leading  p r o f i l e leading  reduce the  points  a  incidence  the  to  are  rotated  camber  "raised"  edge  by  the  in The  incidence, the  about the  of  on  edge-  effective  body r o t a t i o n  edge.is  this  Appendix  theoretical  0.67-NACA-0015  presence of  two  procedure  are  pressure  Figure  same  profile  profile  more t h a n o t h e r  walls.  The  solid quite  i n c r e a s e the  8.3  is  points  airfoil is  pressure 10  degrees  leading  radius of  same d e g r e e t h a t  incidence  actually  results  by  the  edge.  The  magnitude  curvature  the  than  the  this  of  magnitude  any  procedure  of  the  airfoil to  the  negative  for  o c c u r s due  for  in  for  so  are  of  except  larger  a l w a y s be  procedure  distributions  theoretical  still  will  edge  shows a c o m p a r i s o n  satisfactory  peaks  does not  trailing  measured at  m e a s u r e d , and  the  9-  airfoil  actually  edge t o  "raised"  and  peak n e a r t h e  theoretical  layer  from  effective  trailing  details  presented  the  at  about the  i s such  not  be  profile.  The  the  the  rotated  distance  edge. Thus  reduced on  is  to  the  that  leading boundary  there.  The slats  a b o v e p r o c e d u r e was  to  see  circulations test  what there  on  the  airfoil.  20  degrees i n c i d e n c e  configuration,  the  the  applied  effect  to  the  would  be  while maintaining For in  the the  test  the  c a s e of  the  presence  of  airfoil  lift  airfoil-shaped of  full  reducing  Kutta  0.66-Clark-Y a  wall  70%OAH  coefficient  the  circulation airfoil TSOSL was  at  wall  reduced  67  from  3.010  used  a value  on  the  to  2.935, t h a t i s , by of k of  test  wall s l a t s  airfoil  alone  viscous effects  A for  laminar  on  performed  and  Seebohm reduces  profile  layer  is  develops  and  no  layers.  difference calculations  with  the  layer  the  wake.  found  wake  a t two  last  points  edge, two  across  at  to  to  steps are  the  c o u l d be  wake no  iterated  has with  potential the  calculated.  The  layer  displacement wake.  the c i r c u l a t i o n  (which  for  no  boundary  difference across a b o v e and  edges on  of  until  l o n g e r c h a n g e s on  i n c l u d e d i n the  [38]  the  (vertically) outer  of  profile,  represent  fix  even  to a turbulent  usual  are  pressure  the  of  effects  airfoil  the boundary  downstream  the  neglect  agreement  The  f o r the  and  condition  the  Seebohm  good  follows.  by  on  calculations  growth, t r a n s i t i o n  and  effect  itself.  i n v o l v e the  performed  wake) b a s e d on  trailing The  must  i s as  swollen  a  Thus t h e  viscous e f f e c t s  e q u a l - v e l o c i t y Kutta c o n d i t i o n  wake, c a l c u l a t e d airfoil  are  of  calculation  f o r the boundary l a y e r  growth o f  i s extended  to the  airfoil  This  wall slats.  neglect  layer  procedure  2.5%.  comparison  flows  boundary  t h i c k n e s s and  in  test  the  calculations  airfoil  the  calculations  His  the  a l l the  accounting  such  growth of  wall  the  boundary  experiments. flow  of  attached  boundary l a y e r ,  on  i s small  complete  completely  the  0.80  about  present  below  the the  the the  boundary pressure  iteration. theory,  Such  on  the  slats.  A  method  upper s u r f a c e o f maximum  lift,  of  handling an  has  airfoil  the  separated  at  been d e v e l o p e d  high by  flow  region over  incidence, J a c o b and  the  approaching  Steinbach  [23].  68  The  separating streamline departs  surface zero  since the  velocity  observed  s t r e a m l i n e has  variation  along  approximately the  at a  above  trailing  to  the  point at  procedure  of  must be  p o i n t , and  Again  a  wall  the  and  performed  procedure  method, f o r the  trailing  slats,  real  the  by  a  region  the in  equal  second  point  streamline.  The  to be.similar  boundary the  The  pressure  requiring  at  the  position  layer of  the  separating streamline.  incorporated  i f desired.  air  i s seen  to calculate on  non-  to represent  dead  e d g e , and  Again  pressure  c o u l d be  constant  airfoil,  method  upper  to a smooth  separating  the  the  separation point.  accomplished  Seebohm.  the  the  i n the  i s  on  fixed,  separation such  the  edge,  i s thus  calculations  This  the,  approximately  pressure  wake.  pressures  circulation  ah  from  correspond  around  i t s l e n g t h , near  constant  separated  the  pressures  distribution  separating  tangentially  into  the  present  69  8.2  The  Flow  The  i n the  second  geometry  of  actually  occurs  plenum flow  Plenum: The  extension  the  flow  the  actually  of  the  airfoils,  undisturbed  inside  and  8.5  formed  as  air  i n the  the  a  infinite  and  plenum. T h i s s h e a r  zero-energy  stagnant  pressure  f r e e s t r e a m l i n e which of  r e a t t a c h e s to the the along  slotted such  the  wall.  positions,  estimated  from  representation  a of  this  wall.  flow  field  streamline  theory set  the  of  direction  of  the.  the  T h i s shear  flows  plenum  layer  is  upstream  otherwise  the  two  testsection,  as  section the  entirely.  the  plenum  and  end  of  variation  initial  pressures  calculation  flow  at  pressure  unknown. The  and  constant-  downstream  the  and  a  of  stagnant  divides  enters  o f , and  inclinations  a  inside  idealized  wall,  initially  the  plenum. I n a p o t e n t i a l  be  position  a s t r e a m l i n e are  terminal  formed  wall s e c t i o n at the  The  the  same.  leaves the t e s t  slotted  solid  levels  e n t r a i n s the  could  with  present  the t e s t s e c t i o n ,  of the  this  end  layer  which  p h y s i c a l flow  past  which e x i s t s i n the  theory,  upstream  extent,  the  compares  the  layer therefore  flow  shearlayer  with  are the  a i r from  mixes w i t h  flow  8-4  surfaces aligned with  shear  that  testsection,  t e s t s e c t i o n . The  the upper s l o t t e d  high-energy  i s t o make  like  Figure  theory  testsection  outflowing  airfoil,  flows, the  of  theory  the  wall-  the  flat  shows  surrounding  test  and  in  Layer.  more  f l o w . Hence t h e e n e r g y  chamber  the  present  present  in  flow  outside the  Figure  the  the  occurs  r e p r e s e n t s a uniform  Shear  representation  slotted  r e p r e s e n t a t i o n of  multiple  the  experimentally  surrounding  which  of  Bounding  and  could  which  be  omits  70  An  iterative  position  of  this  a p p r o a c h . , The resulting of  the  procedure  streamline  position  flow  a,  but  are  the  the  to  test  t o take  the  position  that the  particular  which p a s s e s  solid  slotted  in. t h e  could  be  vicinity  streamlines  iterated  was in  The  found. vortex  the on Such  wakes,  zero  through  lift-correction  flow  field  in a free-air  will  test  wall  approach  environment.  a streamline i s therefore  s t r e a m l i n e i n the f r e e - a i r  the  two  the  p o i n t s d e f i n e d by  beginning  and  case  the  ends  of  the  end  wall.  pressure  to i n v e s t i g a t e the  f r e e s t r e a m l i n e i n the  dimensional  potential  test  was  airfoil  slats  by  Figure  A6.2  flow  represented  present  by  a single  theory,  set  of  wall,  boundary c o n d i t i o n i s z e r o d i s t u r b a n c e  wall.  condition  On  b o u n d a r y c o n d i t i o n s . On  the c o n s t a n t  of constant  pressure  pressure  can  be  a  boundary, expressed  constant-  analytic developedand  the  pressure  to s a t i s f y  the  an  was  vortex,  the c o n s t a n t  multiply-infinite the  of i n c l u d i n g  representation  a s e t of v o r t i c e s near shows t h e  effect  required  the  assumed-  with the d i r e c t i o n  position  f o r such  wall s e c t i o n s at  In order  to  step-by-step be  calculated  compared  airfoil  position  the  be  overall  obvious  of  would  a  An  occupies  segment  the  time-consuming.  approach  t h a t of  segmented  streamline  c o n f i g u r a t i o n i s made, t h e closely  to calculate  a  would  used f o r l o c a t i n g  are computationally  As  a  developed  t h e r e . These s t e p s  satisfactory  procedures  be  in  s t r e a m l i n e and  streamline i s taxing until  of  inclinations  supposed  could  solid  velocity the via  The wall  boundary-  vortex the  two-  images lower normal  linearized Bernouilli's  71  equation  as  s t r e a m wise'  requiring direction.  image o f a v o r t e x the  image  in  zero Thus  disturbance  in solid  i s a vortex  a  constant  velocity  boundaries,  of equal  the  the appropriate  but o p p o s i t e  pressure  i n  boundary  circulation;  has  identical  circulation.  The are  details  i n Appendix  o f t h e image s y s t e m  6. T h e  results  present  analytic  Havelock  [5], fora flat  in  ground  constant  effect,  pressure  pressure  lifts  above  the  the  airfoil,  so  flow.accelerates  two  low  and a  constant  for  either  the  the experiments i n d i c a t e  i s  a  result,  of  there.  The  C ,  i s  p  the the  the slotted  zero.  By  w a l l upstream  the testsection variation 8.7  of  large  negative  when  i n the- v i c i n i t y pressure  tangential  which of  of  wall this  such  a  variation,  excursions  on  test  of the test  pressure  coefficient  a  leaves  the  along-  a pressure  the  value  tracking  downstream  pressure  shows s u c h  constant  corresponding  w a l l c o n f i g u r a t i o n theory)  Figure  (b)  values.  the  average . value-of.the  than  t o be z e r o  re-enters  walls,  representation i s  requires  through  excluding  l i f t  j e t case;  coefficient,  i s known.  image  which  the t h e o r e t i c a l  streamline  The  -the  analysis of  boundary  (d) b e t w e e n  condition  ( i n t h e TSOSL  and  open  the  lower  involved  compare  two s o l i d  solid  and  jet).  effect  i s  velocity  testsection  airfoil,  and,  the ground  pressure  streamline  boundary,  the  boundary  disturbance  a  8.6  with  (a) b e t w e e n  between  (open  or  the l i f t  pressure  plate  as i t p r e d i c t s a lower  effect  That  of  upper  shown i n F i g u r e  representation  (c)  boundaries  unsatisfactory ground  image  and t h e e g u a t i o n s  as the  slats,  the  streamline i s  72  about  -0.25.  Hence any  r e p r e s e n t a t i o n which  uses a zero  C  value P  is  i n c o r r e c t . \ The  the  wall  slats  circulation, in  the  collectively  since  there  re-entering flow  image  representation  airfoil the  TSUSL w a l l c o n f i g u r a t i o n t h e o r y  lift  s e t of  adjacent  example, f o r a f l a t  and  F i g u r e 8.6), depressed  the  wall slat  boundary plate  the  25%  at  circulation  The  i s 2%  to the  slats  flow. the  on  However  the  wall slats  via  boundary.  (see A p p e n d i x  from  but  the  a Clark-Y  be  the  constant  better  for  calculations  airfoil  the  6  lift  the c o n t r i b u t i o n from  t h a t from would  the.  on  pressure  26.3° i n c i d e n c e  and  immersed  effect  the  constant  a i r value;  comparison  incidence,  values  exiting  predicts  p l a t e at  counter-clockwise  number o f  circulation  the f r e e  vortices  20°  net  image r e p r e s e n t a t i o n p r e d i c t s t h a t  below  i s .23%.  i n the  the  a  a larger  correctly  For  is  are  than  of i n c l u d i n g vortices  have  indicates that  at  20°  pressure the  flat  wall  slat  incidence  and  use  7 0%OAR. However,  any  representation very  shear the  cannot  different  Suppose  pressure  i s known. The  variation  that  the  the  plenum)  flow  energy  flow. vortex  flow  This  of  energy i s zero.  level  sheet,  level  on  On  the  but  layer this  condition. streamline  one other the  could  this  s i d e of side Uniform be  •modelling  two head,  flows  streamline In  ( i n the  the  the  fact  streamline  (in  t e s t s e c t i o n ) the  undisturbed  not  this  satisfies  addition,  must r e f l e c t this  of  correctly.  representing  modelled was  free-streamline  the  i s , total  along  boundary this  of  streamline  flow  i s t h a t of  shear  that  the  along  flow  division  level,  position  tangent-velocity  potential  model t h e  energy  that the layer  single  approach  analytically attempted  by  a  here.  7 3  Physically  the  representative high  energy  of  the  the  plenum. To  low  the  is a  t e s t s e c t i o n flow pressure  the  coincident  leaves  region  shear  usual  of  Vortex to  each  control point  unknown S  set  by,  say,  source  and  a d d i t i o n a l zero  S  rather  to than  velocity  the  the  values  the  flow  condition  in  theoretical distributed  are  strength  densities,  tangential velocity shear are  be  S  i s  on  are  additional  solved.  have  at  an a d d i t i o n a l  and  to  layer  Thus . t h e r e  variation,  equations  formed  the  variation  tangential velocity  and  i s tangential to  equations,  pressure  the  in  densities.  eguations  a  used  Thus i f the  strength  which  so  are  elements  S c o n t r o l points, there vortex  in  on  wall section,  elements  flow  of  l a y e r or  present  different  surface.  p r e s c r i b e the  boundary  solid  Source  that  normal-velocity  the  the  vortex  prescribad-tangential-velocity usual  manner  the  recirculating  a l l of  the  of  and  prescribed on  shear  by  ensure  elements,  used  represented  to  i n the  layer  source  manner  are  is  result  a representative streamline.  surface.  S  variation  streamline  represent  method, along  pressure  the  the  the  Since  i t  surface  tangential form,  from  (3. 1 4 , 3 . 1 8 ) : N  M  R(k)  S 7  % r=l  (B  v  ri  . U . + A . V . )  l  r i i-*  (8.1) =  Here  y  respectively  and  v  on  the  -Ucose. 1  are  the  ±  /(l-C  source  streamline  p.  ),  and  i=l,2,  vortex  representing  . . . s.  strength  the  shear  densities layer.  For  74  calculations squareroot are  i n the term  is  distributed  The  same  sense  positive  two  from  remains only  are  to  to specify  estimates  inclinations  found  the v a r i a t i o n  along  of spreading  be  i f the source  direction,  and v o r t e x  the  elements  streamline  occurs  here  specify  the position  coefficients  of  the  i n t h e plenum, s u c h of  of  wall sections.  pressure  (between  the streamline. f o r unbounded s h e a r  i n the l i t e r a t u r e ,  c o u l d be made  are  a n d p r e s s u r e s c o u l d be  t h e f l o w c o n d i t i o n s on t h e s o l i d  known end v a l u e s ) Values  flow  and t e r m i n a l p o s i t i o n s o f t h i s  known, a n d t h e c o r r e s p o n d i n g  It  the  sequentially.  initial  estimated  as  a  layers  ( f o r e x a m p l e , [ 4 0 ] ) and i f  effect  of  estimates  streamline  confinement  c o u l d then  representing  which  be u s e d the  to  shear  layer.  It  is  proposed  that  the  s t r e a m l i n e r e p r e s e n t i n g the shear the  analysis  assumed  and  streamline the  here.  specified, tracking  variation  position position  and  of  and  pressure as  are  pressure  variation  l e a v e s t h e upstream  the t e s t s e c t i o n  apparantly  computations  the  along the  parameter  is  were  performed.  The  4 was used  streamlines  shown  upper of  solid the  on  t o compute of  similar  wall section  some o f t h e e x t e r i o r  of the  F i g u r e 8.7.  test  in  variations  required f o r specification  downstream  "entraining"  i s a free  o f Appendix  along  layer.  streamline  the  procedure  shape  layer  variation  i s , several pressure  of the shear  This  enters  That  pressure  and r e -  airfoil,  flow.  This  thus  I t would  be  75  expected, the  continuity,  downstream  similar in  from  "leakage"  a right  At  angle  first  e l e m e n t s on sections test  common  a  strength source  the  with  is  "leakage"  about flow  With t h e end variations streamline average Figure  8.8  assumed  the  on  the  show how  the  of C  few  the  flow  source  solid  from  20  wall  from  to  two  source  50.  constant  using  The  elements. net  The  flow  flat  uniform  source  using  curved  by  or p a r a b o l i c v a r i a t i o n s  of  source  "entrainment'••  rate;  flow  i n Hsss's  example  several  pressure  12%.  v a l u e s of p r e s s u r e such  too  t h e number o f  i s eliminated  5% o f t h e  representing  value  experienced  interior  calculations  with  problem  r a t e was  were t r i e d  pressure  on  present. This apparantly i s a  flow  linear  over  t e n , and  still  elements the  were that  increased  interior  source  densities  there  to  was  was  for  densities;  here  of the  end  s e c t i o n s were l e n g t h e n e d  lengths  effect  elements  strength  that  s h o r t . These  section  problem  distributed  should  Hess [ 2 2 ]  i n the c a l c u l a t i o n  w a l l s e c t i o n s , and  chord  "entrainment"  wall section.  appeared  the s o l i d  on  streamline  bend.  were t o o  elements  solid  flows  i t  airfoil  rate  upper  that this  specified,  t h a t the average the  shear  tracked airfoil  layer  pressure was  streamlines. lift  coefficient  along  similar The  the  to  the  results  in  varies  . T h i s r e p r e s e n t a t i o n of the shear  with layer  the is  P used  i n the  measured  following section  v a l u e s of l i f t - c u r v e  to  compare  s l o p e f o r two  the  theoretical  airfoils.  and  75  8^2  Summary... A comparison  of the c u r v e s  that  a t an u p p e r  ratio  of the l i f t - c u r v e  o f F i g u r e s 7.2 and 7.3  w a l l OAS o f 70%, t h e e x p e r i m e n t a l s l o p e s ra/m i s a b o u t  indicates  value  2.8% l o w e r  of the  than the  s  theoretical airfoils.  value, Thus  there  be  accounted  for  in  §8.1 and 8.2. Assuming  Figure  for  by  both  Clark-Y  i s a small residual the  that  the  extensions  there  8.8 o f t h e r a t i o  are  of l i f t  the  difference  to  curves  and  the  case,  increase  in C  value  of  the  residual  /C T F be a c c o u n t e d  of about  P  C  for  Thus  for  reduction  of  shear  in  layer  of  the  the  effective  position  -0.33 t o - 0 . 3 1 .  satisfactory  the  further theory.  by  circulations  and s t r e a m w i s e  preliminary;  between  theory  and  plenum  order  could  a  the  theoretical  be  combination  on t h e w a l l s l a t s  by a s i n g l e  is  and  and 7.3 a t a n y OAR c a n be of  (i) the  by  modifying  ( i i ) by t h e r e p r e s e n t a t i o n . o f  pressure work  same  level  of § 8 . 1 , t h a t i s , by r e d u c i n g  F i g u r e s 7.2  profiles,  in  the  A  slats.  difference  curves  accounted  of  the procedure  on t h e w a l l  the  experimental  adjustment  with  circulations  their  f o r by an  o f -0.31 corresponds to the pressure P by t h e r e c i r c u l a t i n g f l o w i n t h e p l e n u m .  F . e c a l l t h a t an  the  +6%, f o r example f r o m  f o r the  L  C  established  accounted  2.8% c o u l d  outlined  to the curves of  L  present  o f 2.8% t o  theory  similar  coefficients  NACA-0015  variation. required  the  s t r e a m l i n e o f assumed These e x t e n s i o n s a r e to  establish  a  77  9j.  A  two-dimensional  correction-free developed. potential  theory theory  singularities.  The  only  wide  range  the  airfoil  a  effect  on  theory  windtunnel  The flow  Conclusions.  test  is  an  b a s e d on  extended of  which  predicts  airfoil  loadings  of  the  method o f  theory  satisfactorily  configuration  extension the  a  has  been  two-dimensional  distributed  surface  takes i n t o consideration s i z e s and  of  shapes, but  different  not  also  windtunnel  the wall  configurations.  The  r e s u l t s of  dimensional wall  the  airfoil  opposite  transversely  the  t h e o r e t i c a l study  t e s t i n g , a windtunnel c o n s i s t i n g pressure  slotted  wall,  side the  symmetrical  a i r f o i l - s h a p e d s l a t s at  area  between 60  of and air  ratio  the  airfoil,  lift  and  yield  configuration  remain r e l a t i v e l y  chord),  correction-free wall  zero i n c i d e n c e ,  predicts  s i z e s and  carried  in a  p r e d i c t i o n s of  slotted  percent, opposite  within  on  300,000 t o  a few  theory.  consisting when t h e  of  solid  and  a  which  are  with  open  suction  side  distributions of the  free  t h i s low-correction  wall  correction-free for a  wide  percent  incidence.  a  number 1  of  million  Experimental  configuration  two-  a  airfoils (based,  could  be  achieved  was  shaped  a  the  support  work showed t h a t  symmetric a i r f o i l  slotted section  for  on.  two-dimensional t e s t c o n f i g u r a t i o n  the  test  out  the  pressure  that  a n g l e s of  of  of  incidence,  uncorrected  for  airfoil,  portions  zero  will  airfoil  the  solid  theory  R e y n o l d s number r a n g e o f  the  of  The  Experiments  airfoil  70  c o e f f i c i e n t s which are  values.  range of  will  indicate that  with  the a  s l a t s at  surrounded  by  a  78  plenum  chamber.  The effects  above  theory  on t h e w a l l  was t h e n  slats,  w h i c h f o r m s i n t h e plenum  Measurements configuration different  airfoils  from-0.17 air  distribution  with drag  Furthermore,  configuration, measured  the  in size  layer  correction-free  wall  moments  agreement w i t h measurements  airfoils,  wall  the  (chord  with  the  showed good a g r e e m e n t w i t h  in solid  shear  and p i t c h i n g  which r a n g e d  on two  of  f o r viscous  chamber.  t o 0.67, showed good  values.  t o account  and t h e e f f e c t  taken  of the l i f t ,  extended  configurations  for  to height  nine  ratio)  established of  the  free  pressure  correction-free  wall  pressure d i s t r i b u t i o n s  and c o r r e c t e d  by s t a n d a r d  methods.  The l o w - c o r r e c t i o n tested  and v e r i f i e d  developed  to  test configuration  i n t h e work r e p o r t e d  provide  existing  a  airfoils  in  achieve  the low-correction  otherwise  require  correction  procedures.  reliable  windtunnels wall  elaborate  can  o f t e s t i n g high can  configuration. test  which has been  i n this thesis  means which  theory  be  modified  Such t e s t s  facilities  or  be lift to  would  complex  Appendix The  I n t e g r a t i o n o f a Three Dimensional  The density  Flat  potential  Dimensional  Distributed  P o i n t Source  Source  t o a Two  Element  cj> a t a p o i n t P due t o a s o u r c e s t r e n g t h  distribution  a(Q) o v e r a s u r f a c e S i s , f r o m  q(Q) dS ,r (PQ)  <MP)  where r ( P Q ) i s t h e d i s t a n c e f r o m  From F i g u r e A l . l , strength density  1.  ,  the point  the p o t e n t i a l  (3.2),  (Al.l)  P to the point  a t P(x,y,0),  Q.  f o r unit  a, i s As d?  <}>(x,y,0) =  The elements a line  inner integration d£d£; t o p r o d u c e  source element  over a l l such P  As 2  line  due t o a f l a t  2  2  sums o v e r a l l i n f i n i t e s i m a l  that part  of width  of the p o t e n t i a l  source  t o produce  source element  sums  the p o t e n t i a l a t  of width  As, and o f  constant uniform strength density.  The tively,  velocity  components i n t h e x and y d i r e c t i o n s  i n d u c e d a t P by a s o u r c e e l e m e n t  of width  .  a t P due t o  d £ . The o u t e r i n t e g r a t i o n  source elements  distributed  (A1.2)  _„ /(x-a +y +c 2  respec-  As, a r e :  80  As 34) V = x 3x  +  3(j)  V  As 2 As 2 As 2  +  3y  (x-C)I(x,?/y)  2  yl(x,5,y)  d£,  ( A 1 . 3 )  (A1.4)  d£ ,  where I(x,£,y) =  (A1..5) !«»• ( ( x - ? ) + y + ? ) / 2  2  2  3  2  Now  _3  (A1.6) ((x-o +y )/((x-?) +y +c ) 2  2  2  2  2  ((x-£) +y +s )^  .  j  2  2  2  therefore  I(x,£,y)  (A1.7)  = - ~  ((x-0 +y )/((x-£) +y +C ) 2  2  2  2  2  (  x  -  ?  ) 2  +  y  2  Also  ^(-log((x-?) +y )) 2  9  2  5  35  =  " , (x-?) +y +  2  (  X  S  )  2  (tan"  1  i  C-x ^ 1 ) y J  (A1.8)  2  (A1.9)  = (5-x) +y 2  2  Therefore As V  •log( ( x - 5 ) + y ) 2  +log  2  As ' 2  [ ( x - ^ ) 2 +y 2 J  (Al.10)  81  As V  Y  =  r ,As  5-x  2tan  t y  2 (tan  As  -1  X+-—  l y  J  - tan  -1 x —  As  y  J  ).  (Al.ll)  '2 I n a two d i m e n s i o n a l the  stream  incompressible, i r r o t a t i o n a l  f u n c t i o n ty a n d t h e p o t e n t i a l ty a r e c o n j u g a t e  functions.  Thus t h e C a u c h y - R i e m a n n  dty  3y  _  3<j>  Hi  3x'  3x  flow, harmonic  equations require  9(j) 3y  (A1.12)  Hence ty may be e x p r e s s e d a s  ijj(x,y) -  (|i)dy  o  +  where f a n d g a r e a r b i t r a r y  f (x)  - j ( | | ) d x + g(y) ,  (A1.13)  f u n c t i o n s a n d ty i s an a r b i t r a r y 0  constant.  Some u s e f u l  log(y +a ) 2  tan  tan  2  -1  identities are:  = -^(ylog (y +a ) 2  (xtan 3x'  1  —  2  - 2y + 2 a t a n  - ^-log ( x + a ) ) ,  1 rA±B ''"A ± t a n B = t a n ( j ^ g ) •  Therefore,  X  from  x  Al.10,  2  2  1  ),  (A1.14)  (A1.15)  (A1.16)  H  sff"YA + xB  =  +  c  )<  ( A 1  .-17>  where  A = log  (x+%. As  { ( x ~ )  2  + y ^ 2  +  2  B =  2tan  -1  yAs [x  y J 2  2  y -(^) J 2  +  2  (A1.18) C =  2xy  Astan l x  According to equations function  for a flat  and  strength  unit  2  " Y  - ( f )  2  J  (Al.13) and  distributed  density  2  (A1.17),  source element  the  stream  of width  is  (A1.19)  iMx y) = tyo + A - B + C. f  As i n e q u a t i o n  As  (Al.13),  the p o t e n t i a l  <j> may  be  expressed  as  (J)(x,y)  where  - <J)o = -  (|i)dy  f and g a r e a r b i t r a r y  +  f(x) =  functions,  (||)dx.+  g(y),  a n d (J> i s an 0  (A1.20)  arbitrary  constant.  The a p p l i c a t i o n and  (A1.16) t o  of the three r e l a t i o n s  (Al.19) r e s u l t s i n  (Al.14),  (A1.15),  83  (A1.21)  D = flog((x^)V)[(x-^)V!^  Therefore source  the  element of  potential function width  As  and  unit  (J)(x,y) = c f > o + x A  The flat  corresponding  vortex  element can  two-dimensional  4> ( v o r t e x )  results for a be  written  incompressible  =  +  -^(source),  for a  flat distributed  strength  density  yB  2As  +  D-  unit  irrotational  i> ( v o r t e x )  .  strength  immediately  =  i s  (A1.22)  density  since  in  a  flow  +cj) ( s o u r c e ) .  (A1.23)  84  A p p e n d i x 2._ h  Procedure  f o r Block  Commutation  When t h e m a t r i c e s assembled  from  A.,  B  and  C  are  A and B by p a r t i t i o n i n g  Memory  i s allocated  largest  block.  Aj_ 8 and C__  of Matrices  large,  C into  C  blocks  f o r A, B and C a c c o r d i n g  might  be  as f o l l o w s .  t o t h e s i z e "of t h e  * j = l , 2 , . . .N* j=N+l,N+2,-. .N+M* r  test  R(k)  A. .  i=l,2,..NL4  B  - y  r  L  Di  ,  m=l {  .  airfoil,  flaps  *  mi .  and  i  solid  walls  -j  i  R(k)  i=NSUl,NSU2  •  A . .  ;  -  c  ] l  y  L  -,  B  m=l  .  *  mi  f  slats  1  R(k)  ' 'B .  +B.  T  r  r  (A  >• y L  ,  m=l  TT  mil  +A _ r  mL  r-th  r  slat  R(k)  i=NKA,M  ::  B.  nU  •  The  +B . r  jL  'I  r  (A  ^,  m=l  subscript ' i ' refers  mU  +A r  )^  mL .  r  r-th  airfoil  or flap  t o t h e row number i n t h e m a t r i x  C.  The  m e a n i n g s f o r t h e v a r i a b l e s NL4, NSD1, NSB2, NK1, NK2 a n d NKA  are  given  i n t h e programs  Subroutine assemble zero  the  which f o l l o w .  SUB1 c a l c u l a t e s blocks  of  A and B i n b l o c k s a s  C. SUB 1 i s c a l l e d  n o r m a l - v e l o c i t y boundary  condition eguations  surfaces.  It i s called  all  w a l l s e c t i o n s and a second  solid  Subroutine  first  twice  f o r the t e s t time  airfoil  needed  to  t o s e t up t h e on  a l l solid  and f l a p s , a n d  f o r the wall  slats.  SUB2 u s e s A and B t o s e t up t h e K u t t a c o n d i t i o n  85  equations the  test  The The  on t h e w a l l airfoil  FORTRAN  subroutines  (onto)  a  and  slats.  Subroutine  SUB3 d o e s t h e . saiae  for  flaps.  coded  READER  v e r s i o n s o f S U B 1 , SUB2 a n d S U B 3  ( a n d WRITER) r e a d  peripheral storage  device, such  (write)  a  follow.  matrix  as a magnetic  disc.  from  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  C C  C  c c c c c c c c c c c c c c  c  C A L C U L A T E S A £ B IN B L O C K S 6 SETS UP -EON'S ZERO NORM VEL ON A L L S O L I D SURFACES SUBROUTINE S U B 1 ( A , B , C , X X , Y Y , D S , C S , S I , N , L , M f N A , N A F 2 , L 1 , L 2 , N S P S , 1 NSLATtNS-Ul t N K l ) A,B - M A T R I C E S OF I N F L U E N C E C C E F F S FOR SOURCE £ VORTEX ELEMENTS C- M A T R I X FOR S Y S T E M OF E Q N S C*SIG=D S I G ( M ) - UNKNOWNS I N S Y S T E M G*SIG=D - PART IS GAM G A M ( N A F t - N S L A T ) - VORTEX STRENGTHS ON KUTTA BODY. M - T O T A L U UNKNCWS= T O T A L M EQUATIONS NAF = #TE.ST A I R F O I L S I F L A P S WITH KUTTA C O N D I T I O N S A P P L I E D NSU1,NSU2 - SET OF EQN'S FOR ZERO NORMAL V E L O C I T Y ON S L A T S . NK1,NK2 - SET OF EQN'S FOR KUTTA C O N D I T I O N S ON WALL S L A T S . X X , Y Y - CONTROL POINT C O O R D I N A T E S ; DS - ELEMENT LENGTH N S L A T - •# OF S L A T S WITH KUTTA C O N D I T I O N S ; NSPS - U E L E M ' S PER SLAT CS S I - S I N , C C S CF ELEMENT I N C L I N A T I O N N A ( K J - RANGE OF CONTROL POINT #'S FOR K-TH TEST A I R F O I L OR F L A P I . E . 1,50 N - TOTAL # CONTROL PTS WHERE NORM V E L I S ZERO ON A L L S O L I D S U R F A C E S L - BLOCK S I Z E L 1 , L 2 - RANGE OF CONTROL PT H'S FOR BLOCK - SET OF EQN #S ALSO REAL A ( N , L ) , B ( N , L ) , C ( M , L ) , X X ( N ) , Y Y ( N ) , D S ( N ) , C S { N ) , S I ( N ) INTEGER NA(NAF2\ NSU2=NSU1+NSPS*NSLAT-1 NK2=NSU2=NSLAT NAF=NAF2/2 C A L C U L A T E A AND B. DO .1 I = L 1 , L 2 K=I-L1+1 DO 1 J = 1 , N I F { J . N E . I ) GO TO 2 AIJtK)=6.2831927 SUB1  1  eu,K  2  j=o.  GO TO 1 DXJ = X X ( I } - X X { J ) DYJ= Y Y { I ) - Y Y ( J ) XJ=DXJ*CS(J)+DYJ*SI(J) YJ=DYJ*CStJJ-DXJ*SI(J) DSJ2=0S( J ) / 2 . Y JS=YJ*YJ S=XJ-rDSJ2 T=XJ-DSJ2  87  OO LU  o <  00  U-  a. <  r>  u.  1  00 IQ  ft  CM  a oo  00  oj  <  OO  00  o  I  cc  joo  00  I |>3  LU  4-  o  >- — +  -)  •H- - J  OO 00  '  t—t  CM •— 00 Z 00  < u  Z  oo o  <——  _J  CM >-« 00 II II OO <_} X >- I) II  Z  cC CC LU LU  X 00 a. _t oi il X ai 3: ?.  — CC <  CO C£ L J  3  00 2  CJ LU  ro  LU  —<—>—>  zz -> a  cc  a  < CO  —  0£ LU M  a H s: o a e> 2:  a  <J  II -3  • -  —I  < Of  II 00 ^  f-.Z —I LU  lA  ZZ>  a  LU OO  r-  o LU 00  Q  00  LU M  •«•  a  id  00  a.  co  -  s  QT-  O  vf lA on  CM COivf LA 43 ir\ L o l i n i n i r \  2: *  .-t I  00 00 Z Q. + O0 Z ZD > OO  z  LJ  or  ii ii II  w  »—4  LU  co  » Q . CO 00 - 3 II  ^11 _J11Oco a^ .co —0 13— 0a •XL  M  II  00  o  00  LU OO  co 0*  r-i  LO lT\  s0  CM  O  O  i i U-  <t  z  II II —1 CM  ii  ii  LU  r-  to  II CO 00  r-  O  z Qi  _ l ZZ)  Z HI J h w a y— < L U o o >-i o CX oi -3  o a  3  r-  ro <r -0  O-  00 LU OO >—1 II ZD >- a i — Z X ^  < < z z  s:  <t Qi  ii <r> co + ii  —  rO CX >»•  CM ii  z  00 •• CO 1-1  CO —  a  z  a. i i < I  cfl Z l| —- I! O Z I  »  a.  m u  CM - J lT\ O I - CO >i- <j- <r -4- -4-J-  a  a a:  I  O O - I • LL OO Of 2 - 3 LU 00 ••  a . <J-  O  <  ex a z  cc  »—*  LL.  2:  o z  l_) Q I + 00  •»•  o o  -5  00 —I LU >  CC  LU >  OO * >-  oo —• oo +  CD  oo  CO  —>  a  on  cr« c o  r-  co  o  co  CM r— r«-  r > f>- f—  CO  79 80 X f 82 83 84 85 86 ~87 88 89 ~9Q~~-  91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 10 7_ 108 109 110 111 112 113 114 115 116 117 118  SUB2 S E T S UP KUTTA EQN'S DN WALL S L A T S . ' SUBROUTINE S U B 2 ( A , B , C , N A , N , L , M N A F 2 , N S L A T , N S P S t N S U 1 , N K 1 ) ~~~ REAL A ( N , L ) , B ( f l 7 T T 7 C ( K T N S n m INTEGER P » Qt N A ( N A F 2 I C A» B - MATRICES -OF I N F L U E N C E C O E F F S FOR SOURCE £ VORTEX E L E M E N T S C C- M A T R I X FOR S Y S T E M OF EQN'S C*SIG=D C M - TOTAL ti UNKN0WS= TOTAL # EQUATIONS C NAF=#TEST A I R F O I L S £ F L A P S WITH KUTTA C O N D I T I O N S A P P L I E D C N S U 1 , N S U 2 - SET OF EQN'S FOR ZERO NORMAL V E L O C I T Y ON S L A T S . C N K 1 , N K 2 — SET OF EQN'S FOR KUTTA C O N D I T I O N S ON WALL S L A T S . C N S L A T - # OF S L A T S M Jj-_ J_UJ_T A_ C O N D I T I O N S ; N S P S - # E L E M ' S PER SLAT C l ^ T i n n R A N G E ~ O F CONTROL P O I N T # » S FOR K—TH T E S T AIRFOIL OR F L A P I . E . 1,50 C . N TOTAL # CONTROL PTS WHERE NORM V E L I S ZERO ON A L L S O L I D S U R F A C E S C L = N S P S * N S L A T - BLOCK S I Z E ; : NAF=NAF2/2 NSU2=NSU1+NSPS*NSLAT-1 NK1=NSU2+1 NK2=NSU2-fNSLAT C READ IN THAT PART OF A £B WHICH C O N T A I N S THE I N F L U E N C E C O E F F S FOR V E L O C I T I E S C INDUCED AT THE CONTROL P O I N T S ON THE WALL S L A T S . CALL R E A D E R ( A , N , L ) C A L L READER I B , N , L ) SET UP EQNS F_Q_R KUTTA CN WALL SLA_TS_ ' "_TQ i i=i, N SLA T P=1+NSPS*(1-1) Q=P+NSPS-1 C P , Q - T . E . CONTROL PT #S FOR S L A T S C 2 LOOP - T A N G ' L V E L S DUE TO A L L SOURCE E L E M ' S . ' DO 2 J=L ,N • C(J, I )= B(J,P)+BtJ,Q) 2 4 LOOP - TANG'L V E L S DUE TO VORTEX ELEM'S ON S L A T S C DO 4 K S = 1 , N S L A T . . J=NK1+KS-1 KK=NSU1+NSPS*(KS-l) KL=KK+NSPS-i SA = 0. DO 3 K=K'K , KL SA=SA+A(K,P)+A(K,Q) 4 C(J,I)=SA C 6 LOOP - TANG'L V E L S DUE TO VORTEX ELEM'S ON A I R F O I L S £ F L A P S . t  _  — :  :  -coco  119_ _10_ 120 121 122 123 124 125 126 5 12 7 6 128 1 129 C 130 131 . 132 133 C 134 135 136 1.3 7 C 138 C 139 C 140 C 141 C 142 C 143 C 144 C 145 G 146 C 147 -C 148 !_ _ 150'" C 151 C 152 153 154 C 15_5 '_„__.___ '156"" " 157 15 8 9  DO 6 KN- 1, NAF . " K3=2*(NAF+1-KN)-1 K1=NAIK33 K 2 = N M K 3 + 1) ; __ J-=N-NAF + KN SA=0. DO 5 K = K 1 , K 2 . . SA = SA + A ( K , P ) + A { K , Q ) C U » I ) = SA CONTINUE WRITE T H I S PORTION OF C INTO A F I L E . CALL WRITER(C,M,NSLAT) RE TURN ; ' END S U B 3 S E T S UP KUTTA EQN'S ON TEST A I R F O I L S & F L A P S . SUBROUTINE S U B 3 ( A , B , C , N A , N T E , N , M , N L 4 , N A F , N A F 2 , N S P S t N S L A T , N S U 1 , N K 1 ) REAL A ( N , N L 4 ) , B ( N ,NL4) ,C(M,NAF) INTEGER NA{NAF2>,NTECNAF» A B - M A T R I C E S OF IN F L U EN CE COEF'FS FOR SOURCE & VORTEX E L E M E N T S C- M A T R I X FOR S Y S T E M OF EQN'S C*S.IG = D S I G ( M ) - UNKNOWNS I N SYSTEM C*SIG=D - PART I S GAM GAM ( NA F + NS LAT ) - VORTEX STRENGTHS ON KUTTA BODY. N L 4 - M OF CONTROL P O I N T S ON TEST A I R F O I L , F L A P S L S O L I D WALL S E C T I O N S M - TOTAL # UNKNOWS= TOTAL EQUATIONS NA F-ffTEST A I R F O I L S £ F L A P S WITH KUTTA C O N D I T I O N S A P P L I E D NSLAT - # OF S L A T S WITH KUTTA C O N D I T I O N S ; NSPS - # E L E M ' S PER SLAT N A { K ) - RANGE OF CONTROL POINT #'S FOR K-TH TEST A I R F O I L OR F L A P I . E . N - TOTAL H CONTROL PTS WHERE NORM V E L I S ZERO ON A L L S O L I D S U R F A C E S NT E{K I - CONTROL PT # FOR UPPER T . E . ON K-TH TEST A I R F O I L OR F L A P hSU2=NSUl+NSPS*NSLAT-l M<l = NSU2+._ . ' ; . READ IN THAT PART" OF A &B WHICH C O N T A I N S THE I N F L U E N C E COEFFS FOR THE CONTROL P O I N T S ON THE TEST A I R F O I L S I F L A P S . C A L L READER ( A ,N , N L 4 ) : CALL R E A D E R { B , N , N L 4 J SET UP EGN'S FOR KUTTA GN A I R F O I L S AND F L A P S JJOJL J<|_f i _ N A F ; '_ _ _ ' I=KN" " .' " • MT=NTE(NAF+l-KN) MTE = MT+1 . • :  ;  t  __  ____ _  1,50  o CTi  1 5  g  160 16 i 162 163 164 iu• 165 166 167 Jt, 168 169 L70 i w 171 172 173 174 175 176 177 178 179 180 181 18 2 183 184 185 186 18 7 W  c C 2 c  I  1  3 4 c  5  MT t MT E ARE #S FDR T . E . CONTROL P T S . ON TEST A I R F O I L S t F L A P S . 2 LCGP - T A N G ' L V E L S DUE TO A L L SOURCE E L E M ' S . „ . DO 2 J = 1 , N C ( J f n = B(J,Mf)+B(J»MTE) I F { N S L A T . E G . 0 ) GO TO 5 4 LOOP - T A N G ' L V E L S DUE TO VORTEX ELEM'S ON WALL S L A T S . DG 4 K S = 1 , N S L A T J=NK1+KS-1 KK=NSU1-NSPS*IKS-1) KL=KK+NSPS-1 SA = /. DO 3 K=KK,KL SA=SA+A{K,MT)+A(K,MTE) C(J,U=SA 9 LOOP - T A N G ' L V E L S DUE TO VORTEX E L E M ' S ON T E S T A I R F O I L S & F L A P S . DO 9 KM=1 »NAF K3=iNAF-rl-KM)-l K1=NA(K3) K2=NA{K3+1) J=M-NAF-rKM SA^O. DO 8 K = K 1 K 2 SA=SA^A(K,MT)-AIK,MTE) C(J» I ) = SA CONTINUE WRITE T H I S L A S T PART OF C INTO A F I L E . C A L L WRIT ER{C,M» N A F )  ,—,  .  t  ;  8 9 1 C  _  Hfcll END.  O  9i  A£_endix Two  Methods  liaebjcaic ___  Gaussian  savings memory  Solvin_  PORTRAN  operating  dynamic  coded  of  system  storage  notation,  subroutine  a large  Simultaneous  which  Ci20i+C  C i a i +c 3  printout  e m p l o y s some  the system + c  2  provides  Lil^eaE  forsignificant  number o f e q u a t i o n s  allocation.  c i io-.  The  Systems  Elimination.  when s o l v i n g  subscript  the  Ej_uationj__  This  for  of  3_  \ 0  2  To  3  2 3  2  3 3  system reversed  i.a =d . 3  1  20"2 C32a3 = d  a +c  the  virtual  i s written:  ( A 3 . 1)  +  2  a  of paging  illustrate  of equations +c  kind  under  a =d 3  of the subroutine  3  3  ATXB  follows.  92  3__2 S u c c e s s i v e The right  Row  Vector  matrix  C  side  vector  hand  Qrtho_onali_  Process.  f o r t h e s y s t e m C(a,y)=d i s augmented »d' t o f o r m t h e  by t h e  eguations  C i \ 0 i + C 2 lO" +C 3 iO 3 ~ d l t = 0 2  c i 20" i + C 2 2 0 " + c cr - d 2 t = 0 2  3 2  (A3.2)  3  C 1 3CT1+C2 a + C 3 3 ^ 3 - d t = 0 3  A set of N vectors (  The  C  solution  C  C  vector  3  i n (N+1)-dimensional space i s formed:  l i ' 2 i ' 3 i ' * * * N i  C  2  of  ' ~  d  i  i=1,2,...N  )  . equations  (A3.3)  (A3.3)  i s  such  that  the  vector {o  is  orthogonal  solving  to  equations  l  f  a ,a ,..,o^ 2  3  a l l the (A3.2)  vector  having  a s i t s (N+1)st  Let stage is  of  orthogonal  to  (A3.3).  the N vectors  A l l (N+2-j)  t o be o r t h o g o n a l  that i s , a  are orthogonal  v  vectors __A  that  i l . N+2-j  a l l of  (the last  r  f  h  e  first  i n  a l l (N + 2 - j ) the  (N+1)-  (A3.3) a n d  matrix.  f o r  ±= 1, 2, . . . N + 2- j  this  j - t h setare  vectors (j-1)  U^,..  in'this j-th rows  which i s orthogonal 0^, 0  of  the  .U  member o f t h e ( p r e s e n t )  (j-1) v e c t o r s  At each  row o f t h e augmented  first  is", the vectors  V"? ) , i s a v e c t o r o  vectors  to the (j-1)th  -. I n f a c t ,  to  matrix,  Thus  1  of  an  of  component.  #  constructed.  The p r o c e s s  to determining  3  augmented  o  vectors  j , ( j = 1 2, . . . N+ 1) , a s e t o f v e c t o r s  matrix,  t  ( A 3 . 4)  U. d e n o t e t h e j - t h r o w o f t h e a u g m e n t e d  constructed  set  1)  i s equivalent  dimensional unity  t  (by  ... U  j - t h set of construction)  ^, a n d t h u s i s  93  orthogonal  t o a l l of t h e  f i r s t  rows  o f  t h e  augmented  matrix.  Ultimately orthogonal solution  there  i s  t o a l l M rows t o  a  single  o f t h eaugmented  t h e system  o f  N+1  vector matrix.  equations,  which  This  i s  i s t h e  a s t h e (N+1)th  (last)  N+1 component l a s t  w i l l  component To  stage, set  of  o f V^  actually scalar  of  is  t h e  c^  stage  s e to f v e c t o r s  must  1  Thus c ^  c~! i  1  i "  c  l  r  v  1  i s defined i *  j - l  U  i scalculated c  initiate  'j '  t h e  be c a l c u l a t e d  a t t h ej - t h such  that  +  t h e  i ~  1  t h e s e to f " u n i t "  th  component.  procedure  i + i  v  (A3.5)  row o f t h e augmented by t h e s c a l a r  matrix,  that  product (A3.6)  "  = 0  from =  -  (  V  i i i *  the process,  be  This  i =  t o t h e (j-1)th  V  system  a t each  vectors  OJ_-L"  To  In fact,  i s unity.  multipliers  orthogonal  Hence  _j  2  construct  v  is  +  be u n i t y .  vectors  i s best  u  j - i  )  /  (  V  i ~  1  *  D  j - i  )  the s e t of vectors  (  V  1  ( 0 , 0 ,. . . 1. . . 0, 0) , w h e r e  illustrated  a  3  -  7  i s chosen 1 i st h e  by an e x a m p l e .  )  to i  For the  of equations x+y-z=2 2x+y+z=1 x + y-t- 2 z = -  The  augmented  matrix i s  (A3.8)  1  -  94  1  1-1-2  2  1  1  and  0=  (1,1,-1.-2)  For  1-1  1  2  1  U =(2,1,1,-1)  r  j = 1 , N + 2 - j i s 4,  vj-=(1, 0 , 0 , 0 ) , 1 For  c  and  0.= ( 1 , 1 , 2 , 1 ) .  so t h e r e ' a r e  4 "unit"  vectors  (A3-. 10)  v-.,  vj=(0,1,0,0) , ^ = ( 0 , 0 , 1 , 0 ) , V ^ ( 0 , 0 , 0 , 1 ) . z j 4  j=2,  multipliers  ( A 3 . 9)  N+2-j  i s  . Using  i  3,  so  there  the reversed  •V-+*U =a 1 1 1 1  are 3 vectors  so  (A3.11)  V  2  and  3  subscript notation,  and  *U  r ±  i+1  1  =a  1+1,1  ( A 3 . 12)  therefore i and  ^=-1, c^=1, 1 ' 2  ( A 3 . 14)  c =2. 3 1  ( A 3 . 15)  V =c V + V i i 1 i+1  Therefore so  ( A 3 . 13)  11  I+I,1  2  1  1  1  V - ( - 1 , 1 , 0 , 0 ) , ¥^=(1,0,1,0)  and  2  ( A 3 . 16)  ^=(2,0,0,1).  2  All  three  V  are seen  t o be  orthogonal  to 0  i For  j=3, N+2-j  multipliers  c..  •  i s  there  are  2  vectors  and  2  c_=3  ^ 2 2 Vf = c f v f + 1 X 1  2 VT. .  ( A 3 . 1 9)  V = (-1, 3, 0 ,1) .  (A3. 20)  V = (-2,3,1,0)  Both  V  3  are seen  and  t o be  =  1  n  d  C  i '  V  i  +  l *  (A3.17)  a  so  3  so  l*°2 2 and c =3.  Therefore  i  2,  Here  I  V  therefore  . 1  D  2  ( A 3 . 18) x+1  3  orthogonal  to U  and II , 1 2 3  For vector  V  j=4, .  N + 2 - j i s 1, s o  there  i s a single  multxplxer  c^  and  Here V * 0 =3, 1 3 3  V * U =3, 2 3 3  so  c =1. 1 3  ( A 3 . 2 1)  95  Therefore  V^- (T , 0,-1,1)  (A3.8) a s .x=1, The  and  FORTRAN p r o g r a m s  peripheral program  y=0,  storage  is  the s o l u t i o n  vector f o r the system  z=-1. f o r r e a d i n g and w r i t i n g  matrices  devices are h i g h l y system-dependent,  f o r t h i s v e c t o r method i s n o t g i v e n h e r e .  onto so t h e  96  AE_.§____  __  A __________ ________ _________ This specified The  algorithm  is  used  starting point,  given  method  uses the  direction,  From  subroutine  at each  a  and  (xi+Ax,yi+Axtan8!). flow  point  is  to  The  flow  changed  ( x i + A x , y i + A x t a n 0 ) . The and  so  streamlines  increment i n the  THETA  to  from  the  v e l o c i t y m a g n i t u d e , and  pressure  locate  so  flow  the  direction 9  averaged  give  the  direction 9  next  next is  2  8,  and point  there. the  y-  is  now  calculated  o u t l i n e of  the  FORTRAN  coded s u b r o u t i n e  THETA  z' _  at _  w cc  o _|!  z  z X z  0 cc  _  fe:  _!  Ir-t  •  *H  il  _ •  1  io. _ . CO £_ _ z  ^ jr—t *"CO  z  to X 00  z  •«* IO _ >v  II  •. ""^ t i l  _>  i  t_ II to->-+ a: II  II  __  NV  O l» CD  r-l  Hjj-T H  <r«i i r -  'TJ''"-* — — 03| _  -l> ;  -3  _•' < Ml _S Z . II If f-f il II CDl Z . t_ H  _;••<-«  !-« + w CO ^ _5 -rr .<£ »—> _:  0  BR  _3  0 0  _ _ _  CD -*r 11  ZD  • .XI z  *—1 zz —* «* 0 •• * —*>•*_ • &'  <  or. <:  _  -—'.:  0  CS to  r-l. _ ZD  . •«-*•  ir-  *—V _1  nj  0 0  •-CD  CO X-  there,  follows.  fr*:  >  is  point  i s calculated  2  to  that  d i r e c t i o n 81  flow  •s-  Z  flow  on.  An  >< -  a  x-direction.  calculate  (x_,y_), the  used  d i r e c t i o n s are  coordinate  track  point.  starting  calculated,  two  an  v e l o c i t y components,  coefficient  The  to  £_ I!  CO 2  II  l  H  ft':.  _ . «"»  M  • 'ir* •»!  .;—•  ^ - <  >-* a: •r O £_ o O 3 : t_ :  :  Q ' 1_ Z X QL. Ui . "  -^^tal E —  •04-* •  ;  X X •  Z  — • _ - * ] y-i —1 i «-*• rr —* w fc— cc zz : 11O 11_ l <C|<; O H * - 1^. s 2: i — -t> Z5  il w V o T X .XI o  •" ;:  188 189 190 19 1 19 2 193 19 4 195 196 197 198 199 200 201 202 203 204 20 5 206 20 7 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227  C C  c c c c  1  c c c c  S U B R O U T I N E THETA {X,Y,THE,CP,XX,YY,DS,CS,S1,N,VNT,VTT,VM,A,B,SIG, 1 GAM,NA,GAMM,M) REAL X X ( N ) , Y Y I N ) , D S I N J * C S ( N ) , S M N ) , A < N) , B (N) , S IG (N i , GAM { N) I N T E G E R P, Q COMMONOAO N A , N L 4 , N S L A T , N S P S I X , Y ) - C A L C U L A T I N G FLOW PARAM * S H E R E . XX,YY - CONTROL POINT COORDS FOR A L L SOURCE £ VORTEX ELEMS: DS IS E L E M S I , C S - SIN,COS ELEM I N C L I N A T I O N N - TOTAL 3 CONTROL P O I N T S A,BVORTEX - INFL EU LE EN M C' ES .C O E F F S FOR V E L O C I T I E S INDUCED 2 <X,Y) DO 1 J= 1 i N DXJ = X - X X J J ) DYJ = Y - Y Y T J ) XJ = D X J * C S I J ) + D Y J 4 S U J ) YJ=DYJ*CSIJ)-DXJ*SIIJ) DS J 2 = D S ( J ) /2 . YJS=YJ*YJ S=XJ+DSJ2 T= XJ-DSJ2 PHIX-=ALOG( ( S * S + Y J S ) / ( T * T + Y J S J ) PHI Y = 2 . * A T A N 2 ( ( D S 1 J ) * Y J ) , ( X J * X J + Y J S - D S J 2 * D S J 2 ) ) A ( J J =. P F I Y * C S < J ) + P H I X * S I ( J ) BUI = PHIX*CS(J) - P H I Y * S I U ) S I G - SOURCE £ VORTEX STRENGTHS GAM - VORTEX STRENGTHS FOR TEST A I R F O I L £ S L A T S GA MM - TEST A I R F O I L VORTEX STRENGTH V N SVNST T i V T S =T 0 - TOTAL VTST = 0 DO 2 J= I ,N VNST = VNST +  2  C  C  NORMAL £ TANG'L  VELS 3  t X , Y ) DUE  DUE  TO A L L SOURCE  TO SOURCE  ELEMS.  MJJ*SIG1J)  VTST = V T S T t e U ) * S I G ( J ) AS=0. BS = 0. N S A L T , N S P S - # S L A T S , # SOURCE ELEMS PER S L A T I F ( N S L A T . E Q . O ) GO TO 3 CO 4 J = l ,NSLAT NL4 - # OF CONTROL P O I N T S ON TEST A I R F O I L , F L A P S 6 S O L I D WALL P=NL4+NSPS*J  SECTIONS  LENGT  £  CO  228 229 230 231 232 233 234 235 23 6 23 7 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263  264  C=P-NSPS+1 P,Q - 1ST £ L A S T CONTROL PTS ON A S L A T . A P , 8 P - NORM £ TANG V E L S 3 ( X , Y ) DUE TO VORTEX ELEMS ON S L A T S AP = 0. BP = 0. • DO 5 M=P,Q A P=A P+A{ M } 5 BP = BP+B-(M1 AP=AP*GAM-l J ) BP = BP*GAV.{ J ) AS=ASfAP 4 BS=BS+BP C . A T i B T - NORM £ TANG V E L S 3 (X,Y.) DUE TO VORTEX ELEMS ON TEST A I R F O I L . AT=0. B7-0. NA - # CONTROL P T S ON TEST A I R F O I L . C ' DO 6. J - l i N A A T = A T +A f J ) 6 BT=BT+B1J) AT=AT*GAMM 8T=BT*GAMM V N V T r V T V T - TOTAL NORM £ TANG V E L 2. ( X , Y ) DUE TO VORTEX E L E M S . C VNVT=-BS-BT VTVT=AS+AT VNOT.VTVT - NORM £ TANG UNIFORM ONSET FLOW V E L S . C VNOT=0. VTOT=U C V N T i V T T - TOTAL NORM £ TANG V E L 3 ( X , Y ) . VNT=VNOT+VNVT+VNST VTT=VTOT+VTVT+VTST vS=vTT*VTT+vN.T*VNT C THE,CP,VM - FLOW DI R E C T I O N , P R E S S U R E , M A G ( V E L O C ) CP = .1.-VS _ VM= S G R T ( V S ) ~ " ~ THE = A T A N 2 ( V N T , V T T ) RETURN E N C C C  a  99  De_i__  of the Two-Dimensional  The was  design-  based  of the rectangular  on  the  method  makes u s e o f t h e e x a c t  that  i s  created  created  an e l e c t r i c  of  two  a r e s u i t a b l y arranged) i n throat  speed  airflows,  provided  the  along  contracting  the  application  would  swollen  the  by  require boundary  such boundary  increases  slowly.  neglecting  the boundary  Let with  the  (Figure distance is  T be  A5.1),  Normally,  of The  ' b' f r o m  i n  not  the  that  i s  the high  area  the  (when t h e  case  of  pressure  disturbed.  contracting  displacement  layer  the  no a p p r e c i a b l e  displacement  of a ring vortex  vortex  ring  the origin.  The  vortex  ring  real  gradient A  precise  surfaces  be  thickness.  In  i s  e r r o r i s made  i n  thickness.  of r i n g r a d i u s ' a  normal centered  (axisymmetric)  then  to  the  ,  z-axis  on t h e z - a x i s , stream  1  a  function  -  , .  .  Fra  i M r , z ) • = -2pr  where  coils  layers, a r e q u i t e / t h i n and t h e t h i c k n e s s  the strength  plane  layer  field  E s s e n t i a l l y t h e same  favourable  i s  field  a core  known.  occur  that  t h e magnetic  Experimentally,  over  i s well  normally  insert  [ 29 J . The. s o l u t i o n  velocity  field  will  section  c o a x i a l and p a r a l l e l  vorxices.  surface  practice,,  between  and t h e  magnetic  uniformity  a n d Wang  circular  current,  the  contracting  analogy  by two a n a l a g o u s r i n g  uniformity coils  by  Insert.  o f Smith  given  carrying  Nozzle  2TT  0  COS8 _, -  PQ  d0  , - r- ...  ,  ( A 5 - 1)  100  PQ  2  =  =  (z-b)  + a  2  + r  2  - 2ar cos9  2  (A5.2)  ((z-b) +(a+r) ) ( l - k c o s ( f ) } , 2  2  2  2  4ar  (AS:  ((z-b) +(a+r) ) ' 2  3)  2  Then  -4  ar  ( r , z) 4  V ( (z-b) +(a+r) ) 2  k  2  2 j  / (l-k cos a) 2  2  da  0 (A5.4)  2TT  •  (1,-2)  0  da  aF/r  /(l-k cos a) 2  2  r  k  SE/_(<I-_>*-">  where  (A5.5)  a = -  Non-dimensionalizing, l e t  Z = f, cl  R = f, cl -  K  B = c l  _n_  i>  ar '  (A5.6)  and l e t  F(k)  with  =  (1-| )K 2  - E) ,  (A5.7)  1 0 1  ;2  iR  =  (Z-B)  +  2  . (1+R)  ( A 5  _  8 )  2  Tnen  $(R,Z)  = /R F ( k )  where K a n d E a r e t h e c o m p l e t e and  second To  kind,  generate  elliptic  integrals  of  the  first  respectively. the uniform  section,  the  streamlines  identical  ring  vortex  The  (A5.9)  ,  resulting  must  must  stream  u>(R,Z)  =  flow  a t the exit be  of the contracting  parallel  be c e n t e r e d  at  there.  Hence an  (0,0,-b).  function i stherefore  /R ( F ( k J )  F(k ))  +  ,  2  (AS. 1 0 )  where  i 2 k i =•  4R (Z-B)  Reference distribution median  +  2  , (1+R)  , k  2  4R  =  2  (Z+B)  2  2  +  (1+R)  [ 2 9 ] s t a t e s t h a t when b i s 0.46936a over  a  flow  core area  p l a n e between t h e v o r t e x r i n g s  ,, „ (A5.11) c  . 2  the  velocity  o f r a d i u s 0.42241a will  be u n i f o r m  to  i n the within  1 p a r t i n 500.  The polynomial was  elliptic  integrals  approximations  written to search  can  be  evaluated  17.3.34 and 17.3.36  f o r the value  of  R  simply  from  o f [ 4 1 ] . A program  which,  for a  given  102  value  Z  of  gives  (E ,Zo)-  point  t h e same v a l u e o f ty as through  In t h i s way t h e c o o r d i n a t e s  0  s u r f a c e were g e n e r a t e d , R  the stream  u s i n g a v a l u e of ty of 0.10510.  i s chosen t o be 0.30 (see F i g u r e 6 o f [ 4 0 ] ) t o o b t a i n a  0  throat flow uniformity required  r  2  length  within  testsection  0.348m . T h e r e f o r e i s chosen  i s fixed  0  area  ro  i s 0.30a.  i s 0. 34 8m . 2  i s 333mm and 'a'  The  Hence ur } i s  i s 1110mm.  2  The  nozzle  t o be 1.52m, due t o p h y s i c a l r e s t r i c t i o n s i n  at  0.525a. T h e r e f o r e  0.23. Thus  entrance  the e x i s t i n g c o n v e r g i n g Tix^  (R,Z) of  a starting  s e c t i o n , hence t h e n o z z l e e n t r a n c e  1.068m . 2  Hence  area  r , i s 583mm, o r 1 . 7 5 1 r o r 0  S i i s 0.525, z i i s 4.53ro o r 1.37a.  Hence  Zi  i s 1.37. The  t a b l e of n o z z l e c o o r d i n a t e s f o l l o w s . Z , z , R, r a r e a s  above, and w and h a r e t h e e x i s t i n g  width  of  the c o n t r a c t i n g  s e c t i o n and h e i g h t o f t h e new n o z z l e i n s e r t r e s p e c t i v e l y .  z .  0.0 0. 1 0.2 0. 3 0.4 0.5 0.6 0.7 0. 8 0.9 1.0 1.1 1.2 1.3 1.37  z 0 mm 111 222 333 444 555 666 777 888 999 1100 1221 1332 1443 1524  R  .3 00 .300 .300 .301 .304 .309 .318 .331 .343 .370 .395 .4 26 .460 .498 .525  r 333mm 3 33 333 334 337 343 353 367 386 ' 411 440 473  511 55 3 583  A~7T  r  2  =wh  0.34 84m2  .3484 . 3484 . 3507 . 3575 .3706 . 3924 . 4248 . 4700 .5298 .6065 .7023 .8199 .9617 1.068  w  h  9 1 4mm 381mm 915 381 916 381 917 383 920 389 927 400 934 421 942 452 952 • 496 962 554 979 626 1002 7 10 103 6 803 1082 902 1 123 951  103  AH  An.aly_t.ic R e p r e s e n t a t i o n  XEiLHSversexy-Slgtted  The  following  potential between  flow  a solid  boundary  outside  that"divides airfoil which a r e  the  of  apart,  of  wall  is  a  a  constant  A constant  wall represents from  are  vortex of  two-dimensional  model f o r a l i f t i n g  slats.  the  represented image o f  equal  pressure  but  a  Boundary..  airfoil  a transversely-slotted  flow  slats  a Solid  a  free  plenum by  is  flow.  The  vortices  vortex,  opposite  boundary  pressure  streamline  point  a  upper  in  a  circulation; a  vortex  of  circulation.  [41],  point  the  vortices  complex of the  potential  f o r an  same s i g n ,  spaced  infinite a  vertical  distance  'd'  is  the  related  images"  slotted  between  analytic  airfoil-shaped  testsection  F(z)  where  an  and  Vortex  Pressure  " i m a g e d " a p p r o p r i a t e l y . The  image i n  From  of  boundary  the  the  boundary  identical  row  the  ana  describes  lower  consisting  a Lifting  a Constant  "method  boundary  solid  and  of  "central"  t o the  = Klog sinh-r(z-z )  (A6.  0  vortex  circulation  r  is  at  zo,  (positive  and  the  clockwise)  strength K  1)  is  by  (A6.2)  with a  single  reference vortex  to  Figures  between  A6.1,  the  A6.2,  solid  the  and  image system constant  for  pressure  104  boundaries of  Figure  with  z  0  i s t h e sura A6.2,  two  values  sets  circulation,  sets  t h e same  images left  of images.  (a+2b)i.  The  "centered"  spacing,  f o r the s i n g l e  to right)  sets  are of positive  o f a i and  negative have  of four  vortex  has t h e complex  Using  the notation  circulation,  other  two  "centered"  sets  a t - a i a n d - (a + 2b) i .  4 (a + b) .  The  complete  immersed  in a  uniform  are  of  A l l four  system flow  U  of (from  potential  F ( z ) = Uz + K l o g s i n h A ( z - a i ) + K l o g s i n h A ( z - ( a + 2 b ) i ) (A6.3)  - K l o g sinhA(z+ai)  — K l o g s i n h A ( z + (a+2b) i ) .,  where  irS+bT •  A =  The respect  w(z)  complex t o z.  velocity  Hence  w(z)  i s the d e r i v a t i v e  f o r the s i n g l e  = U + KA(cothA(z-ia)  +  (A6  of  *  F(z)  4)  with  vortex,  cothA(z-i(a+2b)) (A6.  - cothA(z+ia)  To airfoil, relation  calculate due  the  force  to the e f f e c t  {[41])  i s used,  on  the  o f t h e two  5)  - c o t h A ( z + i (a+2b) )) ».  vortex  representing  boundaries,  the  the  Blasius  105  D - iL =  where  D  and  L  respectively. only  the To  are  airfoil  forces  this  at i h , the  w (z)  of  the  integral  coefficient  term i n  X and  Y  directions.,  about a contour  using r e s i d u e s , i n the Laurent  1/(z-ih)  1  enclosing  with the.  airfoil  s e r i e s expansion  i s r e q u i r e d . The  -  r—-  . (z-ih) +  -.—-  z-xh  The  +  the  Laurent  of  series  f o r the c o t h f u n c t i o n about i h i s  _„ / -„\ coth(z-xh)  2UK  in  (A6.6)  vortex.  evaluate  expansion  the  (z) dz,  2  I n t e g r a t i o n i s performed  vortex 2  w  3  required coefficient  2K A(cothA(z-i(a+2b)) 2  z-ih -  -—j-=—-  45  ,  ,,r  + ...  (A6.7)  n s  i s  - cothA(z+ia)  - cothA(z+i(a+2b))). . (A6.8)  since  The  residue at  2UK  +  ih i s  2K Ai(+cot(2Ab) 2  + cot(2Aa) + c o t ( 2 A ( a + b ) ) ) ,  c o t h ( i z ) = - i c o t (z) .  (A6.9) (A6. 10)  Therefore  \  •  D - il. =  Substituting  ( ^ ) 2 i r i (Residue (ih) )  .  f o r K i n t e r m s of. r , D i s z e r o ,  (Ab.11)  and  106  L  - L  0  ( l - j j ^ c s c (ka)csc(kb)) ,  (A6.12)  w here  - 2TaTbT '  k  (  4  6  -  1  3  )  ana  L  is  the t u n n e l  This  boundary is  the  by  a  expression single  and a c o n s t a n t  midway  between  (A6.14)  =  T  •c' i s the a i r f o i l  consider  the  vortex upper  reduction  in  lift  between a s o l i d  boundary. I f  the  lower vortex  boundaries,  1  ~ 4UH  '  <  A 6  *  1 5  >  and  °  Now  for  point  t h e two  C  where  ( A 5 . 14)  pressure  Lo  Using  pur  lift.  is  experienced  =  0  the  = T —— , | U c  (A6.16)  2  P  chord,  vortices  Y n  representing  the  airfoil-  107  shaped  wall  notation  of  slats. Figure  From  equations  ( A 8 . 4 ) , (A6.5)  using  . the  A6.1,  CO  w(z)  =  ^J  n  =-oo  k n B n (^ c o t h B n ( z - 2 i h - n r )  + c o t h B n (z-2i(h+£ n ) - n r )  (A6.-18) -  is  the  images  cothB  additional  complex  corresponding to y  vortices  spaced  (z+2ih-nr) - cothB  n  a  velocity  'n*  infinite  horizontal  k  = n  due  ,  n  to the  vertical  distance  B  2TT  (z+2i(h+e  n  = TTOT-I 4 (2h+e  four  rows  ' r ' apart.  n.  )-nr)1 J  sets  of  point  Here  .  r  of  (A6.19)  ) n  and  *r' i s related  Hence r  vortex  vortices  the  to the  complete  representing y  n  slotted  wall  complex the  representing  the  open-area  ratio.  velocity  field  test  wall  slats,  for  the  airfoil, and  the  is  the  uniform flow  U,  .  w(z)  - U + KA(cothA(z-ih) + -  cothA(z-i(3h+26))  cothA(z+ih)  -  cothA(z+i(3h+26))) (A6.20)  CO  +  Y ' k B (cothB ( z - 2 i h - n r ) n n n n=-°° L  v  -  where  6 and  e  are  the  cothB  n  +  cothB  n  (z-2i(h+e  (z+2ih-nr) - cothB  distance  of  r and  y  n  n  ( z + 2 i (h+e  )-nr)  n  respectively  .) - n r ) ) }  from  108  the c o n s t a n t p r e s s u r e boundary. Following coefficient w  2  (z),  2UK  the  of.  previous  1/(z-ih)  the r e q u i r e d  T  L n  the  coefficient  i s , using  k B (cothB -xh-nr) n n n A  =-co  to  calculate  Laurent s e r i e s  + cot(2Ah)  2  2K  procedure  in  + 2K Ai(cot(2A(h+6))  +  '  +  the  expansion f o r  (A6.9),  cot(2A(2h+6)))  + cothB  -x(h+2e  n  n  )-nr) (A6.21)  - cothB  (+3ih-nr) - cothB  n  n  (+i(3h+2e  n  )-nr)K .  wnere  A  Mow  =  T(2h+6T*  ( A 6  *  2 2  >  using  coth(x+iy)  coth(x)csc (y)-icsch (x)cot(y) 2  =  2  ( A  6.23)  coth (x)+cot (y) 2  to calculate  the  r e s i d u e of w  (A6.10) ,  L = pur  "  •—  2TT  (z) a t i h , and  using  (A6.12)  and  "  - £ | ^ - ( c o t (Bh) +  -  and  2  2  cot(8(h+S)))  y y B c s c h ( B n r ) F ( B ,h,e , n , r ) , n n n n n n=-°° 2  L  1  (A6.24)  109  D  pr  2TT  Y  y B coth(B nr)E(B  ^ 'n n " rt y~\ n=-°°  Y\  ,h,_ , n , r ) ,  nn  nn  (A6.25)  where  TT  2(2h+S)'  E(B  n  B  n  ( A 6 . 2 6 )  4(2h+£ ) ' n  ,h,e , n , r ) .= G(B h) + G(B (h+2e )) n n n n (A6.27) + G(3B_h) n  F(B  n  + G(B .(3h+2e_)) , n n y  ,h,e , n , r ) = H(B h) + H (B (h+2 )) n n n n e  (A6.28) H(3B h) - H(B (3h+2 )) n n n e  In  t h e above e x p r e s s i o n s  esc  G(u) =  coth (B 2  (u)  (A6.29)  nr)+cot (u) 2  n  and  H(u) =  cot coth (B 2  When a l l o f t h e y  (u)  (A6.30)  nr)+cot (u) 2  n  are of i d e n t i c a l  the e ( d i s t a n c e o f Y from n n  strength,  and a l l o f  t h e constant p r e s s u r e boundary) a r e  110  equal,  t h e .drag f o r c e  D i s zero,  Numerically ' i t i s , 21  26.3°  incidence,  incidence  r  U,  i s satisfactory t o take  o r more v o r t i c e s y  streamline,  the  c o t h (B_nr)  i s  an  odd  of n .  function  that  since  and  . For the case  average  calculations  for  i n the presence  values a  n greater  than 10,  of a flat  plate a t  of y  £ taken  0.66-Clark-i  o f a 70%OAR.TSOSL  airfoil  wall  from a t . 20°  configuration,  values of the e x p r e s s i o n s a r e  =  .0625,1=0.1,  1.40,  £ = 0.67,  | = 0.67.  (16.31)  H ence  ~  (0.168)(I )  - 1 "  -  ( - 0 . 3 1 7 ) ) u  c  c s  = 1 - 0. 235  and  t h e drag  - 0. 0 2 0 = 0. 7 4 5 ,  force D i s zero.  Hence t h e e f f e c t airfoil boundary  of the wall  i s s m a l l compared on t h e t e s t  Another  slat  circulation y  to the effect  on t h e t e s t  of the constant  pressure  .airfoil.  possible  shear layer, i s a vortex tangential  (A6.32)  analytic sheet,  v e l o c i t y and t o t a l  r e p r e s e n t a t i o n of t h e bounding  across  which  head. T h i s  there  i sa  jump  was n o t a t t e m p t e d  i n  here.  I l l  A p p e n d i x 7. Standard  The  Solid  Wall  Corrections  f o l l o w i n g seven expressions  a r ereproduced  382 o f [ 3 4 ] , a n d a r e t h e e x p r e s s i o n s herein. as  They  The c o r r e c t i o n s a r e t obe  i n c i d e n c e . The s u b s c r i p t s T a n d F  andequivalent free-air  The tions  i nl i f t .  a t t h e measured  measured  used f o rt h e c a l c u l a t i o n s  arewritten f o rthe incidence correction applied  an e q u i v a l e n t change  applied  from page  first  five  t o be added  expressions  values  of windspeed, incidence and l i f t , moment, a n d d r a g  respectively.  are,respectively,  (regardless o f sign)  the correc-  t o t h e measured  quarterchord  values  pitching  coefficients.  AU = U  - U  F  = eU  T  (A7.1)  T  A a = a _ - a _ = 0.  A  A  C  C  L  HC = 4  A C  D  =-  "  C  M = "Sic = 4F 4T  =  C  D  - D C  T  -  = -  2  2  2 E  e  £  C  C  V V K  +  M c 4T  D  + T  !  (A7.2)  2¥ V (  V  ^  C  MC >£?"  <"- > 3  T  W  '^f'  ( A 7  -  4 )  4T  27 L  t h e above e q u a t i o n s ,  4  (  (C  +  4  C  M c  >  (  In  imply  T  C  L ~  If > ' D  < - >" A7  5  T  e i s composed o f t h e c o r r e c t i o n s  112  f o r wake a n d s o l i d  blockage,  1 rc^ e = ffe) D_ C  and i s g i v e n  +  A  K  by  '  (A7.6)  where  K = ^ ( | ) \  and  c/H i s t h e m o d e l  Fig.6.8  of  size.  The q u a n t i t y  (A7.7)  A i s obtained  [39].  In p r a c t i c e , the a - d e r i v a t i v e values graphically.  =  27T,  from  are determined  Otherwise the f o l l o w i n g values  =  0,  _ M ^ = 0 ,  may be u s e d :  (A7.8)  ^ = . 2 .  The f o l l o w i n g c o r r e c t i o n i s a p p l i e d t o t h e m e a s u r e d pressure  distributions  ( a t t h e measured i n c i d e n c e  ct^) :  113  _E___S__  __  _ _______ _______ _ _ _ _ _ i _ _ _ 2 _ __________ ____ In  this  method,  the  usual  Kutta  ___  condition,  v e l o c i t i e s on the  upper and  adjacent  t r a i l i n g edge i s abandoned. The  to  the  lower s u r f a c e s of  t o be determined from the measured To  s i m p l i f y the  between  solid  p o i n t s , and points.  the two  Thus the  therefore unknown  M c o n t r o l points  equal  lifting  body,  circulation  w a l l s by M-N  test  airfoil  additional  density  densities o y on  the  to d e t e r m i n e the M+1  at which the  the  notation  of  control  i s li. There and  airfoil.  a  single Thus  unknowns. There  norma 1 - v e l o c i t y  boundary  are  H+1 are  condition  §3.2,  the t a n g e n t i a l v e l o c i t y a t  a  • i ' is M  N (A8.  The at  is  s a t i s f i e d ; these y i e l d M equations.  In point  of  lift.  t o t a l number of c o n t r o l p o i n t s  strength  Lift.  a i r f o i l i s r e p r e s e n t e d by N c o n t r o l  plane s o l i d  e q u a t i o n s are r e q u i r e d  must be  The  Pi. unknown s o u r c e s t r e n g t h vortex  a  equations, consider a single  walls.  ________  (M+1)st e q u a t i o n u s u a l l y c o n t a i n s the  the . a i r f o i l  expressed  trailing  edge.  The  Kutta  1)  condition  usual Kutta c o n d i t i o n  is  as  =  x  -V  i+ 1  (A8. 2)  114  which  results  i n the equation  M  |  for  N ( B 1  ji  + B  ji+l^ j. .  the control  0  +  Y k  I  ( A 1  ki  + A  ki+l  )  =  - ( U  c  o  s  0  ' i ' and ' i + 1 ' a d j a c e n t  points  i  +  c  o  s  9 i  to  +  1  )  the  (A8.3)  trailing  edge.  The can  resulting  be c a l c u l a t e d  full  from  Kutta  To  determine  Kutta-Joukowsky airfoil  I  V As. i=l \  (A8.h)  1  the circulation  r  from  t h e measured  law e x p r e s s e s t h e measured  l i f t  force,  l i f t ,  the  L, o n t h e  as  The  measured  =  l i f t  C  T  =  PUP  .  'c  1  left.side  U  C  L  i s defined  as  '  T-V" pU c 2  i s the a i r f o i l  T = f  (A8.5)  coefficient  2  The  airfoil  N  C  L  where  ?o a b o u t t h e  the definition  = 4 V-d£ = J  circulation  U8-6)  chord. Therefore  c  C L  -  of equation  (A8.7)  (A8-7)  i s therefore  written  N  i=l  M  t  (A3.  and  C  y  i  i  3  N  a 1  +  Y  3  I A  k=l  K  + UcosG  ^  l  IAs.  (A8.3)  ,  1  become;  a .+ 3  . . A S .  3  i=lAj = l  1  7)  B  I  As. =  N  •  M I B  N  I V  1  x  r l  N  r  N  I  ^  2  i=l  As .  l  k i  A  J  (A3.9) N  = -U  J  cos6.As.  i=l  The  last  C ( o, y)-d,  row  will  of  the matrix  1  ^  +  1  C i n t h e system  of  C  C  L  equations  now be  N  'j ,M+1 =  I B i=l  N  'M+1, M+1  =  y  the last  As  ,  i  As. l  A  L  component  (A8.10)  j=l,2,...M,  N  y  . , L -. k i i = l <-k=l  '  and  r  ±  of the r i g h t  =  ( A 8 . 1 1)  constant,  hand s i d e  vector  will  be  N  M+1  The  expressions  corresponding  -U  J  cosO.As.  iii  1  (A8.10),  expressions  + |UCC  ( A 8 . 12)  L  x  ( A 8 . 11)  and  i n (4.2) and  (A8.12)  (4.3).  replace  the  116  ________ __ _  _______  _______  _i_______2_  __________  k i  _2____i__  ___  Profile  For  a two-dimensional a i r f o i l  conformally  • onto  circulation  T  a.  kr ,  radius  R,  can the  be  mapped  full  Kurta  (A9 1)  Fo =  angle  of  which  i s [37]  0  w h e r e U, a a n d a  circle  profile  4TTRU  sin(a-cto) ,  are the flowspeed, incidence,  0  respectively.  I f  the  circulation  and  i s reduced  zero-lift to a  value  then  0  T = k f  0  = 4 T r R U k s i n ( a - a ) •= 4 T r R U s i n ( a - a - A a ) .  (A9.2)  =  (A.9.3).  0  0  Hence  Aa  is  Kutta  - a r c s i n ( k s i n (a-a ) )  0  the e f f e c t i v e  circulation  0  reduction  (and hence  i n incidence  the l i f t )  required  to the fraction  to  reduce  the  »k»  of the  full  value. In'  the  (a-a )  order  profile  Points .through leading  to  shape  a c h i e v e a' r e d u c t i o n  i s modified by " r a i s i n g "  on t h e p r o f i l e an a n g l e edge.  i n the "effective  are rotated  which  about  i s proportional  The d i r e c t i o n  of rotation  the  t r a i l i n g - edge.  the profile  leading  to the distance i s such  camber,  to  edge  from t h e  reduce  the  117  effective  incidence  coordinates  (x,y) a t  of  the  p r o f i l e . F o r an o r i g i n  raid-chord,  and  flow  from  left  of  profile  to right,  the  expression  0(x)  where the  at  +  ^-(1  *c' i s the a i r f o i l point  point  =  ~ )  chord,  the leading  at the t r a i l i n g  ,  (A9.4)  will  edge  e d g e . The  assign  a zero  and t h e f u l l modified  rotation  rotation  to  Act t o t h e  profile coordinates  are  thus  x'  = xcos0(x)  -  ysin6(x) (A9.5)  y  The to.  the  zero-lift  y  = ycos9(x)  effective amount  of  angle  when a e q u a l s  a  a 0  0  + xsin9(x).  reduction l i f t for  i n camber  i s  (or c i r c u l a t i o n ) the  , Aa w i l l  be  profile zero.  will  roughly being not  proportional developed. change,  The  since  118  Appendix The  Computer Program  The of  the  CPS,  program contains  a  are  subroutine  ATXB, d e s c r i b e d  dependent)  subroutines  call  A,  by  storage  DY, so  writing  i n Appendix  The  which  3.  The  by  the  (system  and  deallo-  memory r e q u i r e d f o r  matrices  and  from/into  U.B.C.  FSPACE a l l o c a t e  real  WR,  i s solved  U.B.C. s u b r o u t i n e use  RE,  matricies  equations  GSPACE and  WALLCO i s u s e d  for a l l test  wall slats. DX,  C.  and  of  calls a l l  the  CALLER i s  used  t h a t h a v e memory .  GSPACE.  subroutine  coordinates  with  and  subroutines  allocated  A  B,  MAIN1, w h i c h  CALCAB, ASSEMA, ASSEMB, ASSEMD,  system of  respectively, blocks  matricies  and  The  Theory  addition, subroutines  required for reading storage.  to  subroutine  MODPRO. I n  peripheral  cate,  Exact Numerical  following subroutines:  FORCES, and  WRD  f o r the  10.  DS,  The CS,  to  airfoils,  create flaps,  the c o n t r o l . p o i n t solid  control point coordinates and  SI,  are  that a l l coordinates  XX  written into may  be  wall  sections  and, YY,  along  peripheral  checked  before  further  calculation.  The  definitions  of  the  v a r i a b l e s used  comment s t a t e m e n t s w i t h i n t h e  The surface  control points such  as  the  are  described  by  f o r an  arbitrary  solid  subroutines.  (XSOLSL,YSOLSL)  plenum boundary  are  read  i n at  execution  119 time  by t h e p r o g r a m w h i c h c a l l s  The VTI,  WALLCO.  control point coordinates  (XM,YM),  and t h e v e l o c i t y  on t h e s t r e a m l i n e r e p r e s e n t i n g t h e s h e a r  in  by t h e p r o g r a m w h i c h c a l l s  in  the c o o r d i n a t e s o f the s l a t  ((XCENT,YCENT))., t h a t each s l a t circulation  As  and t r a i l i n g  sees  layer  are read  MAIN1. T h i s p r o g r a m a l s o l e a d i n g edges edges  (XTE),  (ALF), and t h e f r a c t i o n  (XLE),  reads  centers  the flow  angle  of the f u l l  (Kl) r e q u i r e d .  shown,  this  a s i m i l a r program  program handles  only a single  i s used f o r a f l a p p e d  test  airfoil;  airfoil.  The n o t a t i o n f o r t h e e n u m e r a t i o n o f t h e c o n t r o l p o i n t s i s shown i n F i g u r e A10.1. The l a y o u t o f t h e s y s t e m o f to  equations  be s o l v e d i s shown i n T a b l e A10.1; t h e numbers i n p a r e n - .  theses  indicate  assembles  the p a r t i c u l a r  the corresponding  the equations.  The e q u a t i o n  i n d i c a t e d by  'E*. A c r o s s  of  o f summation  the index  DC—loop  i n t h e program  coefficients numbers  which  f o r t h e unknowns i n  (rows i n t h e m a t r i x  C) a r e  t h e b o t t o m o f T a b l e A10.1, t h e r a n g e ( j , k, p o r q) f o r e a c h c o l u m n o f  the matrix C i s s p e c i f i e d .  A c o m p l e t e sample  run with  the r e q u i r e d c a l l i n g  programs  follows.  The  sample  i s shown f o r t h e C l a r k - Y  airfoil  a t 20°  120  incidence, layer  i n a 70% OAR TSUSL w a l l  i s m o d e l l e d by a s t r e a m l i n e  control points, airfoil  that  b y 50, t h e u p p e r s o l i d  e a c h b y 20, a n d t h e s o l i d  lower w a l l  There a r e 8 a i r f o i l - s h a p e d s l a t s ,  zero  normal-velocity  control points This  leads  unknown in  Thus t h e r e  strength  by 9 where t h e  i s s p e c i f i e d , a n d 20  strength  are prescribed.  d e n s i t i e s a n d 29  d e n s i t i e s . T h e r e s u l t i s 291 e q u a t i o n s  291 unknowns.  The the  v e l o c i t y d i s t r i b u t i o n on t h i s  t h e average value  t h e mean v a r i a t i o n o f p r e s s u r e  The their  profiles  circulation  The  output  distributions Also  streamline  representing  s h e a r l a y e r i s s p e c i f i e d , and c o r r e s p o n d s t o a  c o e f f i c i e n t with  for  sections  points.  each represented  where t h e t a n g e n t i a l v e l o c i t i e s  vortex  end-wall  a r e 262 c o n t r o l p o i n t s  source  b y 20  e l e m e n t s . The  b y 80 c o n t r o l  boundary c o n d i t i o n  t o 262 unknown  The shear  which i s represented  i s 20 s o u r c e a n d 20 v o r t e x  i s represented  control points.  configuration.  o f the wall  o f -0.35, and i s s i m i l a r t o  shown i n F i g u r e  slats  t o 0.8 t i m e s t h e i r  8.7.  are modified full  t o reduce  circulation.  from the program.includes t h e p r e s s u r e .  on t h e w a l l s ,  printed are the l i f t , the a i r f o i l  pressure  the wall drag,  and t h e w a l l  slats,  and t h e a i r f o i l .  a n d p i t c h i n g moment c o e f f i c i e n t s  slats.  MAIN  MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) 0001 000? 0003 OO01 0005  0006 0007 0008 0009 0010 00 11 0012 0013 0011 0015 0016 0017 0018 PC19 0030 0021 0022 0023 0021 0025 0026 0027 0028 0029 0030 0031 0032 0033 0031 0035 0036 0037  C C C C C C C C C  10-22-75  12159113  REAL X X ( 2 6 2 ) , Y Y ( 2 6 2 ) , D X ( 2 6 2 ) , D Y ( 2 6 2 ) , D S ( 2 6 2 ) , C S C 2 6 2 ) , S I ( 2 6 2 ) REAL SlG(291),VTT(262),CP(262),GAM(9),MU(20),GNU<20) REAL V T I ( 2 0 ) , X M ( 2 1 ) , Y M ( 2 1 ) REAL AXX(2a2),AYY(242),ADXC2«2),A0Y(24?),AD3C242),ACS(242) REAL A S K 2 H 2 ) XM.YM •. PROFILE COORDS FOR STREAMLINE REPRESENTING SHEAR LAYER XX,YY - CONTROL POINT COORD3> DX.OY.OS - ELEMENT LENGTHS SIG - SOURCE STRENGTH OENSITlES (ALSO USED AS SOLUTION VECTOR IN SYSTEM ) (GAM,MU,GNU APE PART OF SIG) VTT.CP - TANG VEL, PRESSURE COEFF'. REAL XQ(10),YQ<10),XH(10),YR(10) XQ,YG,XR,YR - MODIFIED SLAT PROFILE COORDS REAL X C E N T ( f l ) , X L E ( 8 ) , X T E ( 8 ) , A L F ( 8 ) , K H 6 ) REAL Y C c N T ( f t ) , D T H I C K ( 8 ) AXX,AYY ETC - CONTROL PTS TO BE READ IN FROM F I L E (PUT THERE BY WALLCO) EQUIVALENCE (AXX,XX),(AYY,YY),(ADX,DX),CADY,DY),(ADS,OS) EQUIVALENCE ( A C 3 . C S ) , ( A S I . S I ) INTEGER CALAH,CALCD,WRAR,WRCD,SDLV,GAUSS,ITER,CALCP,C»LCL,HSIG C O M M O N / e i / NW5,NSLAT,N3U1,NKA,NM2,MS V,NA,NSPS.NTEU,NTEL COMM0M/A2/  303 301  U,CH  C0HM0M/B3/NUl,NWuT,NU3,NWU2,NLl,NKLl,NL3,NWL2iNS0Ll,NS0LSL,NP1, 1 NFLAT,N3PF,Ni1 C0MM0H/H4/CALAB,CALCD,WRAB,wRCD,S0l.V,GAU3S,ITER,CALCP>CALCL,HSIG C0Mn0H/n5> NPi,MP2,NL« REAL T I T L E ( 2 0 ) REAI)C5,303) TITLE FORMAT C20A/I) WRITEr6,304)  FORMAT(IH1) WRITE(h,305) TITLE 305 FOR AT(iX,20A4) RE AD(5,25) CALAB,CALCD,WRAB,WRCO,SOLV,GAUSS,ITER,CALCP, CALCL » HSI G 25 Fn.7.MAT(20I«) WRITE(6,1) CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITCR,CALCP,CALCL,H3IG 1 FORMAT ( 'CALABr I , I 2 , 2 X , ICALCO" , 12, 2X, 'WRABa M 2 . 2 X , 'WRCO=M2,2X, { 'SOLVr',12,2x,'GAUSS=',I2.2X,'ITERai,I2,2X,'CALCP',I2,2X, 2 "CALCL"',12.2X,'HSIG=',12) RE A.i (5.30) NA,NWUl,NHU2,NwLi,NWL2,NS0LSL,NFLAT,NSPF,NSLAT,NSPS, t MSV 30 FORMAT(20I4) WRITE(6,31) NA,NWUl,NWU2,NKLl,NWL2iNS0LSL.NFLAT,NSPF,NSLAT,NSPS» 1 MSV 31 FORMAT<'NAs<,13,2X,•NWU1a•,13,2X, NwU2a',13,2X,'NWL1a 1,13,2X, 1 INWL2=',IS,2X,'NSOLSLs',I3,2X,•NFLATt>,IS,2X,INSPFs',13,2X, 2 INSLAT=',I3,2X,iNSPSai,I3,2X,iMSVat,13) READ(5,32) NfEU,NTEL,CH,U 32 F0RKAT(2I1,2Ffl'.3) WRITE(6,400) NTEU,NTEL,CH,U 400 FORMAT ('NTEU= I , I 3 , 2 X , 'NTELa' , I 3 , 2 X , • CH= ', F8'. 3, 2X, 'U= ' ,F6,1) C READ COCROS FOR AIRFOIL, WALL & WALL SLATS FROM WALLCO F I L E (WALLS & C SLATS HAVE YY = 0'.) READ(2) AXX,AYY,ADX,AOY,AOS,ACS,ASI YWal'8'. . M  1  1  1.000 2.000 3.000 4.000 5,000 6.000 7.000 8,000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 15.000 16.000 17.000 18,000 19.000 20.000 21.0o0 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 12.000 43,000 44,000 45.000 46.000 47,000 48,000 49,000 50.000 51.000 52.000 • 53.000  PAGE P001  MICHIGAN TERMINAL SYSTEM FORTRAN G(H336) 003B 0039 0010 OOH' COM2 0013 0011 eo-'i5 0016 0017 ooie 0019 0050 0051 0052 0053 0051 0055  0056 0057 0058 0059 ooto 0061 0062 0063 0061 0065 0066 0067 0068 0 06? 0070 0071 0072 0073 0071 0075  MAIN  I0-22-73  12159113  NUleNA+J NU3 = NuUNWUl  MU'lsNu3 + NWU2-I WLUNU3 + NWU2 NL3=NLl+NWLl NL'l = NL3 + NwL2-i MS0LlrNL1+l IFCNSOLSL'.EO'.O) NSOLl=NLfl NF1=N|.1+N30LSL*1 IFCNFLAT'.EO'.O;. NFI*NL4+NSOLSL « 1ST C O N P T O N 1ST F L A T SLAT  NF1  NSUi=NL1+NS0LSL*NFLAT*NSPF*i I F CN3LAT.E0'.0) NSU1=NLU+N30LSL*NFLAT*NSPF NSIin = NSUl + N S L A T * N S P S » l I F ( N S L A T . E O ' . O ) NSU2=NSU1 NSU2 - L*ST CON PT ON L A S T S L A T Nil - 1ST CONTROL PT ON STREAMLINE FOR SHEAR LAYER Mil=NSU2+l I F CMSV.LQ'.O) NilsNSU2 WRITER,101) NU1,N';3,NL1,NL3»NS0L1.NF1,NSUI,NSU2,NI1 FORMATC'NU1««,I3,2X,INU3»',13,2X,"NL1"',I3,2X,• NLS»',13,2X, 101 1 iNSOLi"'#i3,2X,INF 1 = 1,I3,2X.'NSU1s',13,2X,'NSU2=1,13,2X,I Nils•, 2 13) NWALL - TOTAL * OF CON PTS ON ALL FLAT SOLID WALL SECTIONS NWALL=NWU1+NWU2+NWL1+NWL2 NW3 - TOTAL * CONTROL POINTS ON AIRFOIL SOLID WALL' SECTIONS & SLATS MWS=NA+NWALL+NSPS*NSLAT+NSOLSL+NSPF*NFLAT NKA « » OF EON FOR KUTTA ON TEST AIRFOIL NKA=NSU2+NSLAT*1 NSVT - TOTAL » SOURCE & VORTEX ELEMS NSVf=NWS+MSV NM2 «• » OF LAST EON FOR ZERO NORM VEL ON INNER EDGE OF S,L« NM2sNKA*M3V NUN - TOTAL « UNKNOWNS NUN=NtaS+2*MSV»NSLAT*l NFL - TOTAL « CON PTS O N FLAT SLATS ( N O K U T T A ) NFLsNSPF*NFLAT NSL - TOTAL * CON PTS O N A L L AIRFblLSHAPED S L A T S NSL=NSPS*N3LAT . WRITE (b.33) NA,NWALL,NSOLSL,NFL,NSL,NWS FORMAT('NA=',13,2X,'NWALL"',13,2X,INSOLSLo',13,2X,INFL"',13,2X, 33 I iNSLs'.njZX^NwSst.IS) WRITE(6,31) MSV,N3VT FORMAT('»3V=',l3,2X,'N3VT»",IS) 31 WRITE(6,35) NKA,NM2,NUN 35 FORMAT('NKA=',13,2X,'NM2s•,13,2X#'NUN«I,13) NA « a CON P T S ON SINGLE TEST AIRFOIL C NlaNA+1 N2=NA+NWU1+NWU2 SET TESTSECTION W A L L HEIGHT DO 2 I=Nl,N2 YW - Y-C00RO FOR UPPER AND LOWER W A L L S YY(i)=YW . N3=N2tl N«3N2*NwH + NKiL2  51.000 55,000 56,000 57,000 58,000 59,000 60,000 61.000 62.000 63.000 61,000 65.000 66,000 67,000 68.000 69.000 70.000 71.000 72.000 73.000 71.000 75.000 76.000 77.000 78.000 79.000 80,000 81.000 82,000 83.000 81,000 85.000 86,000 87.000 88,000 89.000 90.000 91.000 92.000 93.000 91.000 95.000 96.000 97.000 98,000 99.000 100.000 101.000 102.000 103,000 101,000 105,000 106.000 107.000 108.000  PAGE P002  MICHIGAN TERMINAL SY3TCM FORTRAN G(fll336)  0076 0077 0078 0079 0080  0081 0032 0083 OOSfl 0085 0086 0087 0038  C089 0090 009{  0092 0093 Oo9f| 0095  0096 0097 0098 0099 0100 0101 0102 0103 0101 0105 0!06 0107 ClOB 0109 011 0 0111 0112 CMS 0111 0115 0116 0117 0118 0119 0120 0121 0122 0123  MAIN  OO 3 i=N3,Nfl YY(I)=-YW IF(NSLAT.EO'.O) GO TO 500 DO 1 I=NSU1,N3U2 fl YY(i)=YY(I)+YW 5 0 0 CONTINUE IF(NSLAT'.EO'.O) GO TO 15 C XCENT.YCENT - CENTER OF SLATS REA0C5,16) (XCENT(K), Kai,NSLAT) WRITEC6.308) 308 FORMAT('XCENT') WRUE(6,1B) (XCENT(K), Kai,NSLAT) REA0C5,'i6) CYCfNT(K), K»l,NSLAT) 3  309 C 50  WRITEC6.309)  FORMAT('YCENT') KRITE(6,18) (YCENT(K), K«l,NSLAT) MODIFY SLAT PROFILES FOR REDUCED CIRCULATION REAn(S,50) MOD  FORMAT(12) wnifecft.si) MOD 51 FORMAT<'0DPR0a',J2) C MOOiFY PROFILES.IF IMOD' NOT ZERO IF(MOD,CO'.0> GO TO 15 C XLE.XTE - X-COOROS OF SLAT LEADING & TRAILING EDGES READ(5,16) (XLECK), K=l,NSLAT) 16 F0RMAT(i3F6'.l) WRITE(6,306) 306 FORMAT(* XLE ) WRITE(6,18) CXLE(K), Kn1.NSLAT) 18 F0RMATUX,13F6'.l) REA0(5,16) (XTE(K), Kai,NSLAT) WRITE(6,307) 307 FORMAT( 'XTE *) WRlfEt6,1B) CXTECK), K=l,NSLAT) C ALF « FLOW ANGLE AT EACH SLAT READ(5»16) (ALFCK3), KS=1,NSLAT) WRlfEC6,310) 310 FORMAT('ALFI) WRITEC6.18) CALF(K), Kai,NSLAT) C Ki « FRACTION OF CIRCULATION REAO(5.17) (Ki'(K). Kai,NSLAT) «7 F0RMAT(16F5'.3) WRITE.6.311) 311 FORMAT (' K 1 ') WRITE(6,19) (KICK), K»l,NSLAT) 19 F0RMAT(1X,16F5.3) C DTHiCK . SYMMETRIC DISPLACEMENT THICKNESS READ(5.17) (DTHICK (K), Ka1,NSLAT) WRlfE(6,3l2) 312 FORMAJCDTHICKI ) WRITE{6,19) CDTHICK(K), KoJ,NSLAT) NPS=((NSP3-l)/2)-i NSP=NSPS»1 MPsNSPS+1 DO 11 KS=1,NSLAT M  1  12)59113  PAGE P003  109.000 110,000 111,060  112.000 113.000 111.000 115.000 116.000 117.000 118.000 119.000 120.000 121.000 122.000 123.000 121.000 125.000 126.000 127,000 12e,000 129.000 130.000 131.000 132.000 133.000 131.000 135.000 136,000 137,000 138.000 139.000 110.000 111.000 112.000 113.000 111,0Q0 115.000 116.000 117.000 118,000 119.000 150,000 151,000 152.000 153.000 151.Opo 155.000 156,000 157.000 158,000 159.000 160,000 161.000 162.000 163.000  h 00  M.CHIOAN Tt£RM_NAl. 0121 0J25 0126 0127 oi'2B 0129  01*9 0131 0132 0133  0131 0133 0 136 0137 0138 0139 oiiq 0111 0112  0]13  0111 0115 0116 0117 0118 0119 ©ISO 0151 0152 0153 0.51 0155 0156 0157 0158 0159 0160 0161 0162 0163 0|61 0165 0166 0167 0168 0169 0170  SY3TEM  FORTRAN  6(11336)  MAIN  10-22-75  NT»NSllUNSP3»(KS-l) MTsNT+NSP MlnMT.NPS NLaNT+NPS ALZ - ZfRO LIFT ANGLE FOR SLATS ( 0 0 1 5 ) ALZaO. CALL MODPRO(XX,Vy,DX,OY,03,CS,SI,NSVT,XR,YR,MP,NT,MT,ALF(KS),ALZ, 1 Kl(KS),0THICK(K3),XCENT(KS),YCENT(KS),ML,MT,NL,NT,XO,YQ,XLE(KS), 2. XfECKS)) 11 CONTINUE 15 CONTINUE MVS=MSV+1 iF(MSV.EQ'.O) GO TO 36 READ COORDS FOR STREAMLINE FOR SHEAR LAYER READ(5»16) XM REAOC5.16) YM 16 F0RMAT(l'2F6'.l) WRITEC6.17) XM WRITE'C6,18) YM 17 •F0RMAT('XMi,10F7",2:-' 18 FORMAT<'YMi,10F7.2) DO 13 K=1,M3V I=NwD+K J= +1 . ., XX(I)a(XM(K)+XM(J))/2, YYCJ)=(YMCK)+YM(J))/2. DXCI)=XM(J).XM(K) OY(i)=YH(J)«YM.K) D3{n=S0RTlDX(I)*0X(I)*0Y'(I)*DY(I)) CS(I)=BX(I>/nS(I) 15 Si(I)=DY(I)/DS(I) M1,M2 * RANGE OF CONTROL PTS * I S ON SHEAR LAYER C MlaNW9+i M2=NWS+MSV KRITE(6,7) FORMAT('STREAMLINE FOR SHEAR LAYER•) WRITE(6,6) FORMAJ(7X,'XX',6X.'YY',6X,«0X',6X,<DY',6X»'D3',6X,'C3',6X,'SI') WRITE(6,5) (XXCI),YY(I),DX(I),DY(I),DS(I),CS(I),SI(I), IaMl,M2) F0RMATC1X,7F8'.3) READ VELOCITY DISTRIBUTION ON STREAMLINE FOR S.L, REA0C5/22) (VTI(I), Ial,M3Y) 22 FORMA7(i0F8'.3) WRITE.6,23) VTI 23 F0RMAT('VTI',i0F8.3) 36 CONTINUE NrNSVT. MsNlJN NG - a VORTEX STRENGTH DEN'S ON SLATS i TEST AIRFOIL NG=NSLAT+1 NMaMSV NLSsNSLAT CALl. MAiNlCXX,YY,0X,DY,D3,CS,SI.N,SIG,M,VTT»CP,GAM,NG,MU#GNU,NM, l" VTI.XCENT.YCENT.NLS) STOP t  K  12I59US 161.000 165,000 166.000 167,000 168.000 169,000 170.000 171.000 172. 000 173.000 171.000 175.000 176.000 177,000 178.000 179.000 180 . 000 181.000 182,000 183.000 181.000 185.000 186,000 187.000 188.000 189,000 190.000 191.000 192.000 193.00* 191,000 195.000 196.000 . 197,000 198,000 199,000 200.000 201.000 202.000 203.000 201,000 205,000 206.000 207.000 208.000 209,000 210.000 211,000  212,090 213,000 211,000 215,000 216.0 00 217.000 218.000  PAGE P001  MICHIGAN TERMINAL SYSTEM FORTRAN G'(HS36)  MAIN  0171 END •OPTIONS IN EFFECT* 10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAD,NOMAP •OPTIONS IN EFFECT* NAME = MAIN , LINECNT = S7 •STATISTICS* SOURCE STATEMENTS a 171,PROGRAM SIZE n •STATISTICS* NO DiAGNoSTICS GENERATED NO ERRORS IN MAIN NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTiON TERMINATED SR "L0AD+0UJBSL3+ATXB 2 = FlL'E«<l7 3 = «A «o-B 9«*DUMMY* EXECUTION BEGINS  10-22-75  12I59U3 219.000  260  PAGE POOS  ft* CY A«20 N»50 70XTSUSL C/H".66 MSVs20 •** CALAH" t CALCDB 1 WRABa 0 WRCD" 0 SOLV" J GAUSS" 1 NAB 50 NWU1= 2 0 NWU2e 20 NWLl" 10 NWL2" 40 NS0L3L"  ITER" 0 CALCP 1 CALCl" 1 HSIGa 0 0 NFLATa 0 NSPF" 0 NSLATa 8 NSPSa  N T T U » 2 5 NTELB 26 C H = 23.940 U " l'.O NU1 5 1 NU3 = 71 NLt" 9 1 N L 3 * 1 J 1 NSOLlal70 NF1«IT0 NSUl«17i NWALL=120 NFL" 0 NSL" 72 NWS.2D2 NA« 5 0 NSOLSL« NSVT=J62 MSV = 2 0 NK 251 NM2=271 NUNI291 XCF.NJ 46.2 • I'.B .1S'.8 •25*.fl • S 7 . 8 SI.2 YCF/JJ 18.0 8.0 I'S'.O IB'.O IB'.O S  NSU2e2(|2  A  =  XLE ,  <i'i.<i  XTE  io'.2  ie*. o i 32.a 20.1  ,  i's'.o  -3'. 6-15.6  8.1  ie.O  36.0  12.0 21.0  ALF -2.7  -4.8  - 8 . 815'. 2  27.6  • 39*.6  0.0 • 12*.0 21'.0 '15".0  5.5  -36*.  0  11.0 10. T  > . • , . . . . O.BOOOjBOOO.8000.8000.8000.8000.8000,SOC DTHIC* 0 0 0.0 0,0 K  o'.o o'.o o'.o o'.o o  Kao',80000 B L O I S P L T H I C K B Kao',80000 B L D I S P L T H I C K B 0,0 DEl.CPST = -u'.00943(RAD) K = o'.80000 B L O I S P L T H I C K B 0.0 0Fi.EPST = -O^0l678C«AO) K 0,80000 B L D I S P L T H I C K B O E L C P SS TT = -- 0o . RA D ' .0o35 03 82 94 ( (R ADn)) K=0.80000 B L D I S P L T H I C K B On.CPST= o'01924(RAO) 1 1 0 C 0 E G ) Kao ,80000 B L D I S P L T H I C K S 0,0 0ELCPST= 0 0 3 B 7 K R A O ) K=0 2,22<l)EG) ,80000 B L D I S P L T H I C K B 0,0 DELEPSTa O . 0 3 7 6 7 C R A O ) Kao ,80000 B L D I S P L T H I C K B °f.° 2.16C0EG) 0.0 00 7', 00 00 4 7 , 0 0 42,00 3 7 . 0 0 32*. 00 27^00 22 00 17.00 XM 00 •43,00 00 • 3 . 0 0 -8.00 - 1 3 . 0 0 - 1 8 . 0 0 -23.00 -28.00 -33,00 x« -A.e'oo YM 18 20 18 40 18.70 19, 50 20,10 20.70 21,40 22.10 Y M 22 6 0 22.90 22.90 22.60 2 2 . 1 0 21'. 40 20..70 19,70 19.20 18'.50 -0 5 4 ( R E G > -0,96(REG) -1,77(DEG) -3,09(DEG) -3,05(DEG>  Y"  o;o  18.00  STREAMLINE FOR SHEAR XX ' 49' 5 0 0 a'l 5 0 0 39 5 0 0 5";500 29 5 0 0 21,500 19 5 0 0 I'I,500 9 500 '1,500 -0,500 .5,500 .'JO,500 .15,500 -20,500 -25,500 -30,500 -35,500 -40,500 -45.500  YY  100 300 550 900 300 800 400 050 750 350 750 900 750 350 750 050 200 450 850 250  LAYER DX ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 000 ,000 ,000 ,000 ,000 ,000 ,000 ^.000 ,000 ,000 ,000  -5.000  DY  0'.200 0'.200 0,300 0.400 0'.400 0'600 0.600 0'700 0.700 O'.SOO  0'300 0,0 .0.300 .0'.500 .0'.700 -0'.700 -f.000 -0',500 .0.700 -0'.500  DS  5,004 5,004 5,009 5.016 5,016 5.036 5.036 5.049 5.049 5.025 5,009 5.000 5,009 5,025 5.049 5.049 5.099 5.025 5.049 5.025  -0.999 • o!.999 -0,998 -0,997 .0.997 -O'993 -0.993 »0'.990 -0J.990 .0.995 -0',998 .1,000 -0,998 -0,995 -0.990  990 981 .0,995 •0,990 •0.995  SI 0*.040 0'. 0 4 0 0' 060 0.080 O'.OBO 0'. 119 0'. 119 0.139 0' 139 0.100 0'.060 O'.O .0,060 .0.100 -0' 139 -0.139 -0.196 -0'. 100 -0.139 -O'.lOO  Nll"24S  9  MSV" 2 0  nj A J A j A J A J o o o o o o  —#rvj — - * A J o j o o o o c o  —•  — o o  G  t i i t i i r t t i i t UlUJUJU-UJUIUJUJuJaJUJUJ  • Ul  o o  •  o  • • • I UJ tu UJ f\l o cc — r— o(C — — o- j cc o S K I oo f f N ^ o o a o - a o o L n o - i r i r ^ C M C — co j ) c t C ' - K i r - N F « T L n J i n - 0 — — =a ~ cr K I O J =r =r oj Kl K 1 ^ 0 N M i T S Q D ^ O f \ J - £ 0 ' a j 3 o - G P - r v i n c - K i o - o —• cr ^ 1 O C T r-~ — —  ——  o  o  o  o  o  o  o  0  o  »  t  0  i  0  s i r  i  i  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  * • » i  i  i  i  r  t  i  i  UJ U l UJ UJ U l DJ U l U l Ul UJ U l cc —* -O =3 ^ m ^ cr r \ i , " t \ j ^ ) - ' K ' i o £ ^ )  t  I I i  • •  Ul Ul Ul Ul Ul Ul — N f \ J 3 u i - ' C ' C C ( r -o f u  ,  1  (\J-JD30JC007AJO(Miaj 5 0 N N 3 0 J 3 « - * > ^ ' - ' f - ^ C l ! 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Oil 1^030 1.025 1.021 10 19 1.0)6 1.011 •1.013 1.011 1.009 1,007 l'.oos 1.002 0.999 0'.996 0'.991 0'.987 0.981 0.971 0.967 0.958  2,591 2,167 2,317 2,231 2, 120 2,013 1,910 1 809 1,712 1,616 1,522 1,128 1,333 1,235 1.119  -1,828 -5.576  ASUK  BSUH  .8,131 -7'.711 -7,380 -7.013 -6',723 -6J118 -6.125 -5'813 -5 571 -5 312 -5.073 -l'.880  YV  2,176 2,600 2,727 2.858 2J.992 3.129 3'.266  7.191 7'. 9 0 1 8'.356 8'. 8 6 5 9'.165 10,226 11,321 13.588  6,00 8.00 8,00 e.oo 8.0 0 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8,00 8,00 8.00 8.00 8,00 8,00 8,00 8,00  ASUM 3".603 3,181 3,368 3,255 3,117 3,015 2,918 2,858 2,773 2,696 2,621 2.559 2,198 2,112 2,389 2,339 2,290 2,211 2,190 2,136 2.076  BSUM -13-770 -il'.525 -io'.na -9.712 -9.139 • a',662 .8'208 -7,878 -T.511 -7.232 .6'. 9 1 5 -6', 6 7 7 -6.127 • 6' 1 9 3 -5,971 -5!,769 -5,579 -5'.102 -5'.239 -5.088 -1'.918  8.00 6.00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8.00 8,00 8,00 8.00 8,00 8.00 8.00 8,00  1,165  1,261 1,356 1,11" 1,511 1,631 1,728 1,821 1,923 2,026 2,132  0,610 2,566 3,111 1,039 1,511 1,927 5,307 5.669 6' 0 2 3 6,376 6.735  8.00 8.00 8.00 8.00 8,00 8.00 a.oo 8.00 8.00 8.00 8.00 8.00 8,00 8.00 8,00  YY  82.80 80,10 78,00 75,60 73,20 70,80 68,10 66,00 63,60 61,20 58,80 56.10 51,00 51,60 19,20  XX -19,20 -51,60 -51,00 -56,10 -58,80 -61,20 -63,60 .66,00 -68.10 -70,80 -73.20 •75.60 -78.00 -80,10 -82.80 -85,20 -87,60 -90,00 -92,10 -91,80  XX •91,80 -92,10 -90,00 -87,60 -85,20 -82,80 -80,10 -78,00 -75.60 -73.20 -70,80 -68,i|0 -66,00 -63.60 -61,20 -5B.80 -56,10 -51,00 -51.60 -19,20 -16,80  •0.137 -0.139 -0.111 -0.113 -0,116 • 0.150 -0.15« -0.160 -0.167 -0.178 -0.195 -0.228 -0.366 0.197 -0.026  CP •0.350 •0.196 -0 113 .113 0.092 -0 0 7 7 0.066 0.057 '0,050 0.011 0.010 035 031 027 023 019  Oil  008 001 020  CP • 0.081 -0,060 •0.050 -0.013 -0.037 -0.033 -0.029 • 0.025 -0.022 • 0.018 -0.011 -0,009 -0.001 0.002 0.009 0.017 0.027 0.038 0.050 0.065 0.082  0.002 0.002 0.002 0,002 0.002 0.001 0.000 .0,001 -0.002 -0,005 -0,009 -0,018 -0,010 -0.110 0.002 SIG 0.032 0.023 0.019 0.016 0,011 0,012 0.011 0,010 0.009 0.008 0.007 0.0 07 0,006 0.006 0.005 0.005 0.005 0.005 0.005  O.OOU  SIG -0.006 -0.006 -0.006 -0,006 -0.006. • 0,006 -0.007 -0,007 -0.007 • 0.008 -0,008 -0.008 -0,009 -0,009 •0,010 -0,010 •0.011 -0.011 -0.012 -0.013 -0,013  to  130  ^ — < — H ~ - — - — « — - « ^ — - . — < — . 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V. *- «. V • J— > C * 0 0 0 — - 4 0  A:  ft <ft; ft ftt CO  0',0?7 -0,279 0.156 -0.247 SLAT »216 221 VNST VNVT -o',44t 0',397 -0,463 0,3o7 -0,136 0.183 0-'130 o',024 1,251 -0,251 0,703 -0,259 0,268 -0.221 0 101 -0'.228 0.397 -0'.386 S L A T »225 VIJST  -o',414 -0,3l'9 -0,004 0,468 0,969 0,424 0,1Q4 0,036 0.379  S L A T «234 VNST  233  VNVT  0^343 0.148  0,183 0.091  0,560 0.511  VTST 0.001 -0.315 -0.175 0.369 1 .I'll 0.825 0.331 -0.136 -0'.265  VTVT •01203 .0,150 -0.170  VTST .0.255 -0.540 -0.252 0.452 1.910  VTVT  VNT VNOT 0* Oil • 0'. 000 0,156 • O'.OOO -0,0t7 - o ' . o o o -0 454 - o ' . o o o -1,000 - o ' . o o o -0,444 o'.ooo -0,047 o'.ooo 0,126 0.000 .0.010 o'.ooo VNOT O'071 0,171 -0,047 -0,459 -1,000  0,015 -0,439 •0,057 -0,047 0,111 -0.147 •0'.342 -0.038 24 2 VNVT 0'.094  0,000 •0.078 0.000 .0.333  VNT  -o'.ooo -o'.ooo  -0.000 -0^000 -0.000 o'.ooo o'.ooo  i.  0 .484 0' 000 -0.099 0.000 -0.203  •ol190  -0.010 0',265 0,375 0,162 0.466  8* 188 0,209 0,199 0 J209 0,277 0,192 0,201 0.256 0'.268  VTST VNT V NOT VTVT 0,267 O' 069 . o ' . o o o -0.326 0,255 0,171 . o ' . o o o -0.558 0,266 -0.047 - o ' . o o o •0.454 0,141 0,313 0,529 •0,071 ".0*, '158 -0,000 -0.113 0,320 -1,000 -0.000 0,863 0.793 0,137 0.067 o'.ooo 0,245. 0.194 -0,439 0.777 • o'000 -0,047 o'.ooo •0,057 OJ105 0.597 0,006 0,112 o ' . o o o 0.014 0.224 0.022 0.035 0.134 • 0'.098 -0.036 - o ' . o o o STRCAMLINC FOR SHCAR LAYF.RI VNVT VTST VNOT VTVT VNST VNT •0.849 0',089 -0'.049 -0^,040 - o ' . o o o o'.ooo 0.298 •0,319 0,830 -0'.790 -0,040 0,476 -0J416 -0,060 • o ' . o o o 0.281 •0,309 -0.317 -0,080 o ' . o o o 0,397 0.204 -0.261 0,342 • 0',262 -0,080 - o ' . o o o 0.392 • 0'.479 0,328 -0,209 -0,119 -0,000 0.457 -0^573 0,156 -0.037 -0 119 - o . o o o 0.580 -0,732 -0 139 - o ' . o o o 0,074 0 .588 -0,781 -0,139 0'.808 -0,096 -0,000 0.582 0,235 0'.759 -0,274 100 0.000 0.5f'3 0,373 -0,375 o'.ooo 0.349 -0.600 0.435 -0,060 -0,460 o'.ooo 0.215 iO'470 O',460 -0,0 .0,487 0,060 0' 000 0.059 -0.316 -0 421 0.322 o.ooo 0,100 -0.071 - o ' . i e s -0,407 o'.ooo 0'.268 0,139 •0.115 -0'. 136 -0,331 0J192 0 139 0'000 •0.202 -0'.033 -0.262 0' 054 0,066 0 196 0 000 -0.273 0,098 0,031 0,100 -0.000 -0.269 0,111 0,139 - o ' . o o o -0.256 0,019 -0,158 0.072 0.100 - o ' . o o o -0.171 0.087 »0'. 186 -O'163  0.996  i',207 22.870  VTOT .0'.999 -0.988 .0'.999 .0'.891 0'.002 0.896 0' 999 0.992 I'.OOO  VTT -1.201 -1.452 -f.343 -0.712 T.433 1.986 1.704 1.318 1.201  VTOT .0' 997 .0.985 .0".999 .0'889 0.002 0'.899 0.999 0".9?q 0.999  ASUM VTT 5^666 •1.065 4,791 •1^317 5,430 -1.052 10,191 -0'.228 2.190 25.486 2.219 21^553 1,684 18,379 1,151 18 335 1.065 19.559  VTOT »0'.998 -0'.985 • 0'.999 .0.889 0' 002 0.898 0'.999 0'.994 0'.999  VTT -f.057 -1.289 -1.187 -0.688 1.116 1.742 1.595 1.224 l'.057  VTOT -0'.999 • 0'.999 • 0'.998 • 0'.997 • 0'.997 -0'.993 -0'.993  VTT •1,015 •1.020 •1.026 -1.054 -1.084 • l ' . 109 -1.145 -1.183 -1.217 -l'.24I -T.249 -1.255 -1.255 -1.251 • l'.241 -l'.225 -T.200 - l ' . 166 -1.136 -1.095  -0.990 -0'.995 • 0'.998 • I'.OOO -0.998 • 0'.995 »0'.990 • 0'.990 -0'.9B1 -0'.995 -0'.990 -0'.995  . ASUM  3,358 8,175 24.983 23^049 20,332 20,432 21.680  - t ' 173 10.022  BSUM -8' 002 1,149 7.307 i2'.89« 8. 348 -5.410 •fe',44 1 -3'.587 7'. 7 01  18,10  17.92  YY 18.03 17,88 17,76 17.88 18.00 18.12 18,23 18.14 18,03  YY 18,06 17.89 17.76 17.87 18.00 18.12 18.23 18.16 18,06  BSUM -4',866 8,277 -3.238 -3.127 •1.312 -4'.578 0^351 -1,842 0.351 0,326 0.467 t',591 0.869 2.284 1.748 1^846 3,213 -0.417 7^470 2.238  YY 16.10 18,30 18.55 18.90 19,30 19.80 20,40 21.05 21.75 22.35 22.75 22.90 22.75 22.35 21.75 21.05 20.20 19.45 16,85 18.25  -13; i io  •12.416 -12', 223 -12,'123 -10.616 -12', 381 -9 316 -10,026 -9 276 -5.801  XX  -12.18 -13,26 -14,69 -15,40 -15.58 -15,40 -14,69 -13,27 -12,18  XX YY BSUM .7^097 18,06 -24,18 2,175 17.89 -25, • 26 7,784 17.76 -26, ,69 12,464 17.87 -27 ,40 6.486 18.00 -27 ,58 -6,815 18.12 -27 ,40 -7.185 18.23 -26 ,69 .4'. 044 18,16 -25 • 27 6'.869 18,06 -24 .18  BSUM ASUM 6^772 -6^934 5,808 2.417 6,381 7,799 11,032 12.113 25.567 5'.579 20),770 -7'.291 17,431 -7'.315 17,290 • 4'.009 18.449 6'.742 ASUM -1 f,696 •12,560 -6,478 -15.433 -7' 946 -ll,5(/j -11,110 -11,112 -13.140 -ll'.917  -1,26 -0,18  XX •36,18 •37,26 •38,69 •39,40 •39,58 •39,40 •38.69 '37,27 •36,18 XX 49,50  44,50 39,50 34,50 29,50 24.50 19,50 14,50 9,50 4,50 -0.50 -5,50 -10.50 -15,50 -20.50 -25,50 -30.50 -35.50 -40,50 -45,50  •1.146 •0,456  CP  0^084 0,562  SIG  •0,442 -1.108 -0.804 0.493 -1.054 -2.945 -1.905 -0.737 .0.442  0.599 0.099 0.110 0,172 0.134 .0.039 .0,100 -0,134 .0,602  CP -0.133 •0.734 .0.107 0.948 -3.794 -3.925 -1.835 -0.325 -0.133  SIG l'.068 0,172 0.164 0.218 0,124 -0.099 •0.157 -0,204 -1,070  CP •0.117 -0.661 •0.409 0.526 .0,245 -2.033 -1.545 •0.499 -0.117  SIG 0.666 0,095 0.086 0.138 0,121 -0.019 -0,076 -0.128 -0.689  CP SIG -0,030 0.024 -0.040 .0,(106 -0.053 -0.010 •0.111 •0.016 -0.175 -0.009 -0.230 0.010 •0.311 0.018 -0.399 0,011 •0.461 0.061 •0.540 0,020 •0.560 0.197 -0.575 0,080 -0.575 0,0*>6 •0.565 0.059 -0.540 0.050 .0.501 0,028 -0.440 0.005 -0.360 •0.019 -0.291 •0.052 -0.199 -0.0>>8  0'.23810E-Ol-0'.O26ilC-02-0'.1015hE-Ol-0'.16088E-0l.0',,9:»665E-62 0 ' . 9 9 9 ( I 1 E " 0 2 0'.17789E"01 o ' . 1 1 4 3 0 E -  CAM" GAM"»  01  0'.61ia5E-01  Mu< ' ' MU: GNU =  V<U=> 0.-71097 VKL = - 0 ' . 7 l 0 S 0 FORCES ON BODY # i , 5 0 CENTER AT C O'.O , o'.O ) 3 CU 3',013I'1 COT= 0'.06599 CMOs 0'.5703B CM« = - 0 ' . 18295 CIRC« 37'.0638f CLCs 3'.o9640 PERIMs 49,298 FORCER ON BODY # 1 7 1 . 1 7 9 CENTER AT ( 4 6 ' . 2 0 , le'.OO) ilT- 2'.07780 C O r = O'.4o096 CMOs - 0 ' . 3 S 3 4 7 CM4 = -1.02292 Cl«C= l'.R8B9i CLCs l'.0'l911 PERIMs 7 ' . 3 7 7 5 FORCES ON HODY #iflo,l8A CENTER AT ( 3 l ' . 2 0 , IB'.OO) CLT = -o',24827 COT= - 0 * . 0 o 7 5 0 CMOs - 0 . 1 0 2 7 0 CM4* » 0 . 0 4 0 6 4 ClRCs - 0 ' . 4 4 0 1 3 CLCs - o ' . 2 4 4 5 2 PCRIMs 7'.3776 FORCES ON BO')Y #189, 197 CENTER AT ( 2 2 ' . 2 0 , 18'.00) CLT .o'.flasflo CDT = - o ' . 0 5 4 2 5 CMOs -0.20537 CM4 = P0'.09392 ClRCs -0'.7112B CLCs -0'.111B2 PERIMs 7 ' . 3 7 7 6 FORCES ON IIPOY #198,206 CENTER A T ( 10'.20, lfl',00) CLT= -0',7569l CRTs -o'.14861 CMOs -0'.36254 CM4 = - 0 ' . 1 7 3 3 2 ClRCs - 1 . 17703 CLCs -o'.65391 P E R I Ma 7'.3778 FORCES ON BUOY #207,215 CENTER AT ( -f.80, IB'.OO) CLT= -O',5066S CDT= - o ' . 0 2 9 4 4 CMOs -o'.23201 CM4s • 0 ' . 1 0 S 3 7 ClRCs - 0 . 6 * 2 1 3 CLC= -o'.37S96 PERIMs 7'.3778 CENTER AT C -13'.80, IB'.OO) FORCES ON tfnDY #210,224 0',47677 COT= - 0 " . 0 4 0 6 6 CMOs 0'.24933 CM4 = 0 . 1 3 0 1 4 CLT = 0.72990 CICs 0.10550 PERIMs 7 ' . 3 7 7 6 ClRCs CENTER AT < -25'.80, IB'.OO) FORCES ON BODY # 2 2 5 , 2 3 3 0'.16144 0',7C."66 CDl's -O'. 15769 C Os 0'.35735 CMIs f LT = ClRCs 1'.28502 CLCs 0.71390 PERIMs 7 ' . 3 7 7 7 FORCES ON tinOY # 2 3 4 , 2 4 2 CENTER AT C -37'.80, l f l ' , 0 0 ) 0'.07419 CLT = 0',50513 CPTs - o ' . 0 { 7 2 2 CMOs o'.20048 CM4» C IRCS 0'.785lj CLCs 0'.43634 PERIMs 7 ' . 3 7 7 7 EXECUTION TERMINATED :  3  M  I'  isiG S  LO  xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx^  MICHIGAN  0001 0002 0003 9001  TERMINAL  SY3TEM  FORTRAN  0(41336)  0006 0007  OOOB 0009  0010 0011 0012 0013 0014 0015 0016  10*22-75  SUBROUTINE MAiNlCXX.YY,DX,0Y,D3,CS,S!.N»S!G,M,VTT.CP,GAM,NG.MU, GNU,NM,VTI,XC,YC,NLS) REAL XX'(N),YYiN),DX(N),DY'(N),DS(N),CS(M),SI(N) XX,YY > CONTROL POINT COORDSf DX,DY,D3 » ELEMENT LENGTHS CS,SI - COS,SIN OF ELEMENT INCLINATION REAL XC(NLS),YC(NLS) XC,YC - CENTERS OF WALL SLATS N i TnTLA * CONTROL POINTS REAL Slr,(M),VTT(N),CP(N),r.AM(NC),MU(NM),GNU(NM) • •• SIG - SOURCE STRENGTH DENSITIES (ALSO USEO AS SOLUTION VECTOR I N SYSTEM ) (GAM,MU,GNU,' ARE PART OF SIG) M- TOTAL * UNKNOWNS IN SYSTEM C*SIG=D VTT,CP - TANG VEL, PRESSURE COEFF'. GAM - VORTEX STRENGTH DENSITIES ON TEST AIRFOIL t S L A T S MIJ.GNU - SOURCE 8 VORTEX STRENGTH DEN'S ON STREAMLINE REPRESENTING i  S'.L'.  0005  MAIN1  REAL V T I ( N M ) VTI - PRESCRIBED TANG'L VEL ON SHEAR LAYER STREAMLINE CALAB,CALCD,CALCP,CALCL - IF NONZERO CALCULATE A,B,C,D,CP,CL K - R A B, w R C D - IF MONZr-IP WRITE A,B,C,D INTO FILES SOLV - IF NONZERO SOLVE SYSTEM OF EONS C*SIG=0 . GAUSS - IF NONZERO USE GAUSS-ELIMINATION ITER - IF NONZERO U S E ITERATIVE METHOD HSIC - IF NONZERO ALREADY HAVE SIG IN FILE FROM PREVIOUS R U N INTEGFR CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITER,CALCP,CALCL,H3IG COMMON/81/ N»S,NSLAT,NSU1,NKA,N«<2,MSV,NA,NSPS,NTEU,NTEL N*3 - TOTAL # CONTROL POINTS ON AIRFOIL SOLID WALL SECTIONS & S L A T S NSLAT'.NSPS - aSLATS,((CONTROL PTS /SLAT flSU 1 - 1ST CONTROL ON 1ST SLAT NKA - EON M FOR KUTTA CONO'N ON TEST AIRFOIL MSV - » CONTROL PTS ON STREAMLINE FOR SHEAR LAYER ( M U , G N U ) NA - ((CONTROL PTS ON SINGLE TEST AIRFOIL NTEU, M T F L - CONTROL PT * » TEST AIRFOIL TU'. ( U ' . L ) N M 2 - - « OF LAST EON FOR Z E R O NORM VEL ON INNER EDGE OF S.L'. C0MH0N/B2/ U,CH U - UNIFORM ONSET STREAM SPEED CH - SINGLE TEST AIRFOIL CHORO COHMON/P.3/NU1,NWIII NU3,MWU2,NL1.NWL1,NL3,NWL2»NSOL1,NSOLSL,NF1, 1 NFLAT,N3PF,NI1 NU1,NU3.NL1,NL3 - 1ST CON PT ON EACH FLAT SOLID WALL SECTION NWIJ1,NWU2,NWL1 ,NWL2 - « CON PTS ON EACH F L A T 30LIO WALL SECTION N30LSL - * CON PTS ON ARBITRARY SHAPED SOLID SURFACE E'.G', PLENUM BOUNDARY NSOL1 - 1ST CON PT ON ARBITRARY SOLID SURFACE NFLAT,NSPF - « FLAT SLATS, « CON PTS OFLAT SLAT - NO K U T T A APPLIED NF1 - 1ST CON PT ON 1ST FLAT SLAT N i l - 1ST CON PT ON SHEAR LAYER COMMON/B4/CALAB,CALCD,WRAB,KRCD,SOLV,GAUSS,ITER,CALCP.CALCL.HSIG EXTERNAL CALCAB,ASSEMA,AS3EMB,ASSEM0,RE,WR,WRD,ATXB,CPS,FORCES N3VT=N " NUNsM NDA=4*N3VT*NSVT NDBsNDA NDCs4*NUN*NUN (  10119136 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 10.000 11.000 12.000 13.000 14.000 15.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.0 0 0 25.000 26.000 27. 000 28.000 29,000 30.000 31.000 32.0Q0 33.000 34.000 35.000 36.000 37.000 38,000 39.000 40 . 000 41.000 41,000 42.000 .43.000 44.000 45.000 46.00 0 47.000 4 8.000 49.000 50.000 51.000 52.000  PAGE POOl  — I '  4s>  MICHIGAN TERMINAL SY3TEM FORTRAN G(41336) 0017 ooia  0019 0020  0  021  0022 0023 0023  0026 0027 0029 0030 0031  200  0032 O033 0034 0035  0036 0 0 37  I F ( W R A B . N E ' . O ) GO TO 202 IFCCALCD'.EO'.O) GO TO 2 0 3  202  0036  0039 C01« ooii  0042 0013  OOH  0045 0016 0017  204  0050 C051 0052 0053  206  0016 0019  0055 0056 0057 0058 O059 0060 0061 0062 0063 0061 0065 0066 0067 006e  0069 0070  GO T0 204 MsNSVT. M=NSVT LA = 3 CALL CALLER(WR,A,IPTR(N),IPTR(M),IPTR(LA)) N=NSVT M=NSVT LB = 4 CALL CALLER(WR,B,iPTR(N),IPTR(M)»IPTR(LB)) IF(cALCO'.eo'.O) GO TO 205 I F C ^ R A H ' . N E . O ) GO TO 206  N=NSvf MsNSVT LB = 4 CALL CALLER(WR, B, JPTR (N) , IPTR (M) , IPTR(LB)) CALL FSPACE(B,»303) CALL r,SPACE(C,NDC,0,&304) M=NSVT MBNIJN  CALL CAi LERitASSEMA,A,C,IPTR(N),iPTR(M)) IFCWRAR'.NE.O) GO TO 207  0051  LA-::  MsNSVT MzNSVT CALL CALLER(WR,A,iPTR(N),IPTR(M) IPTR(LA)) CALL FSPACE(A,*305) N0B=4*"SVT*NSVT CALL GSPACEitB,NOB,0,8306) LB = 4 N=NSVT MsNRVT CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)> N=N3VT M = NIJN CALL CALLER(A3SEMB,B,C,IPTR(N),IPTR(M)) ND0=4*NUN f  207  10-22-73  CALL GSPACE(A,NDA.0,&30n CALL GSPACE(B,NDH,0,&302) iF(CALAB.Nc'.O) GO TO 200 LA = 3 MsNSVf HaNSVT CALL CALLER(RE,A iPTR(N),IPTR(M),IPTR(LA)) LB = 4 N=NSVf M=NSVT CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)J GO TO 2ol N=NSVf MrNSVT CALL CALLER{CALCAB,A,B,IPTR'(N),IPTR(M),IPTR(XX),IPTR(YY5,IPTR(0X), 1 IPTR(0Y),IPTR(05), IPTR(CS),IPTR(SI» IF'(HSIG),NC,0) GO TO 201 f  0021  oo2e  ..AINl  tOt19136  PAGE P002  53.0Q0 54.000 55.000 56.000 57.000 58.000 59.000 60.000 61.000 62.000 63.000 64,000 65.000 66.000  67.000 68.0 00 69,000 70.000 71.000 72,000 73.000 74.000 75.000 76.000 77.000 78.000 79.000 80.000 81.000 82,000 83.000 84,000 85.000 86.000 87.000 88.000 89.000 90.000 91.000 92.000 93.000 94.000 95.000 96.000 97.000 98.000 99.000 100.000 101.000 102.000 103,000 104.000 103,000 106.000 107.000  Ul  MICHIGAN TERMINAL SYStEM FORTRAN G(«13S6) 0071 0072 0073 0071 0075 0076 0077 0078  Olie 0J19. 0120 0121 0122  208 210  209  900  201 3 214  215  213 .  10-22-73  CALL r,SPACE(D,N0D.0,i307) NsNSVT M=NUN N i = M 51V IFtMSV.CO'.O) Nfal N2=NXE IF(MXE.EQ'.O) N2 = l CALL CALLER(A.1SEMr),0,IPTR(M),IPTR(CS),IPTR(SI),IPTR(N),IPTR(VTn, l" 1 TR(N1).IPTR(VT0),IPTR(N2)) iF(SOLV'.LQ.O) GO TO 208 IF(r,All33'.NE'.0) GO TO 209 IF(ITER.NE.O) GO TO 210 IF(WPCO.I:0',0) GO TO 21 1 LC=7 NsHi.iN -• MsNUN CALL CALLER(WR,C,iPTR(N),iPTR(M),IPTR(LO) LD=8 MrNUN CALL CALLERC<.Rn,D,IPTRCN),IPTR<LD)) IF (GAUSS.EO'.O) GO \»i 212 LU=6 MsNlIN CALL CALLER(ATXB,iPTR(M),C,iPTR(SIG),0,IPTR(LU)) KRITEJ9) SIG wniTE(6,900) M FORMATCSIGCIS.')') • wRITE(h,3) GIG CALL F5PACECC&308) IF(CALCP'.EO'.O) GO TO 216 iF(HSlG'.L'O'.O) GO TO 214 READ(9) SIG WRITE(6,3) SIG FORMATdX, 10C12.5) IF{HSlG>E.O) CO TO 215 NOAs'l*NSVT*NSVT CALL r,SPACE(A,NDA,0,*309) LA=3 . NsNSVT . . . MsNSVf CALL CALLER(RE,A,iPTR(N),iPTR(M),IPTR(LA)) NrNSVf MrNUN KsNJLAT IF (NSLAT.EO'.O) K S J LsMsV iF(MSV.EO'.O) Lsl , CALL CALLER(CPS,IPTR(CP),IPTR(VTT)nPTR{XX),IPTR(YY),IPTR(C3)» 1 IPTR(SI),JPTR(N),IPTRCSIG),IPTR(M),IPTR(GAM),IPYR(K),IPTR(MU)r 2 iPTR(GNU),jPTR(L),A,B) IFCCALCL'.EO.O) GO TO 216 CALL FfiPACE(A,&310) CALL FSPACE(B,&311) NsNSVT Nl=i B  0079 0080 0031 0082 0033 0081 0085 0086 0087 00?8 0089 0090 0091 C092 0093 C091 0095 0096 0097 0098 0^99 0100 010f 0102 0103 0104 0105 0106 0107 0108 0109 0110 C!l{ 0112 ' 0113 C114 0115 0116 0117  MAIN1  10119136 108.000 109.000 110.000 111.000 112.000 113.000 111.000 115.000 116.000 1 17 .000 lie.000 119.000 120.000 121.000 122.000 123.000 124.000 125.000 126.000 127.000 128.000 129.000 130.000 131.000 132.000 133.000 134.000 135.000 136.000 137.000 138.000 139.000 140.000 I'll.000 112.000 143.000 144.000 145.000 146.000 147,000 118.000 119.000 150.000 151.000 152.0Q0 153.000 154 .000 155.000 156.000 157.000 158.000 159.000 160,000 161.000 162.000  •liCHIGAN TERMINAL SYSTCM FORTRAN G<41336) 0123 0124 0125 0126 0127 0126 0129 0130 C 13 i 0132 0133 0131  MAIN!  10-22-75  10119136  N2SNA  XA--0'. YA = o'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(DX).IPTR(DY), 1 IPTP(OS),IPTR(VTT),IPTR(N),IPTR(U),IPTR(CM),IPTR(NI),IPTR(NJ), 2 IPTR(XA),1PTR(YA)) IF(NaLAT'.Ed'.O) GO TO 2 00 { K = 1,NSL AT Nl = MSlll + NSPS*(K-l) N2=M1+NSFS-1 CHrj'.h xs=xc'rK)  YS=i«'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(OX).IPTR(DY), 1 IPTR (IIS), IPTR (VTT), IPTR (N), IPTR (U), IPTR (CH), IPTR (NI), IPTR ( N 2 ) , 2 IPTR(XS),IPTR(YS)) 2 CONTINUE 0135 0136 IF(HRIG'.NE.O) GO TO 217 0137 216 CONTINUE 0138 GO TOI 99 0139 203 STOP 203 oiio 205 3T0P 205 ciii 211 STOP 211 0142 212 STOP 212 0113 217 STOP 217 0111 301 STOP 30 i 0145 302 3T0P 302 0146 303 STOP 303 0147 304 STOP 3 0 1 0148 305 STOP 305 0149 306 STOP 306 0150 307 STOP 307 0151 308 STOP 308 0152 309 STOP 309 0153 . 310 STOP 310 .0154 311 STOP 311 0155 99 RETURN 0156 END IO,EnCDIC,SOURCII,N0LI3T,NODECK,LOAO,NOMAP • O P T I O N S IN EFFECT • , LINECNT a . 37 •OPTIONS IN EFFECT^ N A M E = MAINl 3 J . J , : L E STATEMENTS = 156,PROGRAM SIZE s 69S2 • STATISTICS •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN MAINl  PAGE P004  163.000 164.000 165.000 166.000 167.000 168.000 169.000 170.000 171.000 .172.000 173.000 174.000 175.000 176.000 177.000 ne.ooo  179.000 180.000 181.000 182.000 183,000 184.000 185,000 186.000 187.000 188.000 189.000 190.000 191.000 192.000 193.000 194.000 195.000 196.000 197.000 198, 000 199.000 200,000  LO  MICHIGAN TERMINAL SYSTEM FORTRAN G(41536> 0001 0002 0005 0 0 01 0005 0006 0007 0008 0009 0010 0011 0012 00 1 S 0011  C  C  0015  0016 0017 0016 0019 0020 0021 0022 0021 0024 0 025 0026 <027 0028 0 029 0030  CALCAB  10-22-73  SUBROUTINE CALCAB(A,B,N,M,XX#YY,DX,DY,DS,C3,SI) CALCAB CALCULATES MATRICES A, B OF INFLUENCE COEFFICIENTS REAL XXCN),YYCN),DX(N),DY(N),OS(N),CS(N),SI(N) REAL A(N,M),B(N,M) COHMOM/Bl/ NWS,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS.NTEU,NTEL DO ? I = i,M DO 2 J*1,N IFCl'.EQ'.J) GO TO J OXJ=XX(I)-XX(J) DYJiYYd)-YY(J) X . I . Y J - DIST. riCE OF 'II TO ' J ' IN ' J < COORD'. SYSTEM XJ = DX.!*CS (J)+DYJ*SI ( J ) YJ=DYJ*C5(J)-DXJ*3I(J> l)SJ2 = r>r>CJ)/2'. D3J4 = DS.I2*D5J2 XJS = X.I*XJ  YJS = Y.r*YJ  C C 140 C 141 C 142 143 C c  XP = X.J + DSJ2 XM=XJ-DSJ2 XP3=XP*XP XMS=XM*XM XJ IS ZERO IF ELEMENTS VERTICALLY ABOVE EACH OTHER IFCXJ'.EQ.O.) GO TO 140 PHIX IS VELOCITY IN *IND DIRECTION PHIX=ALOG((XPS+YJS)/(XMS*YJS)) GO TO 141 PHiXrO'. Y J IS ZERO IF ELEMENTS ARE ON SAME FLAT HALL SECTION IFfYJ'.EO'.O'.) GO TO 142 PHIY IS VELOCITY PERP*. TO WIND RIRN PHIY=2.*ATAN2((0S(J)*YJ),(XJS+YJS-DSJ4)) GO TO 143 PHlYao. IFCSI '(.D'.EO'.O'.) GO TO 144 31•J = Sl(i)«CS(J)-CSCI)*SI(J) coj=cs(i)*cs(j)tsi(i)*si(J) A is NORMAL VEL IN 'I' COORD SYSTEM 8 is  TANG'L  VEL  IN  W  COORD  SYSTEM  0031 A(J,I)=PHIY«COJ-PHIX*SIJ 0032 B(J,I)=PHIX*COJ+PHIY*SIJ 0033 GO TO 2 0034 3 A(TJ,I) = 6'.283l05 0035 B(J,I) = o'. 0036 GO TO 2 0037 144 3ij=SI(I)*CS(J) 0038 COJ=CSCI)*CS(J) 0039 A(J,I)=PHIY*C0.1-PHIX*SIJ 0040 B(J,I)=PHIX*COJ*PHIY*SIJ 0041 2 CONTINUE 0.042 RETURN 0013 END ' • OPTIONS I N EFFECT* ID,EBCDIC,SOURCE, NOLIST, NODECK»LOAD, NOM AP • OPTIONS I N EFFECTii NAME = CALCAB , LINECNT » 57 •STATISTICS* SOURCE STATEMENTS s 43,PROGRAM S I Z E B . 1700 •STATISTICS* NO DiAGNOSTTCS GENERATED NQ ERRORS IN CALCAB  10119137  PAGE P001  201.000 202.000  203.000 204.000 205.000 206.000 207.000 208.000 209.000 210.000 211.000 212.000 213.000 214.000 215.000 216.000 217,000 2ie.o6o 219,000 220.000 221.000 222.000 223.000 224.000 225.000 226.000 227.000 22e,000 229,000 230.000 231.000 232.000 233.000 231,000 235.000 236.000 237.000 23e.000 239,000 240.000 241.000 242.000 243.000 244.000 245.000 246.000 247.000 248,000 249.000 250.000 251.000  :  U ) C O  MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) 0001  0002 0003 0001 0005 0 0 06 0007 0008 0009 0010 0011 0012 0013 0011 0015 0016 (.017 0018 0019 0020 0021 0022 0023 0021 0025 0 026 0027 0028 0029 003Q C031 0032 0033 0031 0035 C036 0037 0038 0039 ooio 0011 0012  c  C  \SSEMA  10-22-T5  10119|37  SUBROUTINE AS3EMA(A,C,N,M)  c i s MATRTX F O R S Y S T E M C * S I G » D ASSC.MA ASSEMBLES T H O S E P A R T S OF C T H A T D E P E N D  REAL A(N#N).C(M,M) INTEGER i:,P.O  ON A'.  C0MMON/B1/ NWS,NSLAT,NSUt,NKA,NM2,MSV,NA,NSPS,NTEU,NTEL •  COMMON/B2/ U,CH NSPsNSPS-1 NN2 NM2+MSV NWSV=MWS+M8V =  C » * * « « LOOP 1 - ASSEMBLE NORMAL VEL E Q N S FOR ALL NWS CONTROL P T S  *****  DO 19 1=1,NWS E iS EOUATION « E=I C L00P2 - NORM VELS AT A L L NWS C , P'. DUE TO ALL NWS SOURCE ELEMS DO 2 .1=1, NWS 2 C(J,E)=A(J,i) iF(MSV.EB'.O) GO TO 19 C LOOP 8 - NORM vELS AT A L L NWS CON PTS DUE TO SOURCE ELEMS ( M U ) ON C S'.L DO 8 K=i,HSV J=NKA +K M=NWS+K 8 C(J,E)=ACM,i5 19 CONTINUE IFCNSL AT'.EO'.O) GO TO 12 C****«LOOP '[O - ASSEMBLE KUTTA EONS F O R AIRFOIL-SHAPED SLATS ***** DO 52 KS=1,NSLAT KL = NSUl+N3PS*(K3-'l) • KU=KL+NSP E = N*'S + KS C LOOP 13 - TANG VELS AT T.E*,OF SLATS DUE TO VORTEX ELEMS ON SLAT3 00 13 Ks'l, NSLAT J=NwS+K P = NSU'f*NSPS*<K-l) Q = P+NSP . . . SA = o'. DO 11 M=P,Q C  11 13  SA=SA+A(M,KL)+ACM,KU) C(J.E)=SA  15  DO 15 K=1,NA SA=3A+A(K,KL)+A(K,KU)  SA = o'. C LOOP 15 - TANG VELS AT T'.E', O F SLATS DUE TO VORTEX ELEMS ON TEST C AIRFOIL  C C  C(NKA,E)=3A IF(MSV.EO'.O) GO TO 52 LOOP l"8 - TANG VELS AT T'.E', O F SLATS DUE TO VORTEX ELEMS ON ( G N U ) S'.L DO 18 Kil.MSV J=NM2+K  18  52 12  M=NKS+K  CCJ,E)=A(M,KL)*A(M,KU)  CONTINUE IFtNSLAT.EO'.O) GO TO 20  PAGE  P00I  252.000 253.000 251.000 255.000 256.000 257.000 258.000 259.000 260.000 261.000 262.000 263.000. 261.000 265.000 266 .000 267.000 268,000 269.000 270.000 270.000 271 000 272 000 273.000 271.000 275 000 276, 000 277,000 278.000 279.000 280.000 281.000  282.000 283.000  281.000 285.000 286,000 287.000 28e.000 289,000 290.000 2 9 1 .000 292.000  292.000  293.000  291.000 295.000 296.000 297.0Q0 297.000 298.000  299.000 300.000 3 0 1 ,000 302.000 303.000  CO  MICHIGAN TERMJNAL SYSTEM FORTRAN GCH336)  ASSEMA  10-22-T5  C*****ASSCMfiLC KUTTA EONS F O R T E S T A I R F O I L ***** C LOOP 2{ - TANG VELS AT T E S T A I R F O I L T.E . DIT'. 00 21 Kai,NSLAT -  0013 0015  O032 0 053 0051 0055 0056 0057 0058 0059 0060 0 061 0062 0063 0061 0065  0066 0067 0066 0069 0070  0071 C072 0073 0071 0075 0076 0077 C07B 0079 0080 008 i 0082 0083  ELEMS  ON S L A T S  307.000  3A = u'.  DO 22 MaP.Q  22 SA=SA+A(M,NTEU>+ACH,NTEL) 21 C(J,NKA)=SA 20 3A = o'. LOOP 2 3 - TANG VELS AT T E 3 T C C AIRFOIL DO 2 3 Kil.NA 23 3A = SA + A'CK,NTEU)+A(K,NTEL)  AIRFOIL  T'.E*.  O'.T'. V O R T E X  ELEMS  ON T E S T  C(NKA,HKA)=SA  iFCMSV'.EQ'.O) GO TO 2 1 VELS AT TEST  L O O P 26 - TANG DO 2 6 K = 1,MSV J = M<2 + K  AIRFOIL  T.E*.  o'.T*. V O R T E X  ELEMS  MNwS+K C(J,NKA)=A(M,NTEU)*A(M,NTEL) 26 I F ( M S V . E O ' . O ) GO TO 61 21 C«****ASSEMf)LE NORMAL VELOCITY E O N S F O R MSV C O N P T S ON S,L'. DO 27 KM=1,MSV IsMwS+KM E=NKA+KM C L O O P 2 8 - NORM V E L S A T M S V C O N P T S O'.T'. A L L NWS S O U R C E 0 0 2 8 Jal,NWS  ( G N U ) ON  =  C(J.E)=A(J,I)  LOOP 3 3 - NORM VELS AT MSV ON INNER DO 3 3 Kel.MSV . J = Nk'AtK  C O N P T S O'.T'. A L L  MSV  SOURCE  ELEMS  ELEMS  (MU)  335.000 336.000 337.000  33e.«oo  V E L E O N S F O R MSV C O N T R O L  I=NWS+KM E=NM2+KH IFCMSLAT'.En'.O) GO TO 3 7 LOOP 38 - TANG VELS AT MSV DO 3 8 Kc1/NSLAT PsMSUl+NSP3*CK-l) OsPtNRP J=NwS+K  CON  PfS  D'.T'.ALL  POINTS  ON I N N E R  VORTEX  EiEMS  3A = o'. DO 3 9 M=P,0  39 38 37 C  3A = SA + A(M,I)  C(J,E)=3A SA = o'. LOOP 1 0 » T A N G  C AIRFOIL  310.000 311.000 312.OpO 313,000 311.000 315.OoO 315.000 316,000 317.000 318,000 319.0Q0 320.000 321.OOO 322.000 323.000 321.000 325.000 326.000 327.000 328.000 329.000 330,000 331.000 332.000 333.000 333.000  331.Ogo  MBNHS+K C(J,E)=A(M,i)  33 CONTINUE 27 C ***»*ASSCMBLC TANG'L C SIL*.***** DO 3 5 KM=1,MSV  305.000  306.000 308,000 309,000  OsPtNsP JoNWS+K  28  PAGE P002  301.000 VORTEX  PBU3U1+NSPS*(K-1)  0011  0016 0017 00 I B 0 0 '19 0050 oo5i  10ll9|37  EDGE OF  ON S L A T S  • • VELS  AT MSV C O N P T S D.T'.ALL  VORTEX  CLEMS  ON T E S T  •  339.000 339,000 310,000 311,000 312,000 313,000 311.000 315,000 316.000 317.000 318.000 319.000 350.000  351 . 0 0 0  352.000 353.000 351.000 351.000  O  MfCHICAN TERMINAL SY3TCM 0081 0035 0036  10 C C  0097 0088 0039  .  FORTRAN  C(H336)  K«1,NA 3AaSA + A ' ( K , n C(NKA,E)a3A L O O P 42 • T A N O V E L S DO  t0-22"75  A9SEMA  40  AT MSV C O N P T S D'.T.ALL V O R T E X  ELEHS  s L" *  OO'Q t'O'l  12 35  0092 0093 0094  61  * 0 0 4 2 Kil.MSV  JeNM2+K MsNwS+K C(J.E)=A(M,i) CONTINUE CO'iTlNUE RETURN END  •0PTJOM3 I N EFFECT* ID , EHCDIC , SOURCE,NOLI3T,NODECK#t.OAD#NOMAP •OPTIONS IN EFFECT* NAME = ASSEMA , LINECNT • . 57 •STATISTICS* SOURCE STATEMENTS s 94,PROGRAM SIZE a •STATISTICS* NO DIAGNOSTICS GENERATED NO CRRORS I" ASSEMA  2980  ( G N U ) ON  l o t 10137 353.000 356,000 357.000 35e.00O 358.000 359.000 3 6 0 , 000 361.000 362.000 362.500 363.000 364.000 365.000  PA6E  POOS  MICHIGAN TENMjNAL SYSTEM FORTRAN 0(11336) 0001 0002 0003 0001 0005 0006 0007 0008 0009 ooio e c u  0012 0013  C  tioie  0019 0020 0021 0022 0023 0021 0025 C026 0027 0028 0029 0030 0031 0032 0C33 C031 0035 0036 0037 0038 0039  10»22»75  10U9I38  SUBROUTINE ASSEMBCB,C,N,M) ASSEMBLES THOSE PARTS OF C THAT DEPEND ON B*. REAL B<N,N) ,C ( M , M ) INTEGER C . P . O C O M M O N / B I Z NW3,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS»NTEU,NTEL COMMON/B2/ U,CH —  NSP=NSPS-1 MN2=NM2+MSV NWSV=NHS+MSV C * * * * * A S S E M R L C NORMAL 00 i|9 1 = 1,NWS E=I  VEL EQNS FOR ALL NWS CON PTS  375.OgO  *****  L O O P i\ - NORM VELS DO 1 K=i,NSLAT  J=NWS+K P=NSUi+NSRS*(K-l)  376.000 377.000  SLATS  ON  ,  0=P+NSP 3 B = o'. 00 5 "=P,Q  5  370.000 371.000  372.000 373.OQ0 371,000  IF (NSLAT'.EO'.O) GO TO 3 » ALL NWS CON PTS D'.T'. VORTEX ELEMS  C  CCJ,E)=SB 3 B = o'. LOOP 6 -  C C AIRFOIL  NORM VELS »  386.000  ALL  NW3 CON PTS  6 K=i,NA 3B=SR-B(K,I) C(NKA,E)=OB IF(MSV.CQ'.O) GO TO 1 9 L O O P 9 - NORM VELS • A L L NWS CON PTS  C C S'.L'.  9  o'.T'.  VORTEX ELEMS ON TEST  D'.T'.  VORTEX ELEMS (GNU)  ON  00  9 K = i",MSV J=NM2+K M=NWS+K  C(JiE)=-B(M,I) CONTINUE iF(NSLAT.t'O'.O) GO TO 12 C*****A3SrMP.LC KUTTA EONS FO AIRFOIL-SHAPED SLATS DO 5 2 KS=t,NSLAT KL = NSlll+N3PS*CKS-l)  C C  398.000  100.000  *****  -- -  E=NwS+KS L O O P i l - TANG VELS # T'.E'. OF SLATS D'.T', A L L NWS SOURCE ELEMS • 00 1 1 Jsl.NWS C(J,E)=B(J,KL)+B(J,KU) • IFCMSV.EO'.O) GO TO 5 2 L O O P 1 7 - TANG VELS • T'.E', OF SLATS D'.T. MSV SOURCE ELEMS (MU) ON S'.L.  DO 1 7 K=1,MSV  ooio  001 i 0012  17 52  0013  12  C  101.000  102.000 103.000 IOI.OOO  105.000  106.OOO 107.000  108.000  109,000 110,000 110.000  111.000  J = NK A + K  MsNWStK C(J,E)=B(M,KL)+B(M,KU) CONTINUC LOOP 1 9 - TANG VELS « TEST AIRFOIL T'.E'. DO 1 9 Jal,NWS  391 .000 391.000 395.000 396,000 397.000 399.000  KU-KL+NSP  II  387.000 388.000 389.000 389.000 390.000 391,000  392.000 393.000  19  C  379.000 380.000  383.000 381.000 385.000  DO 6  378.000  381.000 382.000  -  Se=SS-B(M,I)  I 3  PAGE P001  366.000 367.000 368.000 369.000  A3SEMB  con  O0I5 O0I6 0017  'SSEMB  O'.T',  ALL NWS SOURCE ELEMS  112.000 113.000 111.000 115.000 116.000 117.000  to  MICHIGAN 0001 9015  TERMINAL  SY3TEM  FORTRAN  IF (MSV.CO',0) GO TO 24 LOOP 25 - TANG V E L S * T E S T C ON INNER . 1)0 23 Ksl.MSV  C  0016 0047 0048 004") 0 050 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0 062 0063 0064 0065 0066 0067 0068 0 0 69 0070 0071 0072 0073 0074 0075 0076  10110)38  10-22-75  A33EMB  G(41336)  C(J,NKA;aB(J,NTEU)+B(J,NTEL)  19  AIRFOIL  T.E*.  418.000 419.000  o'.T.  MSV S O U R C E  420.000 420.000 421.000 422.000 423.000 424.000 425.000 426.000  ELEMS(MU)  J s N K A +K  M=NWS+K 25 C(J.NkA)nB(M,NTEU)+B(M,NTEL) 24 IF (MSV .Efl'.O) GO TO 61 C**«**ASSEMBLC NORMAL VEL E Q N S F O R 00 27 KMaijMSV  MSV CON  PT3  ON  SHEAR L A Y E R * * * * *  427,  ISNKS+KM  E=NKA+KM IF(NSLAT'.Eu'.O) GO TO 29 - LOOP 30 - NORM V E L S • M S V C O N 00 30 Ks1,NSLAT JrNWS+K P=N3U1+NSPS*(K-1) 0=P+NSP SB = 0'.  C  P T S D , T ' . VORTEX  E L E M S ON S L A T S  DO 31 =P,0 M  31 30 29 C C  3BsSB-B'(M,l) C(J.E)»SB 3B = o'. - LOOP 32 - NORM  •  » MSV C O N P T S D.T.  V O R T E X E L E M S ON T E S T  AIRFOIL  32. C C  ^. VELS  s'  DO 32 K=l,NA. 3B=SB-B(K,I) C(t«A,E) = SB - LOOP 34 - NORM VEL3 •  MSV C O N P T S O ' . T ' .  MSV V O R T E X  ELEMS  (GNU}  L'  DO 34 K s l . M S V J=NM2,K  34 27  M=NW8+K C(J,E)=-B(M,I) CONTINUE  C*****AS3CMRLC TANG'L DO 33 KM=1,MSV  . V E L EONS  F O R M S V C O N P T S ON  . . SHEAR L A Y E R * * * * *  I=NW5*KM  E=NM2+KM  C  36 C S.L  LOOP 36 - TANG V E L S P M S V C O N P T S D . T , A L L N W 3 S O U R C E E L E M S DO 36 Jrl.NWS C(J,E)=B(J,I) LOOP 41 - TANG VELS » M S V C O N P T S O'.T". M S V S O U R C E F! E M S ( M U ) ON  DO 41 Ksl.MSV 0077 JrNKA-tK 0078 0079 M=NwS+K 0080 41 C(J.E)=H(M,i) 0081* 35 CONTINUE 0082 61 CONTINUE 0083 64 CONTINUE 0084 RETURN 0085 END •OPTIONS IN EFFECT* ID,EBCDiCSOURCE,NOLIST,NODECK,LOAD,NOMAP  ON  ooo  42e, ooo 429.000 430.000 431.000 432.000 433.000 434.000 435.000 436.000 437.000 438.000 439.000 440.000 4 41.000 4 41,000 442.000 • 443.000 444,000 445,000. 445.000 446,000 447.000 448.000 449,000 450.000 451.000 452.000 453.000 454.000 455.000 456.000 457.000 458.000 458.000 459.000 460.000 461 .000 462,000 462.500 463.000 464.000 465.000 466.000  PAGE  P002  SSEMO  MICHIGAN TERMINAL SY3TEM FORTRAN G ( 1 1 3 3 6 )  10H9I3B  SUBROUTINE A S S E M 0 C 0 , M C 3 , S I , N , V T I . N 1 , V T 0 , N 2 ) A3SEMD ASSEMBLES R ' . H ' . S . VECTOR FOR SYSTEM C * S I G « 0 REAL 0<M),CS(N),Sl(N),VTI(Ni),VT0(N2) INTEGER P . O C0MM0M/H1/ I J * S , N S L A T , N S U 1 , N K A , N M 2 , M S V , N A , N S P S , N T E U , N T E L C0MM0N/P2/ U , C H NSP=NSPS-1 NWSV=MWS+MSV  0001  oooo  16e.000 169.000 170.000 171.000 172.000  173.000 171,000  NN2=NM2+H3V L O O P 13 - NORMAL ONSET 00 13 1 = 1 , N W S 13  PAGE PO01  167.000  /  0002 OC03  '0005 0006 0007 OOOB 0009 0010 0011  10-22-75  175.000 FLOW VEL AT ALL NWS CON PTS  176.000 177.000  niti)=u*stm  I F ( w S l . A T ' . E O . O ) GO TO 11 L O O P 15 - T A N G ' L ONSET FLOW VEL3 # T'.E*. OF SLATS 0012 D O 13 K s t , N S L A T 00 13 I=NnS+K C011 P =l v S U l + M S P S * ( K - l ) 0015 0=P+NRP 0016 15 n'(I)a.U*CCS'(P)+CS(D)) C LOOP 1/| - T A N G ' L ONSET FLOW VELS • T ' . E . OF TEST AIRFOIL 0017 11 D ( N K A ) = -0*(CS(NTEU)->CS(NTEL)) 0018 I F ( M S V . C O ' . O ) G O TO 16 L O O P 17 - NORMAL ONSET FLOW VEL • ALL MSV CON PTS 6.N 0019 D O 17 K=1,M3V 0020 JsNWS+K I=NKA+K 0021 D(I)=H*SrCJ) 0022 17 L O O P 18 - T A N G ' L ONSET FLOW VEL • ALL MSV CON PTS ON C * PRESCRIBED TAN ' L VEL THERE*. DO 18 K=1,MSV . • 0023 J s h w S+K 0021 I=NM2+K . 0025 48 D(I)=-U*C3(J)+VTI(K) 0026 CONTINUE 0027 16 RETURN J02C END 0 029 IN E F F E C T* I D , E B C D I C , S O U R C E , N O L I ST,NODECK,LOADiNOMAP *OPTJONS NAME = ASSEMD , LINECNT » 5T *OPTIONS IN E F F E C T * SOURCE STATEMENTS = 29,PROGRAM S I Z E n 1191 •STATISTICS* •STATISTICS* NO DIAGNOSTICS GENERATED NO CRRORS IN ASSEMD  i7e.o6o  SHEAR l ' .  SHEAR L ' . •  -  179.000 180.000 181.000 182.000 183.000 181,000 185.000 186,000 187.000 188.000 189.000 190.000 191.000 192.000 193.000 19a.000 195.000 • 196.000 197.000 19e.00O 199.000 500.000 501.000 502.000  u  •  4^  MICHIGAN TERMINAL SYSTEM rORTRAN G(41336) ooof  c ooo?  0003 0001 0005 OC06 0007 OOOB 0009 oo l o 9011 0012 0013 0011 0015 0016 0017 0018 0019  0023 0021 C025 0026 0327 002B 0029 0030 C'031 C032 0033 0031 0035 0036 0037 003B 0039 0040 004 1 0012 •t 043 eon • 0015 0046  10-22-.73  10119t 36  ;  coMMON/nn/ U,CH  C0MMOH/B3/NUl,NWUl,NU3,NWU2,NLl,NWLi,NL3,NHL2,NS0Ll.NSOLSL,Nri, i  C 1 c  2  ;>o2o  0 021 0022  CPS  SUBROUTINE CP3<CP,VTT,XX,YY.CS,SI,N1,3TC,N2,GAM,N3,MU.GNU,N0,A,B) " CPS CALCULATES VEL, PRESSURE » ALL CON PTS'. REAL A(Nl,Ni),H(Nl,Nl) REAL SIG(NJ),GAM(M3),MU(N4),GNU(N4) REAL VTT(Nn,CP<Nl),XX(Nl),YY(Nl),CS(Nl),SI(N15 INTEGER P,0 COPMOM/Hl / NWS,NS1.AT,NSU1,NKA,NM2,M3V,NA,NSPS,NTEU,NTEL  c c  3 4  NFLAT,N3PF,NII  NN2sNM2tMSV NWSV=M«3+MSV NVNsMwS+MSV NSU2=NSU1+MSP3*NSLAT-1 NSP=NSPS-1 iF(N3LAT'.En'.0) GO TO 2 110 1 K = i,N3LAT I=NWS+K  GAM . SLAT VORTE* STRENGTH DENSITIES nAM(K)=srGm  GAMM - TEST AIRFO VORTEX STRENGTH DENSITY GAMMsSIG(NKA) IFCMSV.CQ'.O) GO TO 4 00 3 K=1,M8V IsNKA+K J=NM2+K • MU S'.L'. SOURCE STRENGTH DENSITIES MUCK)=SiG(I) GNU - VORTEX STRENGTH DENSITIES FOR SHEAR LAYER GNUCK)=SIG(J) LL»0  DO } 2 lel.NVN li=I+N3P  ' 52 C  M  5  C c  "  LK="SultLL*NSPS I F C I . C f " . ' , 1) WRITE(6,52) FOIiMATClHl) . , • •• • • VNSf.vTST - TOTAL NORMAL & TANG'L VELS DUE TO SOURCE ELEMS VN3f=0, VT3T=0. AS =ol B3MiO. DO 5 .i=i#Nws • VNSf=VNRT+A(J»I)*SIG(J) VT3T=VTST+BCJ,I)*SIG(J) ASM=A3M+A(J,I) BSM=BSM+B£J,I) A3 = 0. B3 = o'. IF CNSLAT.EO'.O) GO TO 8 DO 6 K=ltNSLAT ' • . P=N3Ui'tNSPS*(K-l) QsP+NSP  P,Q - {ST & LA3T CON PTS OS A SLAT AP.BP - NORM * TANG VELS DUE TO VORTEX ELEMS ON SLATS AP = o'.  503.000 504.000 503.000 506.000 507.000 508.000 509.000 510.000 511.000 512.000 513.000 514.000 515.000 516.000 517.000 518.000 519.000 520.000 521.000 522.000 523,000 524.000 525.000 526.000 527.000 528.000 529.000 530.000 531.000 532.000 533.000 534.000 535.000 536.000 537.000 538.000 539,000 540.000 541,000 542,000 543.000 544.000 545.000 546.000 547.000 548.000 549.000 550.000 551.000 552.000 553.000 554.000 555.000 556,000 557.000  PAGE P001  MICHIGAN TERMINAL SYSTEM FORTRAN G(11336)  PS=ns+BP  5 055 0056 0057 0058 C059 0060 0 061 C  0061 Oo65 0066 0067 0068 0069 0070 0071 0072 0073 0071 0075 0076 0077 0 078 C079 0030 0031 0032 0083 OoSI 0085 9086 0C87 0088 0039 0C90 0091 0092 0093  10-22-75  BP = 0 . 00 7 M=P,Q AP=AP+A(M,I) 8PaflP + r>(M,I) AP.= AP*GAM(K) nP=BP*GAH(K) A3=AS+AP  0017 0018 0019 OcSO 0 051 0052 0053 0051  0062 0063  "PS  •C  S'-L'.  & TANG VELS DUE TO VORTEX ELEMS ON TEST AIRFOIL AT.BT - NORM AT = 0 . BT*o'. n o •> . ! = ! i NA AT=AT+A(J,I) BT=BT+R(J,I) AT=AT*GAMM RTsHTftGAHM t. TANG VELS DUE TO MSV SOURCE ELEMS ON INNER EDGE OF AM,B« - NORM  AM = '. BM = u'. NORM C . AG,BO C EDGE OF S ' L ' . AG = O'. BG = o'. U  & TANG VEL3 DUE TO MSV VORTEX ELEMS GNU ON INNER  IFCMSV.EO'.O) GO TO 11  no  i o K=I,MSV  J=NWS+K  AG=ACH A ' C J , I ) * G N U C K )  10 11 C C  C  flG=HGtB(J,I)*GNUtK) AM = AM+A(.J,I)*MU(K). RM=RM R J,I)*MU(K) VNST=VNST+AM VTST=vTST+BM VNVT.VTVT - TOTAL NORM t TANG VEL DUE TO ALL VORTEX ELEMS VNVTs-HS-BT-HG VTVT=AStAT+AG VNOT,VTOT - NORM & TANG-VEL DUE TO UNIFORM STREAM U VNOf=-U*3ICI) VTOT=U*CSCI) VNT=VMnT+VNVT+VNOT VTT'ci) = VTST + VTVT + VTOT VKL V K U - T E S T AIRFOIL T'.E*. KUTTA VELS IF(l'.r"'.NTEU) vKUrVTTCI) IF(l'.EQ'.NTEL) V K L = VTT'(I) CPCI) = 1'.-VTTCI)*VTT(I) iFti'.r's'.n GO TO ni IFUl'.EC.NUl)' AND*. CNWUl'NE',0)) GO TO 12 IFC'CI Ea'.NU3),AND(HWU2.NE',0)) GO TO 13 +  C  r  IF(CI.EO'.NLI) AND  (NWH.NE.O))  GO TO 11  tFCJl,EO.NL3)lAND.CNWL2.NE'.0)> TO 15 IFCCI EO'.N8OLI).AND'.<N80L3L'NE,0>) GO TO 46 IFC(I.L (!.NF1)'.ANO'.(NFLAT'.NE.O)) GO TO IT IF((NSLAT.EO'.0).OR".(l'.GE',NSU2)) GO TO 70 iFCCl'.Eo'.LKj'.ANn'.CLL'.LE'.NSLAT)) GO TO 18 CONTINUE G  :  ,  70  0  10H9I38 558.000 559.000 560.000 561 .000 562.000 563.000 561.000 565.000 566.000 567.000 56e,000 569.000 570.000 571.000 572.000 573.000 571.000 571.000 575.000 576.000 577.000 577.000 57e.000 579.000 580.000 581.000 582.000 583.000 581.0(10 585.000 586.000 587.000 588.000 589.000 590.000 591.000 592.000 593.000 591.000 595.000 596,000 597.000 598.000 59").000 600.000 601.000 602.000 603.000 601.000 605.000 606.000 607.000 608,000 609.000 610.000  PAGE P002  MICHIGAN  TERMINAL  009q 0095  00  CO'6  0097 00<"8 C099 0100 '0131 01 32 C 1 03 0 101 0105 0106 0107 010B 0109 0110 0 1 11 Ci 12 0 113 0114 01 15 0116 o i 17 0118 0119 0!20 0121  41 30 42 31 43 32 44 33 45 34 46 35 47 36 48  Cl'22  37 67 69  9'24  68  0123  0123 0126 0127 0123 0129  49 38 51 12 13  0130 0131 0132 0133 0134 C135 0136 0137 0138 0139 ciio  0111 0142 0113 0144 •OPTIONS  IN  SY3TEM FORTRAN  G(H336)  CP9  10-22"T5  lOt t<9t3R  iF((I.EO.Nii).AND.(MSV'.HE.O)) GO TO 49 611.000 612.000 GO TO 12 F0RMAT.C4X, 'VNST' ,4X, l VNVT ', 4X, I VNOT t, 4Xi 'VNT ' »5X» 'vfST',4X,'VTVT•,6 1 3 . 0 0 0 614,000 1 IX,'VTOT'»4X,'VTT',5X,'ASUM',4X,'BSUM',5X.'YY',5X,'XX',5X,'CP'i 615.000 2 6X,'SiG') 616.000 WnifE't6,30) 6 17.000 FORMATC'MAIN AIRFOIL') 618.000 GO TO 51 619,oro WRlfE(6,31) 6 2 0 . 000 *^n.i.\,-('o. v-.' Ri«.c-: c w i l w M A L L ' ) 621 . 0 0 0 GO f o 51 622.000 wRITEi'6,32) 623,000 FORMAT('UPPER LEFT SOLID MALL') 624.000 GO fo 51 625.000 WRITE'c6,33) 626.000 FORMAT('LOWER LEFT SOLID WALL') 627.000 GO TO 51 628.000 WRITE(6.34) 629.000 FOfiMATCLOUER RIGHT SOLID WALL ) 630.000 n o fo s i 631 . 0 0 0 WRITE(6,35) 632,000 FORMAT('G0LIO STREAMLINE WALL') 633.000 GO TO 51 634.000 WRlfE(6,36) .. - . 635.000 FORMAT('FLAT SLATS/NO KUTTA') 636.000 GO TO 51 637.000 IFd'.ME'.NSUl) GO TO 67 638,000 wRlfE(6,37) 639.000 FORMATPUPPER SLATS') 640,000 LL=LLtl 641 . 0 0 0 IF(LL',GT'.NSLAT) GO TO 12 642.0Q0 MRITE(6 68) I,II 643.000 FORMAT(/,'SLAT *',I3.2X,I3) 644,000 GO TO 51 645.000 WRlfc'(6,38) 646.000 FORMAJ£'STREAMLINE FOR SHEAR LAYER") 647.000 WRITEC6.40) 648.000 WRI TEC'', 13) VNST, VNVT, VMOT, VNT, VTST, VTVT, VTOT. VTT'(I), ASM, BSM, 649.000 YY(I),XXci),CP(i),SIG(I) 650.000 FORMATdX, 10F8.3,2F7.2,2F8,3) 651 . 0 0 0 IF(wSLAT.CO'.O) GO TO 20 652.000 n  1  f  WRlTi:(6iH)  GAM  20 18  WRlfEC6,18) GAMM FORMAH'GAMMs • ,G12.5)  14 15 16 19 17  WRlfE(6,15) «U KRITEfh,16) GNU FORMAT ('GAMs ', i0G12*.5) FORMAT('U=',10G12.5) F O R M A T ( ' G N U = ' , iOG12'.5) WRlfE(fc,17) VKU,VKL FORMAT (' VKU=' ,G12'.5,2X, ' V K L » ' , Gl2'.5) RETURN END  -  I F ( M S V . E Q ' . O ) GO TO  EFFECT*  19  M  10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAO,N0MAP  653.000 654.000 655.000 656.000 657.000 658,000 659.000 660.000 661.000 662.000 663.000 664.000  PAGE  P003  MICHIGAN TERMINAL SYSTEM FORTRAN G(11336) 0001 0002 0003 0001 0005 0006 0007 000B 0009 0010 0011 0012 0013 CC11 0015  0016 0017 0018  C 5  i-ORCES  10-22-75  SUBROUTINE FORCES(CP,XX,YY,DX,DY,OS,VTT,N.U.CH,Nl,N2,XC.YC) REAL C P C ' ' 0 « X X ( N ) , Y Y ( : N ) , D X C N ) , D Y C N ) , D S < N > , V T T C N ) XCYC - CENTER OF BODY WRITEC6.5) N1.N2.XCYC • FORMAT (' FORCES ON BODY * • , IS, ,',13,3X, CENTER AT (',F7*,2, t >.>.F-:s,n'i 1  1  CLTBO COTBO*,  C 1  C C C  CMO=CMO+CPCI)*((XX(I)-XC)»DXCI)*(YV{I).YC)*DY(I))  CLT - TUNNEL LIFT COEFF'. CLT=CLT/CH COT - TUNNEL DRAG COEFF*. (THEOR »Y ZERO) CDT=CDT/CH C*0 - TUNNEL MIDCHORD PITCHING MOM, COEFF*. CMOiCMO/CH/CH  CM'I - TUNNEL OUARTERCHORD PITCHING MOM'. COEFF*. CMflsCMO-CLT/''. C ciRC - CIRCULATION ABOUT BODY C CLC- LIFT COEFF'. FROM CIRCULATION 0020 CLC=2,*CtRC/CH/U 0021 wRITEC6,2) CLT,CDT,CM0,CM1 0022 2 FORMATCCLTi'.Flo'.S^X.'CDTa'.Flo'.-S^X.'CMOs'.Fio'.-S^Xj'CMfl 1 F10'5). 0023 HRiTEf6,3) CIRC,CLC,PER 0021 3 FORMATCCIRCa'.FlO'.S^X.'CLCo'.FlO.SjZX.'PERIMBljGia'.S) 0025 RETURN 0026 , END •OPTJONS IN EFFECT* 10iEBCDIC,SOURCE,NOL1ST,NODECK.LOAD,NOMAP •OPTIONS IN EFFECT* NAME = FORCES , LINECNT " 57 •STATISTICS* SOURCE STATEMENTS s 26,PROGRAM SIZE B 1276 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN FORCCS 0019  C  CM0=0. CiRCeO. PER=0*. DO 1 I=N1,N2 CLT=CLT-CP(i)*OX(i) CDTBCDT+CP(t)*DY(I) C1RC=CI«C+VTT(I)«0S(I) P E R - ROOY PERIMETER PCRiPCfND3(i)  1  10H9I39 665.000 666.000 667.000 666.000 669.000 670.000 671.000 672.000 673.000 671.000 675.000 676.000 677.000 67e.000 679.000 680.000 681.000 682.000 683.000 681,000 685.000 686.000 687.000 688.000 689.000 690.000 691.000 692.000 693.000 691.000 695.000 696.000 697.000 698.000 699.000 700.000  PAGE P001  14  o o o  J  o o o o o o o o o o o o o o o o o o o o o o o o o o o c - o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O OOO O O O O O O O o o o o oo o o o o o o o o o o o o o o o oo o - o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  OVOJ c o - o to .•^b^o^«(^owfJloc.^o^«(^o* -H r j r j r j r j r j r j r j r j r j r j i o r o M M M KI KI KI KI n c  r " n a u i o V J tr*o- r j i n en* o o o o o o o o o - - « w -  ru  <  ' J'M^rtnc^««^o»^^JW!^^n =r =r ~ — — — — - L O LfVLn L H L O in ,  <, ~  ww  co _ » •>  ULJ« C»JZ O z •»-» o »to > Ci » X O  n Lu »z •» 3  c » a . «~ X z w 2 >- >-  z  *UD » > - —I »_— > - z *-« e  X  O LO LO  -« CO *-  a  Z X X ucx UJ  -  1 -  ru  c a .  -  a.  L J >- > -  u » * L_J X '-x ^ Z » Z CL - r-4 a c - w I H U X ^ — . • — X X If  cc rLO CT fl  r u f\ * r i n e» o o o o 0 0 0 O O O O  .  U * J s < z O *-» CJ Lu » 1 N O I I CT z. « L o c • -K •—*- • • 1— C" u o u u u  r - cr (D D < < < L i t ^ 3 L J L J IU II U_ < r > cr cr cr ti. •--*-«  o o o  e  ru ru r j r j  >- ^ » Z X. Z >- "  h-  z  LJ LU  -  * Z X X X ^ 0 or 01 C O  C O  C  U I  a  a.  Z IU •» U ' H '_> X <  h-  ;< 11 1 It  co o o - — • r o ^n * r  i n o*r*- 0 0  11  t~ a  o O  0 0 — • — « —• G> O O OCTC> C>  O  C ' C ' ^ ' O O C ' C *  o —<; C (  O  —• ^ O O  O  O O O O  ru j ro > «!-«>o : o x » X X  U U  < 'J  11  11 11  11  ^ •  11  o " —« a i KI c IP o r - c o * o ruruojruftJPJfurvjruruM 0  H  CL w o Z LU J H ' XOO+.-C ^ ^ » » Z Z . x >• - • —• Lu '—I "_' ^ U . . • .-. I I X . il- X < X J U I I ^ * < * wv^t-i-k c C 1 H M H i ^ j H h i n i 1 rj. v « ~ ^ L". _ . C U O H h H H H . ^ o « " » » MX>. «^ a o JO: c c . ^ * ' ' '-> a. a. 1 a n w n -w • X X O X X o COUU.li. I ' J O Z 11 o u. u . •H s: x >- C3 X >• L> » x >- x > C X >• X • - • < U « M > - H M  •j-> I— v-  * W Z . I- C. U G C LJ < _J • J II n » - £ c:. O C l - II n • 1- o o cr u LJ G LJ UJ c: c X i <: o  1;  • o X >- + r\! e: t u I I ru r u x . »-i _ J » Z i-t (_) c: u _ H U C cr.n +. x >- x > • I x : I Q _ « — X >- II II  - J '-J CO *  »-N U J _ 4  o o z z H < <  0  0  0  0  0  0  0  c> C C> C- t > i»O  0  0  0  o c» ->  • —< <\r KI c r t n o > r - co  KIKIKIMMMMK\ 0  0  0  0  0  0  0  0  o o o c - o c » o o  cr• r u " » n er m c r c j c c cr T>  & cr> o cz>  O O ' O O O O  MICHIGAN  0045  TERMINAL SYSTEM FORTRAN  C  0046 0047 0048 0049 0050 0051  POINT  10-22-75  DYCI)=YM(K)-YM(J) D3CI)=S0RTCDX(I)*0Xa)+DYCI)*DY'(I))  5  CS(i)sDXCI)/DS'(n 3i(i)=DY(I)/D3'CI) RETURN  END  • OPTIONS I N E F F E C T * 10,CMCDIC,SOURCE,NOLIST,NOOECK.LOAD.NOMAP • O P T I O N S IN E F F E C T * N A M E = MOOPRO , LINECNT * S7 •STATISTICS* SOURCE STATEMENTS = 57.PROGRAM S I Z E s •STATISTICS* NO D I A G N O S T I C S G E N E R A T E D E R R O R S I N MOOPRO  10119t40 756.000 757.000 758.000 759.000 760.000 761.000 762.000 763.000 764.000 765.000 766.000 767.000 768.000 769.000  COORDS.  f  0053  NO  M=MP-I XX.YY - NOW M O D I F I E D C O N T R O L  DO 5 I=L1,L2 J=l-Lt+1 K = .W1 XXCl) = (XH(J) + XM(K))/2' YY(I)=(YMCJ)+YM(K))/2. OX(I)=XM(K)-XM(J)  0052  0054 0C55 0C56 0057  HOOPRO  G(41336)  2700  PAGE  P002  MICHIGAN TERMINAL SY3TEM FORTRAN G(11336)  RE  0001* SUBROUTINE RE(A,N,M,LA) 0002 REAL A(N,M) OC03 READ(LA) A 0001 RETURN 0005 ENO •OPTlGNS IN EFFFCT* 10.EBCOIC,SOURCE,N0LIST,NOOECK,L0AO,NOMAP •OPTIONS IN EFFECT* NA«C = RE , LINECNT • 57 •STATISTICS* SOURCE STATEMENTS * 5.PROGRAM SIZE s •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN RE MICHIGAN TERMINAL SY3TEM FORTRAN G(11336)  WR  10-22-75  WRO  SUBROUTINE WPD(D,M,LD) 0001 REAL D(M) 0002 wRITE(LO) D 0003 RETURN OC01 0005 END • OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOAD,NOMAP •OPTioNS I M EFFECT* NAME a wRD , LINECNT « 57 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN WRD NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTION TERMINATED  13IG  PAGE P001  770.000 771.000 772.000 773.000 771.000 410  10-22-75  C00 1 SUBROUTINE WR(A,N,M,LA) 0002 REAL A(N,M) 0003 WRITE(LA) A 0001 RETURN 0005 END • OPTIONS IN EFFECT* In, EBCDIC, SOURCE , NOLIST, NODECK, LOAD, NOMAP •OPTIONS IN EFFECT* NAME = WR , LINECNT * 57 440 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o *STAT13T1C3» NO DIAGNOSTICS GENERATED NO CRRORS IN WR MICHIGAN TERMINAL SY3TLM FORTRAN G(11336)  lOllRJll  10-22-T5  lOllRHl 775.000 776.000 777.000 778.000 779.000  10ll9t11 780.000 781.000 782.000 783.000 781.000  38B  PAGE P001  PAGE P001  152  Appendix List  11.  o f Equipment Used  Description  Instrument Barocel Type  511  Barocel Type  Pressure (10mm  Sensor  Hg  =  Electronic  10 V o l t s )  Manometer  pressure  transducer f o r  pitotstatic  4-1/2  digit  windspeed  1018B  tube  measurements  voltmeter f o r  from  pitotstatic  tube. Barocel  Signal  Type  1015  Disa  Digital  Type  55D31  Conditioner  amplifier pressure  Voltmeter  3- 1/2  for Barocel' transducers.  digit  windtunnel  voltmeter f o r balance  measure-  ments . Digitec Model  Digital  Voltmeter  4- 1/2  digit  windtunnel  2780  ments  & Northrup  Indicating Model  Amplifier  measure-  strain  measurements,  gauge a m p l i f i e r f o r  Aerolab windtunnel  balance.  9835-B  Druck pressure Model  Microvolt  balance  and S c a n i v a l v e  pressure Leeds  voltmeter f o r  PDCR-22  transducer  for  Scanivalve  surface  airfoil  pressure  measurements  154  157  160  161 -4-  '  i I  1  •14-4 1  1 1  j  1 1  i  i  1 1 i i_! I 1  i a  !  i i 1 > i  ! !  1 >  1  !  i  i  l :  j  i ! !  1  1 i 1 !1 >1 i ( i !1 [  1 I  1 ! 1 1 I  ! i i  i i i i < i I  ...1  1  M l  i  !  1  !  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TTT-s l o t t e d - w a l l open-area r a t i o : Experiment r -j-j-H-F i g u r e 7.1 V a r i a t i o n o f a i r f o i l l i f t c c> e f f i c i e n t w i t h 1  1  J  0  1 1  1 1  U  1  L  L  L  1  s  !  1  1  171  172  on r a t i o o f l i f t - c u r v e NACA-0015 a i r f o i l  slopes  for  dlX' j± — H~h~r  173  Figure  7.4  E f f e c t of a i r f o i l s i z e f o r NACA-0015 a i r f o i l :  on l i f t - c u r v e Experiment  slope  174 1 I.U ! i i i ! 1 i ! 1 i  i  M  I  1  M  | i ! j i l l ! ill! !M Ii l'  -  M i l  -Iii -  1  1  Ll L Ml. ! ' iM 1M l L M i 1 l ! i  i i ! i i  -  1  i  l  i i  i  L. I  i  1 1i i i i 4. T  -  1r  i  L' J  j._  M M  11  I  i I i i i  1  ;  i  -  - P-\t  s J_"L r< M M  1! 1t1 1  pu e  c  i  y  j—  1  v i  i  u  1 1  '  1  l  1  >—\  C  I Jj ta J: tXrV (A  _  *"  i  j j  i  *  1 i  1  1 1 1  I1  f C.  n  1  i 1  i j i  1  -H--  1  1 1  1  1!  i  1 /  1  i !  j  1  ; j '  Y !I  u n !i u  1  | — 1 —  5ir&\  (j  '*!  j  1 O ! t O j I  1!  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'  !  i : \  1 1 !  1  1  "1 ;  |  l  1  1-  1  1S> T1~ P 1 1 1  4  -4  1  11  "7*- i  1  i  i ...  1 1 1  1 1 1 1  1  I i  1 1  j  i  1  1  i  i  1  [_  I 1  j! 1 1i I ! 1  j  44 1 1  1 1 1  1  1—  1  4 11 | 1 I 1 1 1 -)-  1  1 1 1 i 1 1 1  1  4  1  |  1  1  1  1 1 1 I  1  1 1  1 -- 1 1  !  i  " I T  j  N V  «S  1 1 1  1 1  1  i  1 |.  II*"  444--  I1  11 1  1 4 1  1 - 4 . 4i T rV 1--i—ra  ^&  1 ! ""S — — .{. \m J__ 11 J i_ , „,„,... 11 1 1 1 1 1 1 1 II 1 11 1 1 Ti n i n •|" 1 M M 1 11 i 1 1 i ! 1' ! iI M i r 1 j ] "11 1 I 11 • 1i I 1 1 I I I ! 44-44 1 I 1 11 1 1 i l 1 1II TI i 1 I i I : 1 i J_l 1 I 1 1 1 1'i I 1 1 • T~| 1 - i l l . L . i j !.. .1. i_ 1 1 ;  1 1 1  J  TT T j j i 1 1 1 • 1 " ! 1.  44 1 1 41 - 1 1  1  J+t>  |  1 1 1 i 1 -7- 1 ! 1 1 11! I 11 1 1 1  1 1 1  i  1 1  -4 - ~ -1=  1 1  |  1  11 —4 1  1  I  ''•  i L-U  1  v  >  1 1  ;  1  i  —  1  1  1  1t  1  1 1  :  I  t1 1i 1 1 I 1 ! 1 t 1 1 1 I 4—j— ' I  ! ! i  ——  t 1 1 1  1  | | ; 1  1 •1 1 i1 M M  !i 11 1 1 i1  1  1  44—4i • 11  1  1  i  _  ;  11  1  1 1  1 1  i  1 1 1 1 1 1i 1  - +  1  1 !  1 j 1  itf!  -t—  1 1 1 1 1 1 1 1  1  1 1 I 1 i 1 11 I  1  —1i 1  il  1  -4  1  I 1 1 ; IT  i  1  -j-  j •1  t l_  1  1  11 4 - 1 1 11 1__. 1  1 1 I 1  ....  i  1 1  f  I  j1  1 1  —I— 11 ..!., ; 1 1 1 4-44 ; t  1  \  1 ..  T\  t  i i  1  1 1 i | 1 1 1 •KI 1 1 1 111 1 1 1 1 ^ T 41  1 1  i i  i i  Figure  1  1 :  I  1 1 1  i  !  .!..  *>»•  -j  !  1  1  1  1  1 1 1 I I 1 1 1 I F i ~I  1  \i 1 1  I 4  1  I"! 1 1 : 1  1  1  1  1  «1 1 1  1  J  1 "1— 1  t  1  1  1 1  •1 1 ~i—1—r 1! 1 1  t  1  1  i  !  -  1  .1 1 1  1 1 1  1 |  M 1, 1 1 1M 1 ! M  1 i i 1  1  1 i  M I iM M  r  1  1 1  i  1  1  1  1  1 - f  tn  1  1  1' 1 1 i I  !  —  1 1  j 1  !  1"  i 1! | t  1  *>  1  if  i  1  1  1 1 n Li ! 11  1 1  — rl  »-4 1 1  1  i.  i  1  !  1  1 I  1  1 1 1  1 1j ! ! 1 ' i | i 1 1 I i -J—J—  i j >  n u i  | 1  I  1  :  i  I  _  4  s  1  i  L 1  1 1 t i 1 iitrr \ U TT:  1  4  i '!"  1  1! !  1  1 ^ 1  1  1 1  1 j 1  1  i  >wS4_  i  1  !  i  1 ! 1 1  ->  I 1  i i  1  s  1 j I  1  i  U r f  1 1 ' 1 1| 1 1 —L 1 ! i ( 1 1 1 1 - -L. |  v\  I 1  4"  M  1  1 i  T  1  -V  1  ! 1 1 1 1 1 Mi i1  1  1  4  I  1  i  1 1  i  444-  1  III 1 1  1  !"  | 1  I  —  t  ! 1  1  11  \ l  1 |  1  i  1  4  1 !  1 1 1 1 !1 T 1 ! 1 ! 1 1 1 ; 11 :1 Ti i I 1 11 i 1 M M 1 rr 1iI 1 i1 1 1 ~i—p" 1 1  1 T  1 r 1 1 1 1—  ! I 1 1 !1 ! I i -~J~ 7I1 •• 1 • * 4 -r IT •fiV 1 M \i 1n 1X2 LI 1 nJ 0C r - a L V rJ - V J 1O 1I M M Ml . M M M 1 1 1 1 ey f-t Y~ m r+~ 4 4 n 4r-i =41 rr-11 1 ! n 1 J ii r LI 1 1 I 1 1 r ' 1J J L ti  i 1  1  1 |  -4  1 1 1  V  \  i  1  -t  u  - ii^-1 1  -V4-  ! i  c  T  1  y  \  Tt-1 J  1 i 1 I  I  Tl  \  1 1 1  i  1  i 1  1  Ii  —  rr u  V  1  1  _l_  r j  4  - 4f J  ] j 1 1 1 • 1 1 i 1  4+44  !  -t-  1 I] I  t •  1 1  ~4—  •Jfo.t  r  I  — {— • f-— T ~ i  1  1  1 i i i I ! 1  |  Ph—4I •~W -4-44  |  -i_LI_i_ M M — _ M | 1 | _i " 1 1 T: 1 MM M i i ; 1 j ! ' 1 1 M M .L t i i I 1 1 1 1l 1 1 1 ! i 1 1 1 1 1 I 1 1 1 1 1 1 I I i_i I i i - -j1 1 |1 I ! 1 ! 1 ] 1I i I i 1 1 i 1 i 1 |1 1 i 1 1 1 1 i I 1 i 1 \i 11i 1 i 1 1 1 1 1i 1 -}- 1 i | M ' 1 1 I I ! 1 j—r 1 1 | | | 1 1 1 1 1 1 1 i . 1 M 1 I J 1.1 v /• C h 1 /\ iv J t\ \ VSCT: I 1 1 i \J M i 1 111 I1 1 M 4TT — : — r i T—1— - 1  zhLtT _  t  l-H-i  - i !  i  i i  i-S w  r  ——I—I—  1  -i—  ...  id:  i1 T! i 1  I  -L  VX  ; i !  | {  !  1  ' i i  ;  i i i  1i ! | ' 1 1; .  : ; ! i  1  ...LA I.  j lH-hi-  _i i i M .f i ! ^  i  T+ ;  | 1  1 1 1 — i — 1 1  1  i M i  E f f e c t o f r e d u c e d c i r c u l a t i o n on 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 : T h e o r y ( A p p e n d i x 8)  v  1  Figure  8.2  M o d i f i c a t i o n of a i r f o i l p r o f i l e to reduce t h e o r e t i c a l c i r c u l a t i o n t o m e a s u r e d v a l u e : T h e o r y ( A p p e n d i x 9)  03 O  181  (a) Theory  F i g u r e 8.4  Figure  8.5  The s h e a r l a y e r , i n the. p l e n u m surrounding the s l o t t e d wall  chamber  CO  184  Figure  8.6  E f f e c t of d i f f e r e n t types of w a l l boundaries on r a t i o o f l i f t c o e f f i c i e n t s : T h e o r y  00  VJ1  186  4 T1"  III!  M  T ~  -!  ! 1  4  : :  ! i  1  > M  i  TTTT"' rr i r : i  I  !  !  !  1  L -;~T  ...... 4J . j  1  ! 1  1  1 !  i  i  !  !  !  III! M M ,  ; 1  M i l . M M \ 1  1 1 1  1 ! ! :  i  1  ! I  1  1  <  te  1 1 1 ! 1 1  4  1  i  44  Ml  1  -r- r r 1 1 i i  i  i  i  1  !  :  1  1 '  1  i i  ' i i  i i i  1  I  i  i  i  i  -hi  4ir  4 i  i  t  1 1 1 1  1 | ! 1  i i  i  i  i i  i  i  1 1 1 M  l  I  ! 1  i  I  i  ! 1  i  44  1  1  i  i  i  !  : I i I  i  I  1 i 1 1  I  1 ! 1  1 1 j  I  r4  i  11 1 1  l  j  1 1  11 I  1  I  i  1 !'"  -TTT  , !  i  L.....1  i i i  1  i t '  S(  1J  1  TAR i  i  f  J  1  i  i I  1  'I  i  i i  I  ..!_  !  €T  l  1 I i  i  i  i i  1 1 1 1 I  -4--  i  //  i  4—  ( 1  i !  i  i  1 1 j  i t  — | n  Q-  m  4 —r  L  — f f  ! |  i  |  JU  i i  ! 11  1  i1  frr\i  i 1  i  i  I  \f\ 1 10I1 i  I l l "  1  i  1  1  Figure  I  1  1 1  '  i  1  1 i 1  1 ! 1 1 1  I  . M l  8.8  i  I  -  i,.i 1 [  4 "f  i  i i ! 1 1 1 1  — L 1 1  j  1 1  ri  1  I  1  I  1  1  i  i  I  a  --a V-ftp  m  4S  1  1  -  i  1  i i i  1  ' f i  ir  1  I  i  I  4^  1  1 -  s«m  i  I 1 I I I 1  J  —I—j  =u  ti-  i  1  i  ::|  _—_ 1 i t 1  i  — 1[ I  fr --).  1  1  i i i i  _|—  11  i i  u _  i  i  1 1  T T T T - f\ | 1 W T M  l  !  M i l M M M M M M  1I  M  M M  M  i  -  M M  I  ! !'••  i' li ll !l  1 i i  1 1  • -r\ t5 4 r yi . 1 i i  1 1  1 1 1 t  1 i  J i  i  i  i i  1  i  :  -—  i  i  11  I 1  I  -T  i  i  i  ! T  1 1  i  1111  1 1 1 1  I  t i l l ' i l l  11  :  i 1 !  i  i  I ' M I M !  — i  i  1  i !—j — t 1 ' i i —pr  1 I  i  i i  i i  *w  •  i  i  HT- I ;  ' 1  i  i  1 i  i i  i i  ! 1 i i  1  1 i i 1  i i '  M  l  !  M  I  i.J  i  I  1- 1  1 r r r r " 1 4rh m 4~ I ! 1 1 |  r  \ 1 ! | 4-  1  I  jr\ rrr.  1 1  i  i 1  T T T - -4-  X . 1. 1 1 1  1 ! 1 ! )  1  i  1  1  1 1  L.| i  I  i  —  1 i  — !1 1  1  i !  I I  M M M M  M  i i  i  I  i  I  i  i  :  :  i  !  I  i  1  1  TTT •  i  i1  i  1  .j... !  i  1 1  i ._i  _ X J  1 —!—1—1 1 1  1  i  i i  • i  ! 1  4-r^E> ^  | |  t l — 1 — li  T44-  —lT\  r-  -,  ! i  I |  1  I  i  i  i  i  i—  !  111  i  ff 8€)-*r/—•  i i  i  !  1 A //  i  1  i  1 1 1 1 1 1  "A  4+  1 —.—L.  i  -444  4T  1  44-  T - -  i 1  v  I  i i i  i  i  .  —  I  - 4 -  1  i i  1 i  T"  1  i i  ! 1  I  i  i  4  1 1 1  i i  1  i  i i i  i i  n  f  1  1 1 1  -+  i  1 1  1  i  1 1  [4fj 1  I  ~r  1  1  i  . i  !.-  i  i  44 1  4-  i  i  • i • i-  1 -j—  I  1  i i  1  i  1 i • 1  i44  i i  ••  i  1  !  Xt  I 1  .... 4  -4  rf  i 1  i  1  i  i ; i  } I  —r  !  i  l  1  1  1 !  -H  1  1 1  4-  r  i  i  i i i  i  tf  i  i  i  1  r4  i  i i i  i i  i  i i  1 1 i i  1 i  )°/  1  1  i  i 1 1 1 1 1  i  i  4  i  !  1  1 1  l  /  i  1  1  i  i  ' M l TT TT i i  ••  f-j—1 1 11 1 1 1 1  L L4 r  M  11  ! i ;  -fl  1 1 i  _!  i i  11  1  i  Atit i 1 ^ i  i  K  1!  1 1  1  I  1  i i i <  11  ii  i  i  i  i  i  ! I  1  r:TL  1 i  M  1  i  -4  !  i  1  k  fXf  i  ' 1  1  1  WT  i  i i  M  1  +4  1 1  r  M  ,r  4  1 ! 1  Jk  -rC  1  i l l !  • 'i  1  ft-  I i  !  '  I I  i  ' 1  I_I.  11  i  1 1  r  i  i i  i i  i  1 J-4 1 M  i '  1  i i  4M  i  _i  t  1  ;  1  1  !' I ! 1 I I 1 I 1  1 ! 1  1  1 X _ 1 ' 1 1 1  ! i  1  11  r-  +t-h- 11  1  1  i  :  I ' M  i  I 1  i  1 i  i  i  ' I • i M  1 1 |  ' M l  E f f e c t on a i r f o i l l i f t c o e f f i c i e n t s o f assumed p r e s s u r e c o e f f i c i e n t s on a s t r e a m l i n e r e p r e s e n t i n g the plenum s h e a r l a y e r : T h e o r y  i  i  Figure A l . l  Geometry f o r i n t e g r a t i o n  of  a point  source  188  Pfr.O.z)  Axi-symmetric  *~ z  (b) Figure  A5.1  The two-dimensional n o z z l e  insert  Constant-pressure  boundary  t -2  E  h+ S  H  il  F i g u r e A6.1  I  1  1  )  1  1  A l i f t i n g v o r t e x between a s o l i d , a s l o t t e d , and a constant p r e s s u r e boundary: Theory  I  I  I  u>  ( D I <T> Q  I OJ Q  +  cr  a~ Q  +  ro  ro cr  I Q  -a--o-CT  OJ  ro cr  o +  cr  OJ  Q  + cr  a +  cr  cr  F i a u r e A6.2  1  <±)  *-  II Q  W  The image system f o r a l i f t i n g v o r t e x between a s o l i d and a c o n s t a n t p r e s s u r e boundary: Theory  Q  + cr  NSOLI  NSLAT * NSPS  NU2  NUI  U  NTEL ^  NWL2-  NWLI  JH  f i<r i i  NL4  NL2 NL3  NLI i  >  i  ' i'f  » > >1 r 1  1  F i g u r e A10.1  i i i i r'lMiVi * ' '*' > i > > • i i i ) i r } i i Ji / i i it f > i' / i / t / i i i N o t a t i o n f o r the computer of Appendix 10  program  / >Hi  192  Plate l b .  The  o c t a g o n a l t e s t s e c t i o n i n the  windtunnel  193  Plate  3.  The  wall  slats  i n the  side wall  frame  195  Plate  6.  The 616mm NACA-0015 a i r f o i l testsection.  in  the  196  Table 14% X  0. 0. 0. 1. 3. 4. 6. 8. 11. 14. 17. 21. 24. 29. 33. 38. 43. 48. 54. 60. 67. 73. 80. 88. 96. 100.  00 32 96 92 20 80 72 96 52 40 60 12 96 12 60 40 52 96 72 80 20 92 96 32 00 44  1.  Airfoil  profile  Clark-Y YU YL . 4. 19 4. 19 5. 15 3. 15 6. 15 2. 49 7. 24 1. 98 8. 35 1. 54 9. 35 1. 15 1 0 . 26 0. 84 1 1 . 14 0. 60 1 1 . 94 0. 38 1 2 . 65 0. 21 1 3 . 25 0. 09 1 3 . 70 0. 02 1 3 . 94 0. 00 1 4 . 00 0. 00 1 3 . 95 0. 00 1 3 . 74 0. 00 1 3 . 34 0. 00 1 2 . 73 0. 00 1 1 . 85 0. 00 1 0 . 80 0. 00 9. 44 0. 00 7. 83 0. 00 5. 92 0. 00 3. 86 0. 00 1. 45 0. 00 0. 00 0; 00  coordinates.  NACA-0015 +Y X 0. 0. 1. 1. 3. 4. 6. 9. 11. 14. 17. 21. 25. 29. 33. 38. 43. 49. 54. 60. 67. 73. 81. 88. 96. 100.  00 40 00 90 20 80 70 00 50 40 60 10 00 10 60 40 50 00 70 80 20 90 00 30 00 59  0. 1. 2. 2. 3. 4. 5. 5. 6. 6. 6. 7. 7. 7. 7. 7. 7. 6. 6. 5. 4. 4. 3. 2. 0. 0.  00 37 13 88 65 37 02 63 15 60 97 25 43 50 47 32 07 69 22 62 91 09 14 07 84 00  X  0. 0. 0. 0. 1. 3. 4. 6. 8. 11. 13. 16. 19. 23. 26. 30. 34. 38. 42. 46. 51. 54. 58. 63. 67. 71. 74. 78. 82. 85. 88. 90. 93. 95. 97. 98. 99. 100.  Joukowsky 11% : y X U 00 3. 92 0. 05 02 4. 16 0. 40 4. 89 35 1. 09 97 2. 1 1 5. 4 1 6. 28 3. 4 5 91 15 6. 96 5. 14 74 7. 62 7. 10 63 8. 26 9. 4 1 8. 85 77 1 1 . 99 1 4 . 87 17 9. 38 1 8 . 05 86 85 9. 76 1 0 . 25 2 1 . 45 58 89 1 0 . 2 5 . 10 24 1 0 . 83 2 8 . 95 74 1 0 . 99 3 3 . 02 44 1 1 . 07 3 7 . 25 27 1 1 . 06 4 1 . 59 22 1 0 . 98 4 6 . 06 26 1 0 . 80 5 0 . 58 39 1 0 . 55 5 5 . 12 15 1 0 . 19 5 9 . 69 6 4 . 19 75 9. 86 93 6 8 . 59 9. 43 7 2 . 86 05 8. 95 12 8. 44 7 6 . 96 09 7. 92 8 0 . 83 94 7. 38 84. 43 62 6. 85 87. 73 12 6. 34 90. 71 37 5. 85 9 3 . 29 40 5. 40 9 5 . 50 71 5. 07 9 7 . 27 4. 70 29 98. 61 50 4. 39 9 9 . 50 4. 16 27 100. 0 4. 00 61 50 3. 90 0 3. 84  YL 3. 53 2. 90 2. 3 1 1. 77 . 1. 29 0. 88 0. 55 0. 29 0. 12 0. 02 0. 00 0. 07 0. 20 0. 40 0. 65 0. 96 1. 29 1. 66 2. 04 2. 4 1 2. 77 3. 10 3. 40 3. 66 3. 86 4. 0 1 4. 1 1 4. 16 4. 17 4. 14 4. 08 4. 0 1 3. 94 3. 88 3. 84  196A Table  1  cont d. 1  NACA-23012 Main  Flap  Airfoil X  0.00 1.00 2.50 4.00 7.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 67.00 69.00 70.00 71.00 72.32 74.57 75.00 77.82 80.00 82.70  1.00 " 0.00 2.40 -1.10 3.61 -1.71 4.45 -2.10 5.65 -2.55 6.43 -2.92 7.19 -3.50 7.50 -3.97 7.60 -4.28 7.55 -4.46 7.43 -4.53 7.14 -4.43 6.80 -4.35 6.41 -4.17 6.00 -3.92 5.47 -3.65 4.95 -3.35 -3.18 -2.83 4.36 -2.51 -1.98 -1.02 +0.67 3.78 2.30 3.08 2.67 2.64 2.64  u  X  L 0.04 0.00 0.36 -0. 72 0.95 -1.00 1.74 -1.15 2.44 -1.21 3.44 -1.21 4.95 -1.15 6.45 -1.07 7.45 -1, 03 -0.99 8. 46 -0. 94 9.96 12.47 -0. 82 14.98 -0.71 -0.61 17.49 -0.46 20.00 -0.33 22.01 24.02 -0.19 25.52 -0.08 26.53 0.00 Origin of flap coordinates i s (78.87,-0.81) f o r 6 = 2 0 ° . 0.00 0.45 1.08 2.11 3.65 5.17 6.68 7.69 8.69 10.18 12.66 15.13 17.61 20.09 22.07 24. 05 25.54 26.53  y  0.04 0.99 1.59 2.27 2.93 3.33 3.55 3.57 3.52 3. 32 2. 86 2.36 1. 85 1. 35 0.93 0.52 0.21 0.00  Y  Table  2. F r e e  a i rairfoil  coefficients:  Theory.  NACA-0015 a  0 C  L  C  M  da  0  /.  0 3 5 10  0.000 .365 .607 1.210  14% a  0.000 .086 .143 .282  -  0.0000 .0050 .0086 .0204  0.1193 .1229 .1300  •  Clark-Y  0 C  L  0  Mc  -0.087 .012 .101 .244  -0.087 - .0883 - .0901 - .0965  C  M  C  dC da  L  •4  -6. 3 -3 0 5  0.000 .401 .763 1.362  0.1206 .1208 .1203  198  Table  3.  Airfoil  and w a l l  configurations  examined  theoretically.  All  solid  NWL1 The  walls  = NWL2  a r e 4.88m l o n g ,  w i t h MWUl = MWU2 = 20, a n d  =40.  slotted wall  i s 2.44m l o n g ,  composed o f l a r g e  (92mm) NACA-  0015 s l a t s w i t h NSPS = 9. Airfoil and  i s i n the center  NACA-0015;  Further  NA = 81  o f t h e t e s t s e c t i o n ; NA = 50 f o r C l a r k - Y  (46 m a i n a n d 35 f l a p )  n o t e s a r e found a t t h e end o f t h i s a  1. C l a r k - Y  )  c)  M  table. C 0  Mc 4  airfoil  a) F r e e a i r  b  C  f o r NACA-23012.  -8  -0.203  2  1.003  20  -0.137  -0.086  +  .159  -  .092  3.088  .603  -  .169  -0.150  -0.087  +  .014  -  .097  Solid  walls  -8  -0.250  c/H =  0.53  -3  .444  2  1.140  .178  -  .108  20  3.632  .711  -  .197  -8  -0.270  -0.153  NSLAT = 16  -3  +  c/H =  40%  d) 60%  TSUSL  .376  -000  -  .094  +2  1.012  .149  -  .104  20  3.179  .610  -. .185  -8  -0.260  -0.151  -0.086  =10  -3  +  +  .003  -  .092  =0.53  +2  .992  .148  -  .100  20  2.986  .572  -  .175  0.53  TSUSL  NSLAT c/H  -0.086  .377  199  Table  3  (cont'd).  a  Clark-Y 70%  TSUSL  NSLAT  80%  (cont'd)(60%)  =8  TSUSL  NSLAT  =5  NACA-0015  M  C 0  Mc 4  c/H  20  3.061  .579  - .186  0.66  20  2.929  0.570  -0 . 1 6 2  0.25  20  2.907  . 562  - .164  .39  20  2.923  .560  - .170  .53  20  2.970  .563  - .180  .66  20  3.084  .573  - .198  .86  20  3.200  .587  - .213  1.0  -8  -  .256  -  .149  - .085  .53  -3  +  .381  +  .005  - .091  .53  +2  .989  .149  - .098  .53  -8  -0.250  -0.14.8  -0 . 0 8 6  0.53  -3  +  +  .007  - .090  .53  .388  +2  .994  .152  - .097  .53  20  2.888  .554  - .168  .53  -2  -0.243  -0.058  + 0. 0 0 3  +8  +  +  .227  - .015  airfoil  Free a i r  Solid  C  walls  .970  15  1.803  .412  - .038  20  2.382  .530  - .065  -2  -0.305  -0.068  + 0.008  0.67  +8  +1.223  +  .272  - .034  .67  10  1.510  .333  - .045  .67  20  3.074  .675  .093  .67  +3  .371  .087  - .006  .17  3  .388  .090  _  . 34  . 007  200  Table  3 (cont'd)-— a  C  M  c/H 0  4 2. NACA-0015  (cont'd)  b) S o l i d w a l l s  c)  40%  TSUSL  .416  .094  -  .010  .51  3  .453  .100  -  .013  .67  3  . 546  .116  -  .021  1.0  -0.327  -0.072  +0.010  +3  + .364  + .081  -  .010  . 67  8  1.039  . 227  -  .033  .67  20  2.592  . 550  -  .098  .67  TSUSL-  -2  -0.320  -0.070  +0.010  0. 67  NSLAT = 10  +3  + .359  + .081  -  .009  .67  8  1.006  .222  -  .230  .67  20  2. 421  .514  -  .092  .67  +3  0. 355  0. 084  -0.005  0.17  3  . 356  .083  -  .006  .34  3  .358  .083  -  .007  .51  3  . 361  .082  -  .008  .67  3  .365  .082  -  .009  TSUSL  -2  -0.311  -0.068  +0.010  NSLAT = 5  +3  + .367  + .084  -  8  1.000  .224  - .026:  .67  20  2.335  . 500  -  .67  +8  2.442  0. 320  d) 60%  70%  80%  =16  TSUSL  NSLAT  f)  3  -2  NSLAT  e)  (cont'd)  =8  .007  .086  3. NACA-23012 a i r f o i l a) F r e e a i r  -0.291  0. 67  1.0 0.67 .67  201  Table  3  (cont'd)  a  NACA-23012 70%  C  c/H  Mc 4  (cont'd)  TSUSL  +8  2.415  0. 308  8  2. 305  .290  -  .286  '.4..  . .... 8  2.318  .283  -  .296  .6  8  2.296  . 280  -  . 312  -8  8  2.442  .280  -  .331  1.0  NSLAT = 8  Clark-Y  V  -0.295  0.2  airfoil  Compare C  (C ) w i t h C ( T ) , w i t h  NA =  T  J-i  p  110 •and  a = 20 o  Li  L  p  C  M  c (D L  0  4 i) ii)  Free a i r Solid  w a l l s , c/H=.66  3.117  0.548  4.165  .744  Circulation  on w a l l  s l a t s reduced  NSPS = 1 5 ,  a = 2 0 ° , c/H = 0.66. k  70%  TSUSL  NSLAT = 8  C  '  i)  t  3.114  -  3.742  .298  by m o d i f y i n g  L  slat profiles,  C  Mc 4  1.0  3.010  0.569  .8  2.935  .556  -  .178  •  2.610  .502  -  .150  7  S h e a r l a y e r r e p r e s e n t a t i o n . MSV c/H = 0.66, V.  -0.231  = 2 0,  -0.184  a = 20°,  = / ( l - C ).  p  C P  C  L  C  M  C 0  Mc 4  60% TSUSL  0.0  2.420  0. 451  NSLAT = 1 0  -.12  2.686  .502  -  .170  T.28  3. 305  . 572  -  .187  -0.155  202 Table 4c)  3  (cont'd)  C l a r k - Y - Shear l a y e r r e p r e s e n t a t i o n  (cont'd).  C  C  P i)  60% TSUSL ( c o n f d ) - . 3 5  3.188  .600  -  .197  NSLAT = 10  -.44  3.390  .640  -  .207  0.  2. 321  .433  -  .147  -.12  2.591  .486  -  .162  -.28  2.951  .558  -  .180  -.35  3.101  .586  -  .190  -.44  3. 308  .626  -  .200  ii)70%  TSUSL  NSLAT = 8  5. NACA-0015 solid  airfoil  walls;  with reduced c i r c u l a t i o n ,  a reduced  airfoil C  i)  from measured  C  L T  determined. C Mc _4  Mo M  M  1.120  .335  +0.0549  1.120  .280  -  by m o d i f y i n g t h e profile  (k=0.724)  N o t e s : The p o s i t i o n s o f t h e w a l l those i n the experimental beginning with a s l o t sidewall.  The  .0003  correspond exactly t o are spaced  uniformly,  opening a t the upstream end on t h e slats  required  f o r 40, 50, 60,  i s 16, 13, 10, 8, a n d 5 r e s p e c t i v e l y .  effect  airfoil  o f i n c r e a s i n g t h e number o f c o n t r o l p o i n t s  the  test  the  number o f c o n t r o l p o i n t s (k=l).  slats  s e t u p . The s l a t s  The number o f l a r g e  70 a n d 80% OAR  4(b)  circulation  a = 10°,  lift  (k = 0.741) ii)  Mc 4  i s seen i n 1(a) and 4 ( a ) ( i ) ; on t h e w a l l  slats  of increasing i n 1(e) and  on  203  Table  4. A i r f o i l  Joukowsky  Re=.5(10)  t  C  -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 C  C  -.350 -.245 -.137 -.025 .086 . 194 .299 .408 .512 .617 . .721 . 820 .917 1. 0 0 9 1.103 1.192 1.277 1. 3 4 0 1.382 1.408 1.199  L' 5c' D C  C  C  Mc' D 4 C  and e n d p l a t e  L  -.345 -.241 -.134 -.024 .086 .192 .296 .403 .506 .610 .713 .811 .907 .998 1. 0 9 1 1.179 1.265 1. 3 2 6 1.368 1. 3 9 4 1.186  6  Solid C* Mc 4 -.0719 -.0727 -.0728 -.0727 -.0724 -.0716 -.0712 -.0705 -.0693 -.0669 -.0652 -.0639 -.0628 -.0617 -.0600 -.0581 -.0557 -.0536 -.0540 -.0620 -.1611  loadings. Walls  Mc 4 -.0723 -.0728 -.0727 -.0723 -.0717 -.0706 -.0699 -.0689 -.0673 -.0645 -.0625 -.0607 -.0591 -.0574 -.0551 -.0524 -.0491 -.0466 -.0465 -.0546 -.1538  loading  on a i r f o i l  plus  loading  on a i r f o i l  only  c* .0330 .0317 .0309 .0308 .0314 .0323 ' .0340 .0361 .0384 .0410 .0441 .0482 .0530 .0592 . 0652 .0719 .0790 .0869 .0972 .1161 .2551  two e n d p l a t e s  C  D  . 0194 . 0180 .0171 . 0168 .0172 .0179 .0193 .0212 . 0233 . 0259 . 0289 . 0326 .0370 .0425 .0478 .0534 .0594 . 0672 .0773 .0959 .2345  Table  5. Windtunnel balance r e s u l t s -  0.25-Clark-Y ALF  Pt = ' . « 5 ( l O ) 6  CMC/'I  CM0  CD  CL  SOLID WALLS  -0.0916 - 1 0 ' . -0'.3'l5 0'.0260 -.«»'. - 0 . 2 5 1 0'.0?35 -o'. w.o - 0 . 0 9 0 7 -o'. 151 0'.0221 - 0 . 1 3 3 -T. - 0 ' . 057 0'.0215 - 0 . 1 0 7 O'.O'II 0'. 0 2 0 1 —0.080 -6'. -5". 0'. I l l C '. 0 2 0 7 -0'.053 o'.2'U 0'. 0205 - 0 . 0 2 6 -1'. 0'.313 0'.02() '1 0-002 " 2 r (>'.13S 0'.02?1 o'.032 o'.553 0'.0237 0.060 (i'.665 0 .0256 0'.037 0'. 0 . 7 5 3 o'.0278 0'. 1 11 i'. 0. 110 0-815 0'.0302 2'. i'. 0'.938 0'.0330 0'. 168 0.193 T.030 0'. 0373 l'. 5'. '1'. 1 oa O'.03 i7 0'.2 18 6'. l ' . 185 0'.0133 0.211 0.260 l'.256 o'.oi87 l'.321 0'.0532 0 ,286 B'. 0.305 r . 3 ^ 6 0'.05R8 r . ' i 2 7 0-0650 0.321 16'. r . ' i 3 5 0'.0716 0'.331 ir. 0.333 0'. 0788 12'. 13'. r . ' i 2 6 0'. 1 03'l 0.322 0'.308 f. 388 0 . 1218 11'. -  (  -  -  CLARK-Y  RC=.15(l"0)6  Cl, -(>'. 310 • 0'. 2 'i 5 -8'. •O'. 117 -7'. -()'. 052 O', 012 -6; O'. 133 -5'. 0'.237 -1*. 0'.335 -z: . 126 -?-'. O 0*.52H -1. o'.6'll 0". 0'.725 1'. 2\ 0'.812 0'.89 7 3 0'. 9 31 1 l ' . 053 5 6 132 7 ,203 e ,267 9 10 .361 I'. 377 12. 1-379 l'.365 13'. 1/.329 I'.293 15. ALK  -10'.  -<  -  *h  cn  0 0 219 O', 0222 0'.02()0 O'. 0 196 0'.0181 0'.0182 0'.0185 0'.0192 0'.0196 ,0213 ,0212 ,0265 ,0201 0'.0319 0'.0319 03^5 0128 . 016 6 ,0502 ,0550 ,06o'l ,0666 ,0761 ,09i|8 .1165 0.1379  r-0 . 0906 -0.0910 -0.0908 -0.0911 -0.0908 -0.0908 -0.0871 -0.0905 -0.0942 -0.0920 -0.0905 -0.0889 -0.0880 -0.oias -0.0820 -0,0782 -0.0736 -0.07U -0,0655 -0.0572 -0.0513 -0.0638 -0.0682  10%L5+P TSUSL  CHO  CMC/4  - 0 ' . 185 - 0 , 0 9 2 8 - 0 . 159 -0.132 -0.107 -0.081 -0.055 -0.028 -0.000 0'.023 0.051 O',073 0.103 0'. 128 0. 151 O'. 178 0'.201 0.221 0'.216 0.266 0.23 6 0'.302 0.312 0'.315 0'.307 0.290 0'.277  r-0.0921 -0.0917 -0.0921 -0,0921 -0.0919 -0.0923 ~0 . 0909 -0,0875 -0,0391 -0.0951 -0.0937 ^0.0927. -0 , 0899 -0.0888 -0.0353 -0.0827 -0.0796 .-0,0758 *0 ,0715 ^0 ,0618 -0.0586 -0.0551 rO.0591 -0.0676 -0,0715  -Y  airfoils  CL ARKnY  Rti = *.'15(10)6.  'I0ZSS+P TSU  CMO AI.F CO CL 10'. -o',330 0'. 0237 - 0 ' . 187 -9' - 0 , 2 3 7 0'. 0218 - 0 . 1 6 0 - 0 ' . I'll 0'.0195 - 0 - 1 3 1 - 7 ' -o'. 016 0 ' . 0 1 « 9 -0'.107 -6' 0'. 0 •'! 8 0'.0177 - 0 . 0 82 •*5' 0. I ' l l 0'. 0 17 6 - 0 - 0 5 5 0'.2'I0 0 0 1 7 '1 - 0 . 0 2 7 0.000 —> 0'. 5'40 0'.0176 -2' O". 01R5 0'.0 29 0'.533 0'. 020 1 0'.055 -i' 0,081 0'.6'I3 0'.0231 o' ()'.729 o'.0262 0.105 r. z 15 0'. 0 285 0.130 'r 0^8 0.155 0',90l" o'.0315 0'.986 0-0316 0*. 181 0.200 5 » f.062 0'.0389 6 1.133 0'. 0'I26 0.227 7 0.2'19 1.201 0'.0'I62 0-269 l'.?68 0'. 0119 8. ?'. l'.323 0'.0550 0.288 l'. 359 0', 0 6 0 6 0.30a 10 * 0.316 l'.377 O'. 0675 11. 0.319 12 m r . 5 6 2 0'. 0765 l'. 366 0'.09;i 3 0.309 13 • 11 • 1-333 0'. 1 162 0 ' . 2 9 « 0'.280 15. f . 2 9 3 0'.1378  ()'.'I3 0  CMC/1 ^•0.0963 -0'. 0916 -0.0915 -0.0938 -0.0912 -0,0911 -0.0921 -0.0933 -0,0896 - 0 , 0 9 16 -0.0968 -0.0958 -0.0915 -0,0933 -0.0911 -0.0885 -0,0852 -0,0826 -0,0795 -0,0761 - 0 . 06.85 -0 .0609 ^0,0589 -0 . 0636 -0,0701 -0,0718  205  T a b l e 5 - 0.25-Clark-Y Ri: = '.45Cf0)6  50XL.S + P TSUSL  CO ALF CMO CL 10'. -o'.333 0'.0239 -O'. 181 -9'. -0.24U 0'.0210 -.0.158 -8. -0.113 O'.02o5 -o'.131 -7'. -0'. 019 O'. 0131 -.0.106 -6'. O'. 015 0'. 0185 -0.080 -5'. 0'. 110 0'.'173 -0.051 0'.236 0'.0175 -0'.027 "If 0'.335 0'. 0 1 3 I 0.001 *" • 0'.'I22 o'.oiaa 0'.029 0'.525 O'.02o7 • o'.osi 0'. o'.63'l 0'.0230 0.080 0'.720 0'.0251 0'. 105 t. 0.129 2'. 0'.801 0'.0277 3'. 0'.S91 0'. 0 295 0.155 4'. 0'.978 0'.0320 O. 180 0'.203 5'. T.051 0'.0373 6'. .1.127 0'.0102 0.226 0.217 7'. l'. 195 0'.0137 8'. T.257 0'.0474 0'.268 1'. 311 0'.0521 0.236 l'. 10'. 1.35 0 0'.0577 0'.301 0.311 f.372 0'.0613 11'. 12'. 0.315 T.370 0'. 0739 l'.354 0'.09;>5 0.301 13'. 11'. 0.286 1'. 311 0'. 1156 1.282 0'. 1352 0.275 15'. -  CLARK-Y  RE = '.'I5(T0}6  ALF CL -10'. -0'.328 -9'. -o'.23'l O * -0.137 - 7 ' . -0'.013 -6'. 0'.051 -5'. 0'. 146 -4'^ 0'.2'll -3 . 0'.336 -2'. 0'.425 0'.530 -i'. o. 0.637 1'. 0'.722 2' 0'.307 z'.0'.893 1'. 0'.978 5'. 1.055 6. 1.127 7'. I'. 193 8'. l'.258 9'. 1.310 l'.346 10'. 1 r . l',359 l'.359 12' 1 .345 13. 14'. T.307 15'. l',270 :  :  CD 0'.0236 O'. 0210 0'.0191 O'.O 175 0'.0161 O'.O 151 O'.O 158 0'.0161 o'.0166 o'.oiao 0'.02()7 0'.0229 0'.0254 0'.0277 0'.0302 0'.0352 0'.0381 0'.0116 0'.0461 O'. 0509 O'0562 0.0624 0'.0726  CMC/4 -0.0929 -0.0915 -0,0916 -0.0915 -0.0915 -0.09)6 -0,0910 -0.0906 -0.0870 -0.0898 -0.0912 -0.0932 -0.0917 -0.0903 -0.0386 -0.0861 -0.0835 -0,0301 -0.0762 -0,0724 -0,0671 -0.0623 -0.0575 -0,0639 -0,0718 -0,0749  60%L3+P TSUSL  CMO  CMC/4  -0 .183 -0.0931 - 0 .158 -0.0933  -0 .131 -0 .105 -0 .079 -0 .053 -0 .0 26 0 .001 0 .029 0 '.055 0 .080 0 .a 0 4 0 .130 0 .155 0 .180 0 .203 0 .226 0 .247 0 .267 0 .286 0 .300 0 '.310 0 .314 0'.09i'l 0 .302 O'.l 125 0 '.285 o'.1313 0 .274  -0.0929 -0.0930 -0,0921 -0.0925 -0.0915 -0.091 1 -0.0878 -0,0907 -0.0955 -0.0946 -0,0927 -0.0909 -0,0895 -0.0873 -0,0810 -0.0807 -0.0780 -0,0727 -0.0680 -0,0610 -0,0568 -0,0653 -0.0717 -0,0739  CLARK-Y  Rl! = '. '15 (10) 6  70ZL3+P  CMO ALF CD CL -IO'. -0'.324 0'.0273 -0'.133 -9'. -0'.228 0'.023'l - 0 ' . 157 -8'. -O'. 132 0'.0222 -0.130 -7'. -O'. 038 0'.0212 - 0 . 1 0 3 0'.057 O'. 0 196 -0'.077 ~s . 0.153 0 .0191 -0.051 o'.246 O'.O 199 -0.021 -4'. -3". 0'.34 2 0'.0198 o'.ooi 0'.4 27 0'.0202 0'.032 0'.058 o'.533 0'.0219 0'.639 0'.0 239 0.083 0.108 f'. 0.722 0'.0263 2'. 0'.807 0'.0284 0'.133 0.158 3'. 0'.892 0'.0293 '4'. ()'.974 0'.0320 0.132 5'. V.05 0 0'.0354 o'.205 0.226 6'. 1'. 1 17 0 .0379 0'.24 7 7'. l'. 185 0'.0117 8'. l'.243 0'.0462 0'.268 i'.297 0'. 0 5 1 0 0'.236 1. 0.300 IO'. T.330 0'.0563 1.339 0'.0628 0'.309 .11. 12'. l'.34 0 0'.0732 0.310 13'. 0'.293 T.323 0'.0928 14'. T.285 O'. 1 150 0'.233 0.268 15'. i',246 0'.1337 -  f  -  . C\ ARKrY  Rlt = *. 45 C {0) 6  ALF CL -10'. -0',313 - 9 . -0.214 -8'. -0'. 123 - 7 . -0.030 -6'. 0'.063 -5'. O'. 155 0.251 -l'. -3'. 0'.343 -?-'. 0'.131 0'.53S "1". O'. o'.oio 0'.725 i'. 2: 0',806 ()'.890 3'. 0'.967 4'. l'.042 5'. 6'. l'. 167 7'. r.107 l'.230 8. <?'. l'.28l" 10". T.298 t'. 315 11'. 1.310 12'. i'.28 4 13'. 14'. i'.246 15'. f,207  CD 0'.0258 0'.023'l o'.02?6 O'.02l0 O'.O 19 1 O'.O 191 O'.O 138 O'.O 191 o'.oiao 0'.0202 0'.0219 0'.0234 0'.0215 0'.0270 0'.0296 0'.0336 0'.0366 0'.0H3 O'.O'ISI 0'.0500 0'. 0557 O'. 0629 0'.0743 0'.0911 O'.l 122 0'.1278  CMO -O'.l 74 -0.118 -0.122 -0'.095 -0.073 -0,048 -0'.021 0.006 0 .034 0'.060 0.035 o'.l 10 0.135 0.160 O'. 135 0.206 0 ,228 o'.218 0.268 0.236 0.300 0.308 0'.306 0.291 0.275 0.262 -  TSUSL  CMC/4 -0.0925 -0,0929 -0.0924 -0.0922 -0.0920 -0.0926 -0,0919 -0,0914 -0,0372 -0,0910 -0.0952 -0,0939 -0.0926 -0,0913 -0.0895 -0,0877 -0.0349 -0.0326 -0.0778 -0,0741 -0.0637 -0.0613 -0,0603 -0,0679 -0.0734 -0,0773  80XL3+P TSUSL CMC/4 -0.0881 - 0 ,0886 -0,0874 -0.0863 -0.0900 -0,0396 -0.0893 -0,0880 -0.0843 -0,0080 -0,0910 -0.0898 -0,0873 -0,0850 -0.0313 -0,0300 -0,0761 -0.0716 -0,0692 -0,0643 -0.0548 -0.0508 -0,0510 -0.0596 -0,0650 -0,0677  ft—ft—ft—i—* ft— IX. 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I I 1 I I I I I I o o o o o o o o o o o o o o o o o o  o  c  o  o  o  o o o o o 43 0 - i i r u O i i 43 J l O CD I I I I I o o o o o  I o  I  1  c  -• •• -• -r>  o  I o  I o  c  I I o  n  o o o ft ft ft ft o U l J l CO O 1 4 ' J ! CO 14 CO L-J CO r u CO r u I I I I I f I o o o o o o n  ftOOOOOOOOOOOOOOOOCOOOOOOOOOOOO C4343CC-4—J-4—4-4-4'CO 3 ) C Q C O C C 3 ) 3 : C D O i l 5 O - 0 O C 3 3 ) C 3 33Ca'3)\. c ? ? O i - " 3 - C " C > C » » f t W i i J ? - J C o 4 ) c . 0 O - O O 4 ) C 0 C C 3 l 3 : i 4*ftincD-siuli\iru^oo'jirjft4iiL-j.43CP4iroftru-c  CO CO  03 L_ Co  r\  211 Table  5 - 0.53-Clark-Y  CLARK-Y ALF -10'. -9'. -fi'. -7'. -6'. -5'. -4.  Ru='.5(10)6  CL  CD 0'.0225 O ' . O 190 0'.0162 0'. 0 11 0 O'. 0130 O ' . O 123 '0122 0137 0152 -2'. 0 '.409 0164 -r. O'. 4 9 9 0191 o'. 0 ' . 5 9 5 0230 r. O'. 6 8 6 2'. ()'. 7 6 9 0'.0263 0 ' . 8 5 5 0'.0315 3. g ' . 9 1 2 0'.0375 4'. 5'. T . 0 2 4 0'.04?7 1 ' . 1 0 1 0'.0480 6'. 7'. T. 1 7 6 0'. 0511 0'.0605 l'.25f e'. 9'. 1 '. 3 1 9 " 0688 0763 T.373 10. 0856 1 '.4 0 5 i r. 1 ' , 4 1 2 0967 12'. 1101 l'.4 69 13'. 1279 l'.4«8 14'. 1 ' . 4 9 0 O'. 1466 15. l ' . 4 R 8 O'. 1681 16'. " l ' . 4 8 3 0'. 1885 17'. l ' . 4 6 3 0'.2093 lfi'. l'.O 4 1 0'.2283 19'. 0'.2500 1.408 20'. - 0 . 3 1 4 -0'.?52 -0'. 157 -o'.063 0.030 0.124 0'.221 0.319  CLARK-Y ALF -10'. -9. -8'. -7'. -6'. -5'. -4'. -3. -2 o'. i'.  2'. 4. 5. 6'. 7'. 8'. 9'. 10'. ll  'r  13'. 14'. 15'. 16'. 17'. 18'. 19'.  CMO -0.179 -0.151 -0.129 -0.105 -o'.os 1 -0.056 -0.031 -0'.005 0'.021 0.016 0.070 0.095 0.118 0.113 0.166 o'. 189 0.211 0'.232 0'.252 0.272 0 .289 0'.302 0'.310 0.318 0.319 0.313 0'.309 0'.300 0.292 0.279 0.265'  Rir = ' . 5 " c i 0 ) 6  CL CD -0'.336 0'.0216 -0'.215 0'.0212 -0.151 0 ' . 0 l - i 5 -0'. 057 O'.O 177 O'. 0 34 0'.0171 0.127 O'.O 168 0'.221 O'. 0169 0.315 0'.0175 0'. 10 9 O'.O 188 O ' . l 93 o'.02ol O'. 589 0'.0228 0'.678 0'.0266 0'. 762 0'.0310 0'.813 0'.0352 0'.932 0'.0402 l'.O05 0'.0 4S3 i ' . 08 7 0'.0503 !'. 151 0'.055 3 1.223 o'.06()5 ~l'.282 0 ' . 0677 0'.0718 ['.333 1 '.379 o'.osn i'.'113 0'.0960 '['.4 37 0". 1078 1 .151 " 1228 ['.4 53 14 00 l ' . l 16 1583 1 '.131 1768 1 '. 4 0 7 1 9 i | 9 ' 1.375 2145  5 0 XLS +P  CMC/1 -0 .0874' -0 . 08'69 -0.0872 -0.0801 -0.0381 -0.0086 -0.089-5 -0.0904 -0 .088 t -0.0076 • -0.089 1 -0.O890 -0.0076 -0.0861 -0.0858 -0.0851 -0.0035 -0.0817 -0.0813 -0.0791 -0.0770 -0.0720 -0.0729 -0.0713 -0.0758 -0.0824 -0.0364 -0.0911 -0.0976 -0.1061 -0.1121  60XLS + P  CMO  -0.179 -0'. 155 -0. 129 -o'. 104 -0.030 -0.055 -0.031 ••0'.005 0.022 0'.016 o'.071 0.095 0.118 0'. 1 4 2 O'. 165 0.188 0.209 0'.230 0.250 0'.268 o'.281 0.296 0.305 0'. 3 1 1 0'.312 0'.307 0'.300 0.289 0.281 0'.268  TSUSL  -  TSUSL  CMC/1 0 . 0 8 7 8 0 . 0 0 0 3 0 . 0 8 7 0 0 . 0 8 8 1 0 . 0 3 8 6 0 . 0 8 9 3 0.0901 0 . 0 9 0 5 0.0391  -0'.0H75 - 0 . 0 8 8 9 - 0 . 0 0 9 7 - 0 . 0 3 9 0 - 0 . 0 8 7 0 - 0 . 0 8 7 7 - 0 . 0 8 5 1 -0.0861 - 0 . 0 0 3 2 - 0 . 0 0 1 3 - 0 . 0 7 8 9 - 0 . 0 7 6 1 -0.O756 - 0 . 0 7 5 5 - 0 . 0 7 5 8 - 0 . 0 7 8 3 - 0 . 0 8 3 8 - 0 . 0 8 9 8 - 0 . 0 9 7 5 - 0 . 0 9 8 7 -0.1041  CLARK-Y ALF -10'. -9'. -8. -7'. -6'. -5'. -4'. -3'.  RC = . 5 ( 1 0 ) 6  CD CL -0'.331 0 ' . 0 2 3 7 -0'.240 0 ' . 0 2 0 1 -0'. 143 O ' . O 1 7 3 -0.053 O'.O 1 6 1 0'. 037 0 ' . 0 1 1 0 0'. 130 0 ' . 0 1 2 8 0'.224 0 ' . 0 1 2 9 0'. 3 1 6 0 . 0 1 3 2 0.109 O ' . O 1 3 6 0'. 19 <| 0 ' . 0 1 1 2 0'.587 O'.O 1 6 4 0.672 O ' . O 1 9 2 0'. 756 0 ' . 0 2 2 4 0'.836 0 ' . 0 2 6 2 0'.924 0 ' . 0 2 9 9 ()'.992 0 ' . 0 3 4 3 {'. 068 0 ' . 0 3 9 2 l ' . 137 O'. 0 1 5 9 l ' . 198 0 ' . 0 5 1 4  -V. o. r.  1'.  3. 4'. 5'. 6'. 7'. 8'. • 9'. 10'. 12. 13'. 14'. 15'. 16. 17'.  T.256  l'.302 ['.342 l'.37i i'.393 1'. 4 0 1 1'. 4 01 '['.383 l'.361  CLARK-Y 1 ALF CL •10'. •^0.315 -9'. -0'.224 -0'. 132  0'.057l O'.O 6 3 9 0'.0712 o'.ooii o'.0952 O'.l 1 i l 0'.1260 0'. 1 4 3 9 0'.162t  70XLS+P  CMO - 0 . 176 - o . 151 -o'. 125 - 0 . 101  -o'.0 7 6 -o'. 0 5 2 ' -o'.0 2 7 -o. 0 0 3  CMC/4 - 0 . 0 8 6 1 - 0 . 0 3 6 0 - 0 . 0 852 - 0 . 0 8 6 0 - 0 . 0 8 5 8 - 0 . 0 8 6 6 - 0 . 0 8 7 3 - 0 . 0 3 8 0  0. 024  - 0 . 0 0 6 5  o'.0 1 8  - 0 . 0 8 5 3  0 . 071 o'. 0 9 5 ()'. 1 1 8 o'. 1 4 2  -  0 0 0 0  o'.1 6 5 o'.1 8 6  -  0 . 0 R 4 7 0 . 0 8 1 9 0 . 0 8 1 t 0 .0799  o'. 2 0 7 o'.2 2 6 0. 215 o'. 2 6 3 o'. 2 7 8 o'. 2 8 9 o'.2 9 7  . 0 . 0 . 0 .C  8 8 O O  7 6 5 3  2 5 0 9  - 0 . 0 7 7 2 - 0 . 0 7 4 1 - 0 . 0 7 1 3 - 0 . 0 7 0 8  0 .  - 0 . 0 7 0 1 - 0 . 0 7 0 3  o'.2 8 6 o'.2 7 4  - 0 .00.05 - 0 . 0 8 3 5 - 0 , 0 9 0 0  302 0 . 293 •o.2 9 1  - 0 . 0 7 6 1  80%LS+P  CD CMO ' 0 213 - 0 . 1 7 1 0179 - 0 . 1 1 6 0157 - 0 . 1 2 2 -(>'. O i l 0115 - 0 . 0 9 7 -6. 0.047 0132 - 0 . 0 7 3 -5'. 0'. 137 0125 - 0 . 0 5 0 0'.227 -4'. 0 133 - 0 . 0 2 5 ()'. 320 O ' . O 137 0.000 -3'. 0'. 4 1 1 O ' . O 111 0.027 -2'. 0'. 493 0.051 "0155 0'.075 0165 o'. 0'.588 0'.671 0.099 0191 1. 0.122 0214 2'. 0'.751 0'. 1 4 6 3'. 0'.830 o'.022<l O'.l 6 8 4'. 0'.908 0'.0216 O'. 1 8 9 5'. (t'.982 0'.0269 ['.052 0'.0302 0.209 6'. 0.228 7'. 1'. 11 2 0'.0317 0.215 fi'. l ' . 168 0".0331 0'.26t 9'. l ' . 2 1 60'.0424 f.24 3 0'.0477 0'.272 'r "['.282 O ' . O S I O 0'.28 0 11. T.296 0.234 ,0623 12'. 1 .304 0'.2S3 ,0727 13'. T.29 0 0.277 .0866 14'. r.265 , i o i o 0 . 2 6 6 15'. '['.238 0''.254 16'. ,1172 i0  TSUSL  TSUSL  CMC/4  -0.0310 -0.0843 -0.0854 -0 .0851 -0 .0855 -0.0370 -0.0375 -0.0877 -0.0060 -0.0350 -0.0873 -0.0871 -0.0053 -0.0836 -0.0828 -0.0321 -0.0313 -0.0787 -0.0762 -0,0733 -0,0703 -0.0711 -0.0698 -0.0727 -0.0751 -0.0788 -0.0843  212  T a b l e 5 0.53-Clark-Y CLARK-Y RE = '.5C10)6 CD ALF CL -10'. - 0.3.'17 o'. 0234 - 9 . - 0 . 2 5 4 o'. 0 195 - s ' . - 0 . 159 o'. 0165 -7'. -()'. 065 o'. 0151 -6'. ()'.028 o'. 0143 -5'. O'. 122 u'. 0110 -4'. 0'. 21 7 o'. 0145 -3'. 0'.313 o'. 0 159 0'.4 07 o'. 0177 -2'. 0'. 4 9 '1 o'. 0135 -r. o'. 0'.591 o'. 0205 r. 0'.678 o'. 0211 2'. 0'.763 o'. 0280 0'. 8 'I 8 o'. 0322 3'. 0'.936 o'. 0366 4'. 1'. 0 1 2 o'. 0 413 5'. 6'. T.092 o'. 0463 1. 162 o'. 052'! 7'. e'. T.235 o'. 0584 l'.302 o'. 0668 9'. 10'. l'.357 o'. 0758 1 '.401 o', 0860 11'. l'.'HO o'. 0972 12'. l'.4 66 o'. 1097 13'. 1.481 o'. 1262 14'. 15'. l'.490 o' 1454 16'. l'.4a6 o'. 1616 f . 4 7 6 o' 1869 17'. 18'. T.457 o'..2012  50XSS+P TSUSL  CMO -0.181 -0.157 -0'. 131 -0.106 -o'.OR2 -0'.057 -0.032 -0.006 0.020 0.015" 0'.070 0'.091 O'.l 17 0'. 112 0'. 166 0.188 0'.210 0.230 0'.252 0.271 0'.288 0.30 0 0'.3i0 0.316 0'.318 0'.310 0'.309 0'.299 0'.291  CLARK-Y RII = .5C10)6  CMC/4 -0 .0076 -0 . 0882 -0.0875 -0.0885 -0.0386 -0.0092 -0 .0900 -0.0905 -0.0392 -0.0081 -0.0397 -0 .0889 -0 .0883 -0.0867 -0.0863 -0.0846 -0.0839 -0.0820 -0.0800 -0.0779 -0.0756 -0.0741 -0 .0750 -0.0753 -0.0770 -0.0880 -0.0874 -0.0961 -0.0993  ;  :  70%SS*P TSUSL  CMO CD CL -0'.331 o',0225 - 0 . 1 7 7 - 0 . 2 1 2 O'.O 195 - 0 . 1 5 3 -()'. 149 0'.0173 - 0 . 1 2 8 - 0 . 0 5 7 0'.()157 - 0 . 1 0 2 ()'..0 31 o'. 0152 - 0 . 0 7 8 0'. 125 0'. 0 151 - 0 . 0 5 1 ()'.?13 O'. 0152 - 0 . 0 2 9 ~f ' 0'. 3 1 1 O'. 0158 - 0 . 0 0 5 -2'. O'.l 06 ()'. 0 167 0'.0 22 0'.489 O'.O 179 0.016 o'. 0'.581 O'.02o3 0'.070 0'.669 0'. 0234 0.093 1. 0'.751 0.0270 2'. O'.l 17 ()'.834 0'. 0 3 0 7 o'.l H 3'. 0'. 9 1 6 0'.0319 0'. 161 . 4'. 0.993 O'. 0 38 6 0'. 185 5'. 6'. T.0 63 0'.0433 0.206 0.225 l ' . 130 O'.045'l 7'. 8'. r. 199 O'.05l 4 0.215 0'.26 3 l',257 0'.0577 9'. 10'. • i".304 0'.0652 0'.277 r . 3 i i 0'.0738 0'.288 11 '. l'.369 0'.083l o'.295 12'. 13'. T.307 0'.0911 0'. 3 0 1 14'. 0.299 1'. 4 01 0'. 1096 15'. 1 '.39 3 0'. 1255 0.292 16. 1.38 0 0'. 1433 0'.281 i',362 0'. 1603 17'. 0.273 0'.262 l'.331 0'.1791 ie;  ALF -10'. -9'. -8'. -7'. -6. -5'. -4.  CLARK-Y R E = . 5 C l 0 ) 6  60ZSS+P TSUSL  CMC/4 CMO CL CD -0'.342 0.0236 o'.ioo - 0 . 0 3 7 7 il'. -0'.247 0205 0'. 155 -0.0881 -3. -O'. 15.5 0103 0.130 -0 .0882 -7'. •o'.06J 0U.5 •o'.105 -0 . 0886 -6'. 0.030 0155 '0.081 - 0 . 0 8 8 6 -5. 0.124 0152 0.056 -0'.0892 -4'. 0'.218 0154 0.031 - 0 . 0 3 9 9 0'.315 0165 •0.005 -0'.0903 0'.406 0174 o'.021 - 0 . 0 3 0 8 -2. 0135 0.015 - 0 . 0 8 3 1 -r. ()'.492 0'.587 0214 0'.069 - 0 . 0 3 9 6 0. ()'.676 0241 0.093 - 0 , 0 9 0 0 f. ()'.760 0277 0.117 - 0 . 0 8 3 5 2'. ()'.812 0321 O . I H -0.0863 3'. o'^o 0362 0.164 - 0 , 0 8 6 7 4'. 1', 0 01 040 7 0 187 - 0 . 0 3 4 3 5'. 1.035 0 454 0'.209 - 0 . 0 3 3 9 6'. l". i55 0508 0'.230 - 0 . 0 8 1 3 7. 1 .224 0558 0.251 - 0 . 0 7 9 0 ' s . 1.286 0627 0'.270 - 0 . 0 7 5 9 1.336 0708 0.235 - 0 . 0 7 3 8 10'. 0304 0'.29 7 - 0 . 0 7 4 1 i'.soi 11'. 09 11 O'.305 - 0 , 0 7 4 3 12'. 1 .416 1019 0.312 - 0 . 0 7 3 4 13 . T.4 39 1132 0.312 - 0 . 0 7 6 9 14'. r.45i 134 0 09 - 0 . 0 8 0 5 0 15'. f.453 1533 0,298 - 0 , 0 0 8 9 16'. l'.4 43 1711 0.289 - 0 . 0 9 1 5 17'. T.4 29 1901 0.230 -0 ,0996 18. 1 '.413  ALF -10.  CLARK-Y RC=.5(10)6  ALF CL 10. - 0 . 3 2 1 -9'. -0.237 -8. -0'. i l l -7'. -b'. 047 -6'. o'.0 39 -5'. 0'. 132 -4;. 0'.222 0'.315 3. -2'. 0'.4 06 ()'. 493 -f. 0'. (l'.583 (]'.669 i. ()'.749 2'. f' 0'.827 0 '.909 4'. 0'.978 5. 1'.053 6'. 1.12 0 7'. e'. i'.182 1 '.237 9'. r.28o 10'. 11'. r . 310 12'. t'.334 13'. 1.349 14. 1.353 15'. r.34o 16'. 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