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An investigation of the theoretical and experimental aerodynamic characteristics of a low-correction… Malek, Ahmed Fouad 1983

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c  AN INVESTIGATION OF THE THEORETICAL AND EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF A LOW-CORRECTION WIND TUNNEL WALL CONFIGURATION FOR AIRFOIL TESTING BY AHMED FOUAD MALEK B.Sc.(Engineering), Alexandria University, 1970 B.Sc.(Mathematics), Alexandria University, 1974 M.A.Sc.(Mechanical Engineering), University of B r i t i s h Columbia, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering)  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1983  (Q)  Ahmed Fouad Malek, 1983  ,1  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  of B r i t i s h Columbia,,I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by  department or by h i s or her  the head o f  representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department of The U n i v e r s i t y of B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3 x-n  DE-6  (3/81)  Columbi  written  - i i-  ABSTRACT  This thesis deals with a new  approach to reduce wall corrections  i n h i g h - l i f t a i r f o i l t e s t i n g , by employing two s i m i l a r non-uniform transversely s l o t t e d walls.  The s o l i d elements of the s l o t t e d wall are  symmetrical a i r f o i l s at zero incidence, and the spaces between the s l a t s are non-uniform, increasing l i n e a r l y towards the rear.  This wall configuration provides the flow conditions close to the free a i r t e s t environment which lead rections.  to n e g l i g i b l e or small wall cor-  The theory uses the p o t e n t i a l flow surface vortex-element  method, with " F u l l Load" Kutta Conditions and wall s l a t s .  s a t i s f i e d on the t e s t a i r f o i l  This method i s very well supported by physical evidence  and i t i s simple to use.  The surface v e l o c i t i e s can be calculated d i r -  e c t l y and the aerodynamic l i f t and pitching moment are determined by numerical integration of the calculated pressure d i s t r i b u t i o n s around the a i r f o i l contour.  This method can be developed to include a simulation of  the flow i n the plenum chambers i n the analysis.  •Also, the performance of this new wall configuration was experimentally.  Two  examined  d i f f e r e n t sizes of NACA-0015 a i r f o i l were tested  i n the e x i s t i n g low speed wind tunnel a f t e r modifying both the configurat i o n of the side walls and the test section to accommodate the new t e s t . Pressure d i s t r i b u t i o n s about the test a i r f o i l s were measured pressure  taps around their contours.  Also the l i f t s and the p i t c h -  ing moments were obtained by integrating the measured surface The experimental,results  using  pressures.  show that the use of the new wall configuration  - iii with AOAR = 59% would produce wind tunnel t e s t data very close to the free a i r values.  - iv-  TABLE OF CONTENTS P a  Abstract  g  e  i i  Table of Contents  .  iv  L i s t of Figures  vi  L i s t of Tables  ix  Symbols  x  Acknowledgement  ..  xi  I.  INTRODUCTION  1  II.  POTENTIAL FLOW ANALYSIS  6  11.1  Introduction  6  11.2  Surface Singularity Theory  7  II. 3  The Kutta Condition  10  I I I . ASSEMBLING AND SOLVING THE BASIC EQUATIONS  13  IV.  RESULTS OF THEORETICAL INVESTIGATION  19  V.  EXPERIMENTAL INVESTIGATION  24  VI.  EXPERIMENTAL RESULTS  30  VII. CONCLUSIONS AND RECOMMENDATIONS  33  REFERENCES  35  - v(Continued) Page APPENDIX 1 - AN ANALYTICAL REPRESENTATION OF A SURFACE VORTICITY  37  APPENDIX 2 - EVALUATION OF THE INTEGRAL IN EQUATION (5). REF. [10]  40  THEORY:  APPENDIX 3 - CALCULATION OF THE VELOCITY COMPONENT INDUCED AT A POINT IN THE FIELD OF VORTEX DISTRIBUTION  45  APPENDIX 4 - AN ANALYTICAL APPROACH TO THE FLOW IN THE WIND TUNNEL PLENUM  48  A.  Introduction  48  B.  The Induced Tangential and Normal V e l o c i t i e s due to V o r t i c i t y D i s t r i b u t i o n Simulated Shear Layer Tracking  48 50  C.  APPENDIX 5 - LIST OF EQUIPMENT USED IN THE EXPERIMENTAL INVESTIGATION 53  FIGURES  54  TABLES  91  - vi-  LIST OF FIGURES Page FIGURE (1)  COMPARISON OF AIRFOIL PRESSURE COEFFICIENTS: THEORY KEF. [9]  54  FIGURE (2)  VORTEX REPRESENTATION OF TWO COMPONENT AIRFOIL: REF. [10]  55  FIGURE (3)  NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS: REF. [10] 56  FIGURE (4)  STREAMLINES CONTOURS AROUND AN AIRFOIL:  FIGURE (5)  LOCATION OF TRAILING CONTROL POINT:  FIGURE (6)  AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL: APP. 4  59  FIGURE. (7)  NOTATION USED TO CALCULATE INDUCED VELOCITIES:  60  FIGURE (8)  LOCATION OF ELEMENTS ON AIRFOIL SURFACE:  FIGURE (9)  COMPARISON OF HIGHER ORDER METHODS:  FIGURE (10)  COMPARISON OF AIRFOIL VELOCITY DISTRIBUTIONS: REF. [10]  FIGURE (11)  GEOMETRY FOR CALCULATION OF HIGHER ORDER TERMS:  FIGURE (12)  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION  65  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION  66  COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION  67,  FIGURE (13)  FIGURE (14)  REF. [12]  57 58  REF. [10]  REF. [10]  REF. [10]  61 62  REF. [10]  63 APP. 2  64  FIGURE (15) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SINGLE UNIFORMLY SLOTTED WALL TEST SECTION  68  FIGURE (16) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE UNIFORMLY SLOTTED WALL TEST SECTION  69  - Vll (Continued) FIGURE (17)  Page COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN LIFT COEFFICIENTS  70  COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS  71  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION  72  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION  73  COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION  74  COMPARISON OF THE EFFECT OF TWO DIFFERENT ANGLES OF ATTACK ON THE RELATIVE ERROR IN LIFT COEFFICIENTS FOR NACA-0015 AIRFOIL  75  GEOMETRY OF THE IDEALIZED FLOW IN THE PLENUM CHAMBER FOR NACA-0015 AIRFOIL  76  COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS: FLOW IN WIND TUNNEL PLENUM IS INCLUDED IN THEORETICAL ANALYSIS  77  FIGURE (25)  CALIBRATION OF NOZZLE AND TEST SECTION DYNAMIC PRESSURES  78  FIGURE (26)  WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT  79  WALL EFFECT ON THE PRESSURE' COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT  80  EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT  81  EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT  82  COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS: EXPERIMENT  83'  FIGURE (18)  FIGURE (19)  FIGURE (20)  FIGURE (21)  FIGURE (22)  FIGURE  (23)  FIGURE (24)  FIGURE (27)  FIGURE (28)  FIGURE (29)  FIGURE  (30)  - viii (Continued) FIGURE (31)  FIGURE (32)  FIGURE (33)  FIGURE (34)  FIGURE (35)  FIGURE (36)  Page E F F E C T OF WALL CONFIGURATION  ON L I F T C O E F F I C I E N T F O R  NACA-0015 A I R F O I L ,  EXPERIMENT  E F F E C T OF WALL CONFIGURATION  ON L I F T  NACA-0015 A I R F O I L ,  EXPERIMENT  C/H=.67:  84  COEFFICIENT FOR 85  EFFECT OF WALL C O N F I G U R A T I O N ON P I T C H I N G MOMENT COEFFICIENT FOR N A C A - 0 0 1 5 A I R F O I L O F SIZE C/H=.34: EXPERIMENT  86  EFFECT OF WALL C O N F I G U R A T I O N ON P I T C H I N G MOMENT COEFFICIENT F O R N A C A - 0 0 1 5 A I R F O I L O F SIZE C/H=.67: EXPERIMENT  87  EFFECT OF A I R F O I L S I Z E ON E ^ F O R NACA-0015: COMPARISON OF E X P E R I M E N T A N D THEORY  88  AIRFOIL  SECTION CHARACTERISTICS AS AFFECTED BY VARIATIONS  OF T H E REYNOLDS  FIGURE (37)  C/H=.34:  NUMBER:  U.B.C. MECHANICAL WIND TUNNEL  R E F . [17]  89  E N G I N E E R I N G LOW-SPEED C L O S E D - C I R C U I T 90  LIST OF TABLES  TABLE 1  AERODYNAMIC CHARACTERISTICS OF AIRFOILS EXAMINED THEORETICALLY  TABLE I I EFFECT OF THE PRESSURE ALONG THE SHEAR LAYERS ON THE NEW WIND TUNNEL THEORETICAL DATA TABLE III LOCATIONS OF THE PRESSURE TAPS TABLE IV  THE EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF NACA-0015 AIRFOIL  TABLE V  THE LIFT-CURVE SLOPES OBTAINED EXPERIMENTALLY FOR NACA-0015 AIRFOIL  SYMBOLS  Definition Open Area Ratio; OAR = Slot s i z e / ( s l a t size + s l o t s i z e ) . Uniform Open Area Ratio. Average Open Area Ratio. Slat chord:  a i r f o i l chord r a t i o .  A i r f o i l chord:  tunnel test section height r a t i o .  Uniform flow v e l o c i t y . Stream function. Vortex density or induced surface v e l o c i t y . Total head. Static pressure head. Dynamic pressure head. Pressure c o e f f i c i e n t The average pressure c o e f f i c i e n t along the upper constant pressure streamline. The average pressure c o e f f i c i e n t along the lower constant pressure streamline. L i f t coefficient. Leading edge pitching moment c o e f f i c i e n t . Quarter chord pitching moment c o e f f i c i e n t . L i ft-curve slope. Free a i r data. Wind tunnel data. Relative error i n l i f t  coefficient;  * 100 = (m T F m  Relative error i n pitching moment c o e f f i c i e n t s ; M  - C, M c/4  * 100  - xi -  ACKNOWLEDGMENT  This i s to acknowledge Dr. G.V. Parkinson's  expert  supervision, invaluable advice and h i s academic as well as f i n a n c i a l support to me.  These were very important factors i n  helping me carry out this research i n p a r t i c u l a r and achieving a successful graduate work i n general.  Also, I acknowledge the  help I received from the s t a f f of the Department of Mechanical Engineering at U.B.C. during the time I was working on my research.  I especially acknowledge my wife's moral support and her  wonderful e f f o r t i n typing my t h e s i s .  - 1 -  CHAPTER I INTRODUCTION In recent years, much attention has been devoted to a v a r i e t y of problems associated with wind tunnels as simulators of f l i g h t .  I t i s im-  portant to test the largest possible model i n any given f a c i l i t y , not only to maximize Reynolds number but to improve the accuracy of the model i t s e l f and the forces and moments measured i n the tunnel.  Today's a i r c r a f t designers  are demanding that wind tunnels should accommodate testing of a new breed of h i g h - l i f t wings using large models which produce r e a l i s t i c a l l y high Reynolds numbers, and that wind tunnel data should be measured with great accuracy. Aerodynamicists are then pushed towards t e s t i n g large models and that often involves large flow distortions i n a flow regime where wall effects cannot be accurately estimated and where t h e i r effects on the measured aerodynamic data cannot safely be accounted  for by the standard l i n e a r corrections.  An obvious solution to the above problem could be achieved by using wind tunnels with very large test cross-sections. However, this approach i s not always possible since the cost of construction of wind tunnels of s u f f i c i e n t size i s very high, and few such large tunnels are available. Another solution has been suggested by Sears  [1] and others [2], i n which  the test section of the conventional wind tunnel i s modified by i n s t a l l i n g a two-dimensional  test section with perforated top and bottom walls, and  providing for a u x i l i a r y blowing and suction at the top and bottom walls. The model flow f i e l d inside the t e s t section i s detected by means of pressure sensors, then i t i s adjusted by a u x i l i a r y suction (or blowing) through the control valves u n t i l the flow f i e l d agrees with a desired unconfined value.  - 2V i d a l and others [3] have introduced some modifications to Sear's concept of the s e l f correcting wind tunnel.  The basis idea i s to approxi^  mate a continuous d i s t r i b u t i o n of v e l o c i t i e s , i n a stepwise fashion, by segmenting the plenum surrounding the porous walls, and c o n t r o l l i n g the flow through the walls by applying suction or pressure to the plenum segments.  In t h i s way i t should be possible to approximate the unconfined  disturbance flow f i e l d about a two-dimensional  airfoil.  Unfortunately,  a l l the above s e l f correcting wind tunnel arrangements need a complex system of sensors and expensive computer hard ware as well as the a u x i l i a r y equipment such as pumps or blowers.  Pollock [4] has introduced the concept of the s e l f streamlining wind tunnel, i n which a set o f tensioned membranes replace the s o l i d walls and under the action of the model pressure f i e l d , deform i n the same way as a streamline i n an i n f i n i t e flow f i e l d .  In order to prevent the c o l -  lapse of the membrane towards the model i t i s then equipped with a number of equally spaced pressure tappings which communicate with f l e x i b l e bellows with r i g i d connecting tubes.  The volume behind the wall membrane and  containing the f l e x i b l e bellows i s subject to a backing pressure which i s adjusted so that the sum of the normal wall force components at the two ends of the wall i s equal to zero.  One of the basic problems  with such conformable walls i s that the membrane tends to bulge between the pressure tappings under the action of the pressures acting on either side; also i t i s d i f f i c u l t to manufacture.  - 3 -  An alternative to the above arrangements seems to be needed f o r modern subsonic t e s t i n g .  Such a method should enable us to modify our  e x i s t i n g wind tunnels with wall arrangements which are not only r e l i a b l e and cheap to operate, but also can answer the c a l l for r e a l i s t i c measurements and accurate corrections.  Contributions to such a method are pre-  sented i n t h i s t h e s i s .  I t i s well known that most corrections to data i n wind tunnels with open j e t s are opposite i n sign to those i n wind tunnels whose walls are s o l i d , Ref.  [5].  These opposing e f f e c t s suggest the strategy of  employing  p a r t l y s o l i d , p a r t l y open walls i n pursuit of cancelling the corrective e f f e c t s of the two types of w a l l . purpose have been considered. slots.  Recently, two such designs for t h i s  One has walls with narrow longitudinal  The other has walls patterned with small holes.  Using the l i n e a r theory to investigate these two types of wall configurations, Parkinson and Lim  [6] and others have found that there i s  a lack of agreement between the experimental results and those which are predicted by the theory for the longitudinal s l o t t e d w a l l .  Also they have  found that the "porosity parameter" i s not simply an empirical function of the open area r a t i o but i t must be determined empirically for each a i r f o i l under t e s t , an impossible s i t u a t i o n f o r the p r a c t i c a l use of porous wall configurations.  - 4 -  Parkinson and Lim [6] have attributed the lack of success of the longitudinal s l o t s and porous wall theories to the occurrence experimentall y of separated flows i n the s l o t s and holes.  Such flows either are not  accounted f o r i n the theories, p r i m a r i l y as they add undesirable nonlinearit i e s to them, or are dealt with semi-empirically. separations may  In addition, these flow  seriously degrade the main flow i n the v i c i n i t y of the  walls.  Williams [7], using a surface source method, has investigated another type of wind tunnel, one which has an upper transversely s l o t t e d w a l l , with uniform gaps and  airfoil-shaped s l a t s , and a lower s o l i d w a l l .  His t h e o r e t i c a l analysis shows that using t h i s type of wall configuration w i l l improve the performance of the tunnel test section.  The analysis  predicts that such a test section w i l l produce l i f t data within a small percentage error of the f r e e - a i r values, while the pressure d i s t r i b u t i o n and the p i t c h i n g moment data are of lower accuracy, F i g . (1).  Also h i s  experimental investigation shows that such a wall configuration provides a flow free of separation on the transverse s l a t s .  In the present thesis, a d i f f e r e n t surface s i n g u l a r i t y analysis, using a surface v o r t i c i t y method, i s used to investigate a wind tunnel with both walls transversely s l o t t e d as another approach to modify the wind tunnel wall configuration i n order to recreate the f r e e - a i r streamline patterns about the t e s t a i r f o i l , which would then experience the corresponding freea i r loading.  - 5 -  The approach here uses symmetrically  transversely-slotted upper  and lower walls, with symmetrical a i r f o i l - s h a p e d s o l i d s l a t s at zero i n cidence.  The flow i n c l i n a t i o n s near the wall w i l l be small, for a l l  p r a c t i c a l cases envisaged.  Hence a l l the wall s l a t s w i l l operate within  t h e i r unstalled incidence range, so that the a i r which flows near the wall w i l l be free of separated wakes.  A uniform spacing of the wall s l a t s shows that (see F i g . (16))  the  upper surface of the a i r f o i l section i n the presence of the s l o t t e d wall tends to experience a s l i g h t l y lower negative pressure d i s t r i b u t i o n than that of the f r e e - a i r conditions near the leading edge and s l i g h t l y higher further a f t .  The effects tend to cancel for l i f t but lead to appreci-  able errors i n p i t c h i n g moment.  A solution to this problem i s to use  graded, narrow gaps upstream of the test a i r f o i l and wider ones downstream, rather than uniform spacing between the wall s l a t s .  Also the surface s i n g u l a r i t y analysis has been used here to simulate the shear layers which enter or leave the t e s t section from or to the upper and lower plenum, respectively, i n order to represent the flows i n there.  The success of the above wall configuration depends on the experimental v e r i f i c a t i o n of the performance of the wind tunnel which i s equipped with the proposed wall arrangements. investigation are presented here.  The'results of such an experimental  - 6 -  CHAPTER II POTENTIAL FLOW ANALYSIS II.1  Introduction An e f f i c i e n t , r e l i a b l e method for c a l c u l a t i n g the v e l o c i t y d i s t r i -  bution on the surface of a i r f o i l sections i s required. formation methods such as that of a r b i t r a r y shape.  Conformal trans-  Theodorsen [8] can analyze sections of  These methods are based on the theorem which states that  i t " i s always possible to transform the p o t e n t i a l f i e l d around any closed contour into the p o t e n t i a l f i e l d around a c i r c l e .  Such methods are not .  simple and, as there i s no such theorem for transforming the p o t e n t i a l f i e l d around multi-component sections, one looks to surface s i n g u l a r i t y methods of analysis.  These methods replace the p o t e n t i a l flow f i e l d  side the a i r f o i l contour with that about a set of s i n g u l a r i t i e s ,  out-  sources  or v o r t i c e s , which s a t i s f y the same boundary conditions.  The surface s i n g u l a r i t y methods can deal e a s i l y with multi-component sections and are of comparable accuracy with conformal transformation methods for  single-component cases.  The most widely used surface s i n g u l a r i t y method  [9] employs sources and sinks on the surface of the a i r f o i l section combined with a v o r t i c i t y d i s t r i b u t i o n to generate c i r c u l a t i o n .  Williams  others, has employed t h i s to investigate his wind tunnel model. has, however, some drawbacks.  [7], among The technique  One p a r t i c u l a r problem arises from the  application of the Kutta Condition, i n the form of equal v e l o c i t y magni-. tudes at the control points of the upper and lower t r a i l i n g (U  edge elements  = ~U )• This can be c a l l e d the NO LOAD Kutta Condition since t_ t 1 u  - 7 -  i t eliminates any l i f t from the a i r f o i l near the t r a i l i n g edge which i s i n c o n f l i c t with the aim of examining sections with large rear loading. To overcome t h i s problem a better method must be developed.  Recently a d i f f e r e n t surface s i n g u l a r i t y method was developed by Kennedy [10]. This method uses a d i s t r i b u t i o n of vortices on the surface of the a i r f o i l section.  The vortex density, which i s determined d i r e c t l y ,  i s equal to the surface v e l o c i t y (this i s shown i n Appendix 1). The boundary condition which i s applied here i s that a l l s o l i d surfaces are streamlines on which the stream function i s required to be constant.  II.2  Surface S i n g u l a r i t y Theory In two dimensional,  incompressible,  function must s a t i s f y Laplace's  i5l 9x  2  +  i 9y  2  i =  i r r o t a t i o n a l flow the stream  equation,  0  ( 1 )  For the flow over a i r f o i l sections there can be no normal v e l o c i t i e s at the s o l i d surfaces, and thus each surface i s a streamline of the flow. Since the stream functions Y s  (S=l,2,....,M) on the surfaces of M com-  ponents on a multi-component section are constants, the boundary condition for  equation  (1) can be written as,  f  ¥ , on the surface S  =  (2)  The stream function f o r a uniform stream incident to the p o s i t i v e X axis at an angle cc i s given by,  m = y cos a - x s i n a  (3)  - 8 which s a t i s f i e s equation (1).  This equation, and a l l subsequent equations  are i n dimensionless form.  The distances are dimensionless with respect to the chord length C, the v e l o c i t i e s with respect to the free system v e l o c i t y u<» and the stream functions with respect to the product  U^C.  The point vortex of strength T, located at ( X , Y ) has the stream q  function, Y = - r _ £n ( r ) ,  ( 4 )  1/2 where  r = <(x-x \  o  )^ + (y-y )^ i  |  equation (4) also s a t i s f i e s (1) , except at r=*0.  Because of the l i n e a r i t y  of equation (1) any c o l l e c t i o n of point vortices or any continuous d i s t r i b u t i o n of them as i n F i g . (2), that l i e s on the a i r f o i l surface, S, w i l l s a t i s f y equation (1) i n the region outside of S.  Then the stream  function at a general point P due to v o r t i c i t y having a density y(S') at S' and continuously d i s t r i b u t e d over the a i r f o i l surface S, i s given by, T  p  = ^  ^Y(S')  In  r (P , S-') d S'  (5)  s Applying the boundary condition, equation (2), to the combined flow due to a uniform stream plus the above d i s t r i b u t i o n of the v o r t i c i t y , one obtains, ¥  S  =  y  S  cos a - x  S  sin a - i 2TT  lY(S') /  In r(S,S') d S'  (6)  - 9The a i r f o i l surface i s divided up i n some manner into N small surface elements.  On each of these there i s a control point, C^, located at  (x^, Y^), at which the boundary condition, equation  (6), i s made to apply.  Each element j has v o r t i c i t y of density Y(Sj) d i s t r i b u t e d on i t s surface. The i n t e g r a l i n equation  (6), over the whole surface S, i s then replaced  by a summation of N i n t e g r a l s over the N surface elements. equation  Applying  (6) at the control point, C^, one obtains,  ^  N E  +  j=l  r  -T-  H(S\) £ j  ' ^  In r ( C , S\) d S! = y.cos a - x. s i n a 1  3  (7)  3  The r e s u l t s required of an a i r f o i l analysis method are the surface v e l o c i t i e s .  Kennedy [10] shows that the tangential v e l o c i t y at the  i n t e r i o r of the s o l i d surfaces has to be zero so that these surfaces become streamlines, which also r e s u l t s i n the discontinuity i n the tangential v e l o c i t y across a vortex sheet being equal to the density of the vortex sheet. equation  Thus Y (S^) i s equal to the surface v e l o c i t y .  i n solving the  (7) one therefore solves d i r e c t l y for the v e l o c i t i e s on the a i r -  f o i l surfaces.  At t h i s point i t i s necessary to make some assumption about the section geometry, the location of the control points and the form of Y (S_.) over each element j .  The simplest approximation i s to assume that  the elements are s t r a i g h t l i n e s with control points at the element midpoints and Y (S.) i s a constant over each element. 3  Using the above approximation and applying equation control point y i e l d s the system of equations,  (7) at each  - 10 -  ¥  N I  + S  "  j=l  K.. Y. = R1 3  3  > (i =  N)  (8)  1  where K . i s the influence c o e f f i c i e n t of the element j on the control point i , R. i s the right hand side of equation (7) evaluated at control point i and Y  i s the stream function for the a i r f o i l component.  Using the notation of F i g . (3), the BASIC influence c o e f f i c i e n t s can be written, K  . . = * ' {(b+A) Jin (r?) - (b-A) In (rj?) + 2a t a n " ip 4TT 1 2  1  ( , ^ ,) a +b -A 2 a  4A}  (9) The d e t a i l s of the calculation of t h i s equation are provided i n Appendix 2, which i s extracted from Ref.  [10].  The K.. and R. are purely functions of the geometry of the surface elements and the angle of attack.  The system of equations  (8) i s a set of  N equations for the N unknown Y- and M unknown ¥ , where there are M ponents.  com-  The M additional equations required for a solution to t h i s  problem are termed the Kutta Conditions and there i s one for each component i n the test a i r f o i l section, and each a i r f o i l s l a t i n the w a l l , for the cases to be considered l a t e r .  II.3  The Kutta Condition Kutta and Joukowski were concerned with a i r f o i l sections whose geo-  metries are calculated by a conformal transformation technique which maps the flow over a c i r c u l a r cylinder into the flow over an a i r f o i l section with a cusped t r a i l i n g edge.  These sections have two stagnation points,  - 11 -  one located near the leading edge and the other near the t r a i l i n g edge. Also the v e l o c i t y at the t r a i l i n g edge w i l l be, i n general,  infinite.  They both proposed that the c i r c u l a t i o n around the c i r c u l a r cylinder be adjusted so that one of the stagnation points i n that flow be located at the point which w i l l map into the a i r f o i l t r a i l i n g edge. In t h i s case the i n f i n i t e v e l o c i t y and the stagnation point, occurring together at the t r a i l i n g edge, cancel and y i e l d a f i n i t e , non-zero v e l o c i t y there.  I t has been shown by Milne-Thomson  [11] that a consequence of  t h i s assumption i s that the stagnation streamline leaves the cusped t r a i l i n g edge tangent to i t and photographs of flow v i s u a l i z a t i o n studies of Prandtl and Tietjens [12] show this e f f e c t c l e a r l y , see F i g .  (4).  This condition can be modelled by providing an additional control point just o f f the t r a i l i n g edge. f u l l y by Bhateley and Bradley  [13].  Such a Kutta Condition was used successThe bisector of the t r a i l i n g edge i s  extended into the free stream and a control point placed a small f r a c t i o n of chord downstream of the t r a i l i n g edge, as i t i s shown i n F i g . (5). I t i s then assumed that the streamlines through the other control points of that component  also pass through this control point.  applies to these t r a i l i n g control points, C  Equation (8) then  , and the Kutta Condition can  be written as, N Y  S  +  m  . \ p ,j i*=l "-m E  Y  1  j  =  \p  , (m = S=1,2,....M) m  (10)  There are M such t r a i l i n g control points, one for each component, and hence M Kutta Condition  equations.  - 12 Thus the problem of p o t e n t i a l flow over an a i r f o i l section has been reduced to that of solving (N+M) (8) and  equations, prescribed by equations  (10), simultaneously to get N vortex densities, y , and M stream  functions  - 13 -  CHAPTER III ASSEMBLING AND SOLVING THE BASIC EQUATIONS OF THE POTENTIAL THEORY The f i r s t step i n the solution i s to define the elements which describe the a i r f o i l surface.  One obvious method of doing t h i s i s to l e t  the supplied co ordinates be the end points of the surface elements. has the disadvantage that there may or that they may  be i n s u f f i c i e n t co-ordinates available  be i r r e g u l a r l y spaced.  To overcome these problems the  a i r f o i l i s divided up, from i t s leading edge at x=0 at x=l. Fig.  This  to i t s t r a i l i n g edge  The end points of the surface elements are located, as shown i n  (8), at x x  0  co-ordinates given by,  = 1 (l-cos<J>,J ,  (£ = 0,1,2,  ,N)  (11)  where 4>„ = 2IT*.  Here N must be an even number i n order that the end point be located at the a i r f o i l t r a i l i n g edge.  This d i s t r i b u t i o n of points pro-  vides, i n general, a more accurate solution because i t concentrates  the  control points near the leading edge and t r a i l i n g edge where the largest v e l o c i t y gradients generally occur.  The corresponding  co-ordinates y^  of the element end points are determined by i n t e r p o l a t i o n on the given a i r f o i l data.  The use of a cubic spline function has been found to be  the most r e l i a b l e method, since i t gives smooth curves through the given points and can be e a s i l y and e f f i c i e n t l y computed.  Here the U.B.C. com-  puter subroutine SAINT has been used for i n t e r p o l a t i o n .  - 14 -  The control points are taken as the mid-points of each surface element, as shown i n F i g . (3), then the b i s e c t o r of the t r a i l i n g edge i s extended, as shown i n F i g . (5), and the control point i s located on t h i s extension a distance O.Olt from the t r a i l i n g edge.  This distance was  found to give the most r e l i a b l e r e s u l t s for a wide range of a i r f o i l sections.  The a i r f o i l system of axes x-y should be rotated clockwise an angle ot, the angle of attack, then the co-ordinates of the element end points with respect to the wind system of axes X-Y w i l l be given by,  X. = x. cos a+ y. sin a 1 1 i  (12)  Y. = y. cos a - x. s i n a i i i Having determined  the co-ordinates of the element end points and  control points one can proceed to calculate the BASIC influence c o e f f i c i e n t K..,  which i s given by equation (9), and R,, which as a r e s u l t of the  rotation of the axis should be given by, R  ±  = Y  (13)  ±  As the vortex densities are i d e n t i c a l to the surface v e l o c i t i e s , counter-clockwise about the a i r f o i l section, the co-ordinates of element end points and control points should be taken i n that order around the polygonal contour.  For a single-component a i r f o i l i n free a i r , the system of equations (8) and  (10) can be written i n the matrix form as,  - 15 -  i=l  K  K, 1,N  1,1  1  Y  l  R  l  (14)  i=N i=N+l  ^,1  •  •  • •  N  Kutta Condition  Y  N  R N  1  R tp  The above system of equations i s then solved f o r the unknown N vortex densities Yj  a n <  ^ the stream function  When this technique i s extended to multi-component a i r f o i l sections the  point d i s t r i b u t i o n , given by equation (11) i s f i r s t scaled to the  chord of each i n d i v i d u a l component before being applied.  -  I t i s then nec-  essary to move each component to i t s correct location.  This i s done by  specifying the amounts by which the leading edge of the component i s translated and the angle through which the component i s rotated.  With the  geometry thus defined, one can calculate the K^^ and R^ from equations (9) and (13).  The multi-component case gives r i s e to a d i f f e r e n t stream function for each component and each component has i t s own Kutta Condition.  A two-  component a i r f o i l with N elements on each component gives r i s e to a system of equations which can be written:  - 16 K  i=l  1,1  • •  K  i=N  1,2N  *Sj,2N  i=N+l  V i , i  i=2N K  1  X  * '  2N,2N °  K  0  Y  1  Y  Y  1  l=2N+2 Kutta Condition, component  2  =  V i  R  2N  Kutta Condition, component  1  N  N+l  1  R  l  Y  -  Vl,2N°  2N,1 '•  0  (15)  2N  1 ^2  R t P  2  The U.B.C. computer subroutine FSLE has been used here to solve the system of equations (14) and (15) .  The solutions are the dimension-  less surface v e l o c i t i e s at the control points and the dimensionless functions of each component.  The pressure d i s t r i b u t i o n , l i f t  stream  coefficient  and the leading-edge p i t c h i n g moment can be calculated from the v e l o c i t i e s as follows: 2 C P  i  = 1-Y. N  C  L  C„  =  I  . 1=1  N = - Z  C Ax. p. l 1  C  (x. Ax, + y. Ay.)  (16)  - 17 -  and summations are performed counter clockwise around the polygonal contours (as i t i s shown i n Appendix 1).  The technique described so f a r makes the simplifying assumptions of straight l i n e elements and constant vortex density on each element, which i s referred to here as the BASIC method.  Kennedy [10] however, has  studied the effects of including the higher order terms due to surface curvature and a l i n e a r l y varying vortex density on each element. s u l t s are shown i n F i g . (9).  The re-  These results show that the inclusion of  element curvature raises the v e l o c i t i e s while including the l i n e a r vortex density decreases the v e l o c i t i e s .  Also from Appendix 1, one can notice  that the two terms which introduce the l i n e a r v e l o c i t y d i s t r i b u t i o n and surface curvature into the influence c o e f f i c i e n t s are of the same magnitude but of opposite signs, thus t h e i r effects tend to cancel when they are combined.  I t i s therefore recommended that only the BASIC method with straight  l i n e elements and constant vortex density be used.  Kennedy's investigation [10], has shown that the BASIC method with " F u l l Load" Kutta Condition gives accurate r e s u l t s for most a i r f o i l s .  An  example of a section with a f a i r l y sharp peak i n the v e l o c i t y d i s t r i b u t i o n i s given i n F i g . (10).  This i s a Joukowski a i r f o i l with a cusped  trailing  edge for which 40 elements were used to describe the section, and there i s excellent agreement with the exact solution.  In the current investigation the single-component a i r f o i l sections, NACA-0015 and CLARK-Y 14% have been represented by 40 elements while the two-component a i r f o i l , NACA-23012 with 25% s l o t t e d f l a p deflected 20 degrees,  - 18 has been represented by 70 elements (40 elements f o r the main a i r f o i l and 30 elements f o r the f l a p ) .  The wind tunnel with s o l i d walls i s represented  by 128 elements while the wind tunnel with s l o t t e d walls i s represented by 64 elements f o r the s o l i d parts of the wall and 8 elements f o r each o f the 16 s l a t s (8 s l a t s for each wall, with c/C = .15).  The e a r l i e r part of t h i s research was reported as a University of B r i t i s h Columbia M.A.Sc. Thesis (Ref. [15]).  I t was i n t h i s t h e o r e t i -  cal study that the optimum AOAR of 59% was established by considering values ranging from 50% to 70% f o r the same three a i r f o i l p r o f i l e s considered here.  Also i n t h i s e a r l i e r study an attempt was made to simulate  the shear layers bounding the plenum flows.  This model was extended i n  the present investigation, and i t i s described i n Appendix 4.  To include the flow i n the wind tunnel plenums i n the current analysis, each bounding streamline (two of them are considered, one f o r each plenum, with two d i f f e r e n t prescribed constant pressure c o e f f i c i e n t s ) i s then divided into 8 elements, which adds 16 vortex elements to the previous 232 elements i n the case where NACA-0015 i s the test a i r f o i l .  In a l l of the above numerical analyses, the t o t a l extent of the wind tunnel wall has been taken as four times the test a i r f o i l chord, C. Also the gaps i n the non-uniform s l o t t e d walls, with AOAR=59%, are increasing l i n e a r l y towards the rear of the test a i r f o i l , with 2% increment.  The test  a i r f o i l was i n each case located with i t s mid-chord point at the tunnel center-line, and rotation for angle of attack was about this point.  F i n a l l y , FSLE i s a subroutine which uses the Gaussian method to solve a general r e a l matrix backward s u b s t i t u t i o n ) .  Elimination  (with p a r t i a l pivoting, forward and  - 19 -  CHAPTER IV RESULTS OF-THE THEORETICAL INVESTIGATION The t h e o r e t i c a l curves presented next were calculated by the method of Ref.[10]. theoretical  Figures (12), (13) and  (14) show comparisons of  pressure d i s t r i b u t i o n s for the a i r f o i l s , of chord C, NACA-  0015 at a =10? NACA-23012 at a = 8°with 25.66% s l o t t e d f l a p at 5 = 20°and CLARK-Y 14% at a = 10? respectively, i n free a i r and between the s o l i d walls of a conventional wind tunnel t e s t section of height h, with C/h = .8. tunnel l i f t c o e f f i c i e n t s C  The  are 1.6873, 3.6448 and 3.0131 which are 37.66%,  T 30.36% and 41.11% higher than the free a i r l i f t c o e f f i c i e n t s C_ L  of 1.2257, F  2.7959 and 2.1353 respectively.  The very large errors i n the a i r f o i l l i f t c o e f f i c i e n t s developed i n the wind tunnel with s o l i d walls shown by the above r e s u l t s , prompted a search of a wall configuration that would exhibit the known cancelling effects of p a r t l y open, p a r t l y closed walls, and which would therefore provide n e g l i g i b l e or small errors.  From the above r e s u l t s i t can be seen that nearly a l l of the i n creased values i n the tunnel l i f t c o e f f i c i e n t s are due to the greater suction over the top surface of the a i r f o i l .  Therefore, the f i r s t step towards the  modification of the conventional wind tunnel was to change the wall c o n f i guration of the upper s o l i d wall, and t h i s was done here by using a uniformly transversely-slotted upper w a l l .  The pressure d i s t r i b u t i o n for the a i r f o i l  NACA-0015 at a = 10° for such a wall configuration of 59% UOAR, with s o l i d lower walls, appears i n F i g . (15) along with the corresponding pressure  - 20 d i s t r i b u t i o n f o r free a i r .  This figure shows some improvement i n the  pressure d i s t r i b u t i o n opposite to the upper uniformly s l o t t e d wall, but i t also shows that there are differences i n the lower surface  pressure  d i s t r i b u t i o n s , and i t suggests that the flow there i s experiencing  lower  induced v e l o c i t i e s i n the presence of the lower s o l i d wall than i n free air,  and a possible solution to t h i s problem i s to use a s l o t t e d lower wall  as well.  Accordingly, the wind tunnel with double uniformly  transversely  s l o t t e d wall has been investigated here and the r e s u l t s , as i n F i g . (16), show that the pressure d i s t r i b u t i o n s for NACA-0015 at a = 10°and o f C/H = .8, opposite to the lower s l o t t e d wall i s closer to the free a i r values than before, while the upper surface suctions s t i l l ing  tend to be low near the lead-  edge and s l i g h t l y high near the t r a i l i n g edge, for the same UOAR 59%. A _  solution to t h i s problem would appear to be to use graded, wider gaps rearward and narrow ones forward, rather than uniform spacing between the wall slats.  All  the above support the motivation  for using the double s l o t t e d  wall test section with nonuniform spacing, F i g . (6a). Several average OAR have been examined to look for the most suitable wall configuration which would develop the l e a s t errors i n the tunnel l i f t and pitching moment coe f f i c i e n t s f o r the three d i f f e r e n t a i r f o i l s , mentioned before, and of d i f ferent s i z e s .  I t has been found that AOAR=59% i s suitable for t h i s wide  range of a i r f o i l shapes and s i z e s .  Figure  (17)shows comparisons of the e f f e c t on the t h e o r e t i c a l l i f t  - .21 c o e f f i c i e n t s o f the r a t i o - f o r d i f f e r e n t a i r f o i l s and t e s t section wall H configurations.  The a i r f o i l s and t h e i r angles of attack are those used  before, and the t e s t section walls are either s o l i d or double uniformly s l o t t e d with AOAR=59%.  I t can be seen that with the s o l i d walls, for the  calculated range of -, the corrections can exceed 40% of the free a i r values, while with a s u i t a b l e AOAR of the s l o t t e d t e s t section the preC dieted errors can be kept within 4% for the three a i r f o i l s , and for -.sr .8. Also, F i g . (18) shows comparisons of the e f f e c t on the t h e o r e t i c a l C quarter chord p i t c h i n g moment c o e f f i c i e n t s of the r a t i o - for the same a i r H f o i l s and t e s t section wall configurations.  I t can be seen that with the  s o l i d walls the corrections can exceed 3% of the free a i r values, while with the s l o t t e d t e s t section of AOAR=59% the predicted errors can be kept withC in  1.2% for the three a i r f o i l s , and for -«r.8.  Figures  (19), (20) and (21)  show comparisons of the t h e o r e t i c a l pressure d i s t r i b u t i o n s , for the above a i r f o i l s and angles of attack, i n free a i r and i n the double s l o t t e d t e s t section of AOAR=59%, again with C_=».8. H  I t can be seen that there i s quite  good agreement between the two d i s t r i b u t i o n s for the three d i f f e r e n t a i r foils. Figure  (22) shows comparisons of the e f f e c t on the t h e o r e t i c a l l i f t  C c o e f f i c i e n t of the r a t i o - for NACA-0015 a i r f o i l at two d i f f e r e n t angles of H o o attack, a = 10  and a = 20 , and with two d i f f e r e n t wall configurations;  s o l i d walls and double nonuniformly s l o t t e d walls of AOAR=59%.  The two wall  configurations exhibit the known blockage e f f e c t of increasing the error i n l i f t c o e f f i c i e n t s as the angle of attack increases, however the e f f e c t i s not very appreciable when the s l o t t e d walls are used. that the s l o t t e d walls with AOAR=59% s t i l l  Also, F i g . (22) shows  i s the most suitable wall configuration  - 22 not only for a wide range of a i r f o i l sizes and shapes but also for a wide range of angles of attack.  For a more accurate representation of the flow i n this double slotted wall test section the l i m i t i n g upper and lower streamlines AB and CD, as shown i n F i g . (6)a, should be considered.  Physically, the stream-  l i n e s are shear layers and they could be idealized as streamlines at constant pressure.  Using the method which has been developed here (see  Appendix 4) f o r tracking streamlines, t h e i r geometries can be described and consequently  they can be included i n the present analysis.  Figure (23) shows the geometry of the idealized flow i n the plenum chambers, the two bounding streamlines of a test section with the new wall o c configuration of AOAR=59%, and the test a i r f o i l NACA-0015 at a = 10 , - = .8. It also shows the values of the stream function, Y , and the average pressure s c o e f f i c i e n t C ; for each streamline. Note that Y > V and > Y„ because p u 1 Z 2 of the approximate numerical procedure which has been used i n determining the geometry of these streamlines.  Figure (24) shows comparisons of the t h e o r e t i -  c a l pressure d i s t r i b u t i o n s , for NACA-0015 a i r f o i l at angle of attack a = 10°, i n free a i r and i n the double slotted t e s t section of AOAR=59% with - = .8, H  and the flow i n the plenum i s included; the average pressure c o e f f i c i e n t along the upper streamline i s C  =-.2  the lower streamline i s C  = .11.  (19) and  while the average pressure c o e f f i c i e n t along I t can be seen from a comparison of Figures  (24) , that the proposed a n a l y t i c a l approach to the flow i n the wind  tunnel plenum does not produce a test a i r f o i l pressure d i s t r i b u t i o n i n as good agreement with free a i r values as does the simpler model of Figure (19). Hence the plenum model analysis i s relegated to Appendix 4, as useful background information.  - 23 Table I contains a l l the t h e o r e t i c a l data, l i f t and pitching moment c o e f f i c i e n t s , presented i n t h i s chapter.  Also, Table II has the t h e o r e t i c a l  l i f t and pitching moment c o e f f i c i e n t s for NACA-0015 a i r f o i l , tested i n the double non-uniform  transversely slotted walls when the flow i n the plenum i s  included i n the analysis (the two shear layers i n the upper and lower plenum are simulated by two constant pressure streamlines).  - 24 -  CHAPTER V EXPERIMENTAL INVESTIGATION The success of the proposed transversely s l o t t e d upper and lower walls with the non-uniform gaps, depends on the experimental examination of the performance of a wind tunnel t e s t section devised with the new wall configuration.  V.1  Such experimental v e r i f i c a t i o n i s described as follows.  Test Section Specifications The experiments were performed i n a two-dimensional test-section  insert designed and b u i l t for an e x i s t i n g low-speed c l o s e d - c i r c u i t wind tunnel.  This i n s e r t i s 915-mm wide by 388-mm deep i n cross-section, and  2.59-m long.  The t e s t a i r f o i l s were mounted a t the midpoint of the t e s t  section and spanned the 388-mm depth.  The side walls  (two of them) were  surrounded by .39 by .3 by 2.44-m plenums and they could be f i t t e d with shaped s l a t s of NACA-0015 section and chords of 92-mm, at zero (8 s l a t s were used for each s i d e ) .  airfoil-  incidence  A f u l l range of wall average open area  r a t i o (AOAR) could be tested, as the s l a t s were f i t t e d with metal s l i d e r s which i n turn were separated by wooden spacers i n an aluminum channel recessed i n the side wall foam.  Modifications to the e x i s t i n g wind tunnel consisted of an inserted nozzle and d i f f u s e r section i n addition to the 388 by 915-mm t e s t section. The t e s t section walls are p a r a l l e l and s o l i d for  (or s l o t t e d ) ; no provision  compensation for boundary layer growth i s attempted.  Williams' [7]  experimental research, using the same wind tunnel and t e s t section showed that the test section wind speed i s s p a t i a l l y uniform to within 0.3%  i n the central "core" flow outside the wall boundary layers.  Also,  boundary layer pitot-stati.c tube measurements i n the empty test section (over the range of wind speeds covering the range of Reynolds numbers for a l l a i r f o i l tests) indicated a displacement thickness of the order of 8-mm, where the test a i r f o i l s would be mounted.  Pankhurst and Holder  [5—ii3  show that a wind tunnel test section, with the above wind q u a l i t y , would produce a i r f o i l pressure measurements (taken a t the mid-span) with very small boundary layer e f f e c t s .  They indicate that a wall boundary layer,  as the one specified before, w i l l not cause any contamination of the area around the a i r f o i l pressure taps.  V.2  A i r f o i l Models Tested Two d i f f e r e n t a i r f o i l s have been tested.  The surface pressure  measurements were made on the two a i r f o i l s , i n addition the l i f t and pitching moment data were obtained by the integration of the pressure c o e f f i c i e n t s . The a i r f o i l s for which r e s u l t s are reported here are two a i r f o i l s of NACA-0015 section, 383-mm span and 307 and 616-mm chord, which were machined from s o l i d aluminum b i l l e t s to close tolerances. Each a i r f o i l was mounted on a c i r c u l a r spar which passed through a c i r c u l a r hole i n the test section f l o o r with 3-mm clearance a l l around. were less than 2.5-mm on a l l t e s t s .  The f l o o r and c e i l i n g t i p clearances The test a i r f o i l s were mounted ver-  t i c a l l y on the turntable of a 6-component balance, which was not used i n obtaining the l i f t s and the moments i n the present work; the reason w i l l be explained l a t e r i n the chapter.  Both a i r f o i l sections were f i t t e d with a number of center span pressure taps.  A l l pressure taps are flush mounted and have 5-mm diameter  - 26 orifices.  P l a s t i c tubes of 1.6-mm inside diameter and approximately  1-m  length transmit the surface pressures through the mounting spar to a location external to the test section.  Pressures were measured v i a a 48 port  "Scanivalve" manual scan pressure transducer.  The signal coming out of  the "Scanivalve" was  system "NEFF", which was  fed to a data acquisition  hooked to the computer PDP-11.  With the help of a computer program, along  with the "NEFF" system, the surface pressures were measured, then the pressure c o e f f i c i e n t s  were calculated and recorded instantaneously i n a computer  f i l e which could be c a l l e d any time l a t e r .  V. 3  Test  Procedures  The test section wind speed was deduced from a p i t o t - s t a t i c tube mounted on the flow centerline i n the tunnel nozzle midway between the s e t t l i n g chamber e x i t and the test section entrance.  Located thus, the  p i t o t - s t a t i c tube would be far enough up-stream to be r e l a t i v e l y unaffected by the test model "blockage" e f f e c t s .  Also the p i t o t - s t a t i c tube  would be s u f f i c i e n t l y far downstream i n the nozzle that the flow speed would produce numerically large p i t o t - s t a t i c tube readings (mm of water). Thus a s u f f i c i e n t l y accurate reading on the "Betz" micromanometer could be obtained.  The nozzle p i t o t - s t a t i c tube was  calibrated against a second  p i t o t - s t a t i c tube mounted i n the empty s o l i d walled test section, on the flow centerline, where the test a i r f o i l s would be located.  The second  p i t o t - s t a t i c tube was connected to a "Barocel" pressure transducer; i t s output i s read using a d i g i t a l v o l t meter.  During tests, the t o t a l head  i n the nozzle i s measured (with the same p i t o t - s t a t i c tube) using a  - 27 Barocel E l e c t r o n i c Manometer; i t s output signal was fed to the data acquis i t i o n system "NEFF".  The t o t a l head was used as a c a l i b r a t i o n pressure  for the "Scanivalve" pressure transducer.  The reference wind speed and s t a t i c pressure used to reduce the pressure measurements are determined as follows:  Since the t o t a l head at  the nozzle p i t o t - s t a t i c tube (when there i s a t e s t a i r f o i l i n place, a t incidence) i s e s s e n t i a l l y the same as the t o t a l head i n the test section i n the v i c i n i t y of the test a i r f o i l , the reference wind speed and s t a t i c pressure can be deduced from the nozzle p i t o t - s t a t i c tube measurements of the t o t a l head and dynamic pressure.  This equality of t o t a l heads can only  be v e r i f i e d i n the empty test section (solid walls) with no test a i r f o i l i n place; such measurements indicated that the test section t o t a l head was lower than the nozzle t o t a l head by 1 part i n 200.  Let  H, P and q be the t o t a l head, s t a t i c pressure head and dynamic  pressure head respectively, and l e t the subscripts "N" and "T" refer to the nozzle and test section respectively.  In the empty test section ( s o l i d  w a l l s ) , according to the above, H = P + q = H = P + q . N N N T T T  (17)  v  If K i s the empty test section (solid walls) calibrated r a t i o of q  N  to q^, then since only  and ^  are measured while under t e s t , the  equivalent empty test section (solid walls) dynamic pressure head and s t a t i c pressure head are respectively as follows:  <  1  8  )  - 28 -  and  P  T " \  -  K  %  l l 9 )  Thus the reference wind speed and s t a t i c pressure used to reduce data taken for any test a i r f o i l , at any incidence, i n the presence of any  slotted-  walls AOAR, are the equivalent values i n the undisturbed uniform stream conditions that would actually occur i n the empty (solid walls) test section corresponding to that measured pressure c o e f f i c i e n t , C  and q  N  i n the nozzle.  To evaluate the  , of the test a i r f o i l at any point x along the P  x  a i r f o i l center-span contour  (where there i s a pressure tap), the following  was considered; C  = <V T P  x  Using equations  C  px =  { P  ) / C  *T  ( 2 0 )  (18) and  x"<V  K  V  (19) along with equation (20); therefore  }  /  K  * N  ( 2 1 )  Where P^ i s the t e s t a i r f o i l surface pressure head, i t was measured d i r e c t l y by means of the "Scanivalve" and the "NEFF" systems.  Figure (25)  shows a t y p i c a l nozzle test-section wind speed c a l i b r a t i o n curve using the two p i t o t - s t a t i c tubes.  The r a t i o K i s determined  from a straight l i n e  least-squares curve f i t t e d through the o r i g i n .  In the i n i t i a l experiments, the l i f t and moment data were taken from the 6-component balance, but that data was much less than the expected one, f o r the same t e s t a i r f o i l and angle of attack.  Sugiyama [16]  attributed  t h i s trend to the existence of a clearance flow, between the wind tunnel  - 29 c e i l i n g and f l o o r and the a i r f o i l t i p s , t h i s flow causes a t r a i l i n g vortex sheet i n the wake o f the a i r f o i l , and the l i f t o f the a i r f o i l i s then dercreased through the downwash induced by the t r a i l i n g v o r t i c e s .  In l a t e r experiments the clearance between the t e s t a i r f o i l t i p s and the wind tunnel c e i l i n g and f l o o r  was sealed o f f completely by ad-  hesive tape t o eliminate the e f f e c t o f the clearance flow mentioned before. This way the 6-component balance was not used a t a l l to get any of the l i f t and the moment data; these data were obtained then by the numerical integrat i o n of the pressure c o e f f i c i e n t s around the t e s t a i r f o i l contour. The following equations describe how these data were obtained numerically.  .4. N P p  V  c  d  x  ( 2 2 )  T and  m  C  Where C  =  -(h>  c p  (x d x + y d y)  (23)  and C, are the l i f t and the pitching moment c o e f f i c i e n t s , r e T 0 . spectively. L  T  M  The U.B.C. computer subroutine QINT4P has been used i n the evaluat i o n of equations (22) and (23) . This subroutine f i t s the best polynominal to the "n" experimental data points (C^ vs the pressure tap coordinates) and then i t integrates to get the C_ and the C . The error was i n the T 0 —4 M  order of 1.9 x 10  on the average,  "n" i s the number of the pressure  taps at the center-span o f the test a i r f o i l .  There were 48 taps for the  a i r f o i l with the 616-mm chord and 47 taps for the other a i r f o i l with the 307-mm chord, t h e i r locations are l i s t e d i n Table I I I .  - 30 -  CHAPTER VI EXPERIMENTAL RESULTS  In t h i s chapter the experimental r e s u l t s of t e s t i n g two d i f f e r e n t sizes of NACA-0015 a i r f o i l section are presented.  The two sizes were C/H = .3'  and C/H = .67, where C/H i s the t e s t a i r f o i l chord to the wind tunnel ratio.  height  In a l l tests the Reynolds number "Re" was 500,000, while the wind  tunnel t e s t section was devised with two d i f f e r e n t wall configurations; the s o l i d walls configuration and the double non-uniform transversely s l o t t e d wall configuration. was  Three AOAR were examined, AOAR = 50%, 55% and 59%, because i t  f e l t that the experimental optimum AOAR might be lower than the t h e o r e t i -  cal one.  Also a wide range of angle of attacks were used.  Figures  (26) and (27) show comparisons of pressure d i s t r i b u t i o n s  for two NACA-0015 a i r f o i l sections, C/H = .34 and C/H = .67, respectively, at a = 10° and tested i n the wind tunnel with two d i f f e r e n t wall configurations; the conventional  s o l i d walls and the new double non-uniform trans-  versely s l o t t e d walls.  These two figures i l l u s t r a t e the expected greater  suction over the top surface of the t e s t a i r f o i l when the wind tunnel i s devised with the conventional  Figures  s o l i d walls.  (28) and (29) show comparisons of the pressure d i s t r i b u t i o n s  for the two d i f f e r e n t s i z e s , C/H = .34 and C/H = .67, respectively, of NACA-0015 a i r f o i l at the same angle of attack a = 1 0 ° .  Each size was tested  i n the wind tunnel with the proposed double non-uniform transversely s l o t t e d walls, but with three d i f f e r e n t average open area r a t i o s ; AOAR = 50%, 55% and 59%.  Again, i t can be seen from these two figures that the smaller  the AOAR the greater the suction over the top surface of the t e s t a i r f o i l .  - 31 -  Figure (30) shows the pressure d i s t r i b u t i o n s for the two d i f f e r e n t sizes of NACA-0.0.15 a i r f o i l , C/H: = *34 and C/H = .67, at a = 10°; both sections were tested i n the wind tunnel with the same nonconventional wall configuration, the double non-uniform  transversely s l o t t e d wall with  AOAR => 59%. Because the data are collapsing together, t h i s figure indicates that the new wall configuration i s very much independent of the size of the t e s t a i r f o i l .  Figures (31) and (32) show the C - a curves f o r the two d i f f e r e n t tested sizes of the NACA-0015 a i r f o i l , C/H = .34 and C/H = .67, respectively. Both figures show the e f f e c t s of the wind tunnel conventional s o l i d walls and the nonconventional ones, the double non-uniform  transversely s l o t t e d  walls with AOAR = 50%, 55% and 59%, on the a i r f o i l l i f t c o e f f i c i e n t s C L  T  As expected, the conventional s o l i d walls cause more blockage, thus the wind tunnel l i f t c o e f f i c i e n t s with this wall configuration are higher than the wind tunnel l i f t c o e f f i c i e n t s with the new proposed wall configuration, also, the smaller the AOAR the more the blockage e f f e c t i t has. S i m i l a r l y , Figures (33) and (34) show the C c/4  a curves f o r  the same two a i r f o i l sizes and the same wall configurations as mentioned i n the previous two figures. mental values of E  Figure (35) shows the corresponding experi-  for the tested sizes of NACA-0015 a i r f o i l , C/H = .34  L  and C/H = .67, i n the presence of the conventional s o l i d walls and the double non-uniform  transversely s l o t t e d walls of d i f f e r e n t AOAR; 50% 55%  and 59%, and i n comparison with the corresponding t h e o r e t i c a l curves from Figure (17).  A l l tests were run at the same Reynolds number of 500,000.  The experimental value of m^ used i n determining E  t  was taken as 0.0915  - 32 -  from Figure (36), extracted from Jacobs and Eastman [17].  Agreement i s  seen to be quite good, except for an apparently high experimental value for C/H = .34 i n the presence o f s o l i d walls.  Table IV contains a l l the experimental data, l i f t and pitching moment c o e f f i c i e n t s , presented i n t h i s chapter.  Also, Table  V contains  the values o f the l i f t - c u r v e slopes, m, f o r the two NACA^0015 a i r f o i l sections, C/H = .34 and C/H = .67, tested i n the low^speed wind tunnel with the s o l i d walls and the proposed double non^-uniform transversely s l o t t e d walls, with the three examined AOAR; AOAR = 50%, 55% and 59%.  The values of m were obo - a data between  tained by f i t t i n g least-squares s t r a i g h t l i n e s to the C L  T  a = (0°, 10°).  Appendix 5 l i s t s the equipment which has been used i n the experimental investigation.  Figure (37) shows an outline of the U.B.C. low-speed wind  tunnel which was used i n the experimental work presented here.  - 33 -  CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS A two-dimensional  theory which predicts a s a t i s f a c t o r i l y correction-  free wind tunnel t e s t configuration has been developed. p l i c a t i o n of the two-dimensional  The theory i s an ap-  potential flow theory based on the method  of d i s t r i b u t e d surface s i n g u l a r i t i e s .  The extended theory takes into con-  sideration not only a wide range of a i r f o i l sizes and shapes, but also the e f f e c t on the a i r f o i l loadings of d i f f e r e n t wind tunnel wall configurations.  The above p o t e n t i a l flow theory was then developed to account f o r the flow i n the plenum chambers, i n order to represent the flow i n the t e s t section as close as possible to the physical s i t u a t i o n .  The r e s u l t s of the t h e o r e t i c a l study indicate that for a i r f o i l testing, a wind tunnel consisting o f two symmetrically non-uniform transversely slotted walls, with the gaps between the s l a t s l i n e a r l y increased downstream, and the s l a t s symmetrical  airfoil-shaped at zero incidence  with average open area r a t i o 59%, w i l l y i e l d uncorrected pressure d i s t r i butions , l i f t c o e f f i c i e n t s and pitching moment c o e f f i c i e n t s which are within a few percent of the free a i r values.  The theory predicts that t h i s  low correction wall configuration w i l l remain r e l a t i v e l y correction-free for a wide range of a i r f o i l sizes and shapes.  In the present analysis, the flow i n the plenum chambers i s idealized by two constant pressure streamlines, but actually there are two shear layers.  Each one divides two flows, the high-energy  flow which  - 34 -  exists i n the t e s t section, and the low-energy stagnant flow outside plenum.  the  The model as developed did not lead to a prediction of better agree-  ment of t e s t a i r f o i l loadings with free a i r values. .  Experiments c a r r i e d out on two NACA-0015 a i r f o i l sections for Reynolds number of 500,000, i n a two-dimensional test section support very much the predictions of the theory.  Experimental work showed that the correction-  free test configuration could be achieved with double non-uniform transversely s l o t t e d walls, with AOAR = 59%, consisting of symmetric a i r f o i l shaped s l a t s at zero incidence, when the s l o t t e d sections were surrounded by plenum chambers.  Measurements taken with the correction-free wall configuration of pressure d i s t r i b u t i o n s , l i f t s and p i t c h i n g moments for the NACA-0015 a i r f o i l of two d i f f e r e n t s i z e s , C/H = .34 and C/H = .67  (chord to height r a t i o ) ,  showed good agreement with established free a i r values.  The low-correction wall configuration concept which has been examined t h e o r e t i c a l l y and v e r i f i e d experimentally  i n the present thesis can be  de-  veloped to provide a r e l i a b l e means of t e s t i n g high l i f t a i r f o i l s with high Reynolds numbers i n the e x i s t i n g wind tunnels.  Such a t e s t would otherwise  require expensive and complex t e s t f a c i l i t i e s .  I t would be desirable to con-  tinue the experimental program by t e s t i n g a variety of a i r f o i l p r o f i l e s i n the new  t e s t section, including h i g h e r - l i f t p r o f i l e s such as the NACA-23012  with s l o t t e d f l a p , examined t h e o r e t i c a l l y i n Chapter IV.  - 35 -  REFERENCES  [1]  Sears, W.R., "Self Correcting Wind Tunnels." Aeronautical Journal, February/March, 1974.  [2]  Sears, W.R. , V i d a l , R.J., Erickson, J.C., J r . and R i t t e r , A., "Interference-Free Wind Tunnel Flows by Adoptive-Wall Technology." J. of A i r c r a f t , November 1977. V o l . 14, No. 11.  [3]  V i d a l , R.J., Erickson, J.C., J r . and C a t l i n , P.A., "Experiments with a Self-Correcting Wind Tunnel." AGARD No. 174, Wind Tunnel Design and Testing Techniques 1978.  [4]  Pollock, N., "Self Streamlining Wind Tunnels Without Computers." 7th Australasian Hydraulic and F l u i d Mechanics Conference, Brisbane, 18-22 August, 1980.  [5]  i . Pope and Harper, "Low Speed Wind Tunnel Testing", Wiley, 1966. ii.  Pankhurst and Holder, "Wind Tunnel Technique", Pitman, 1952.  [6]  Parkinson, G.V. and Lim, A.K., "On the Use of Slotted Walls i n TwoDimensional Testing of Low-speed A i r f o i l s " , CASI Trans. 4, Sept. 1971.  [7]  Williams, CD., "A New Slotted-Wall Method for Producing Low Boundary Corrections i n Two-Dimensional A i r f o i l Testing", Ph.D. Thesis, Oct. 1975, University of B r i t i s h Columbia.  [8]  Theodorsen, T., "Theory of Wing Sections of A r b i t r a r y Shape", NACA Report No. 411,1931.  [9]  Hess, J.L. and Smith, A.M.O., "Calculation of Potential Flow about Arbitrary Bodies", Prog, i n Aero. S c i . , 8, Pergamon, 1966.  [10]  Kennedy, J.L., "The Design and Analysis of A i r f o i l Sections", Ph.D. Thesis, 1977, University of Alberta.  [11]  Milne-Thomson, L.M., "Theoretical Aerodynamics", Dover, 1973.  [12]  Prandtl, L. and Tietjens, O.G., "Applied Hydro- and Aerodynamics", Dover, 1957.  [13]  Bhateley, T.C. and Bradley, R.G., "A Simplified Mathematical Model f o r the Analysis of Multi Element A i r f o i l s near the S t a l l " , AGARD-CP-102, 1972.  [14]  Hess, J.L., "Higher Order Numerical Solutions of the Integral Equation for the two-dimensional Neumann Problem", Computer Methods i n Applied Mechanics, V o l . 2, pp. 1-15, 1973.  - 36 -  [15]  Malek, A., "A Theoretical Investigation of a Low-Correction Wind Tunnel Wall Configuration f o r A i r f o i l Testing", M.A.Sc. Thesis, August, 1979, University of B r i t i s h Columbia.  [16]  Sugiyama, Y., "Aerodynamic C h a r a c t e r i s t i c s of a Rectangular Wing with a Tip Clearance i n a Channel." Journal of Applied Machanics, D e c , 1977.  [17]  Jacobs, E.N, and Sherman, A., " A i r f o i l Section Characteristics as Affected by Variations of The Reynolds Number". Report No. 586, National Advisory Committee for Aeronautics, 1937.  - 37 APPENDIX  1  AN ANALYTICAL REPRESENTATION OF A SURFACE VORTICITY The f o l l o w i n g d e s c r i b e s an a n a l y t i c a l the s u r f a c e v o r t i c i t y  i n the f i e l d o f a c o n s t a n t complex v e l o c i t y  Let {W(Z)=dF(Z)/dZ} A small c i r c l e  approach to  problem u s i n g the complex a n a l y s i s :  The contour C-| r e p r e s e n t s an a i r f o i l  C-|.  two-dimensional  s u r f a c e , and Z Q i s any p o i n t W,.  be the complex s u r f a c e v e l o c i t y  a t the contour  i s drawn around the p o i n t Z Q a t which the complex v e l -  o c i t y W ( Z Q ) i s to be f o u n d ,  The f u n c t i o n the contour  {W(Z)/(Z-ZQ)>  {C ,(C +Cr,)} n  1  t  is analytic  i n the r e g i o n e n c l o s e d by  --38 -  therefore;  r L  o  L  r l  u  r 2  C o n s i d e r the 1 s t term o f the i n t e g r a l W(Z)  I as Z approaches ° ° ,  then  = W-j the f r e e stream complex v e l o c i t y , which i s c o n s t a n t , t h u s :  h  =  W  (  Z  J>TTT  =  )  r  W  ^  l  2 l T i  (  2  )  *  while  *  in C  (3)  l  and  h*fr£h C  ^  Q  2T  ri  (4)*  2  t h e r e f o r e e q u a t i o n (1)  I  = W(Z )  = 2TH W  ]  -  becomes,  Jjffi-) C  dZ - 2TTi W(Z ) Q  (5) *  = 0  l  or  *  Now i f we r e p l a c e the a i r f o i l density Y U ) » is written  t  as,  n  e  complex v e l o c i t y  s u r f a c e by v o r t e x d i s t r i b u t i o n , w i t h  a t the p o i n t  Z  Q  would be  W ( Z  Q  ) ,  and  it  - 39 -  where  i s the complex v e l o c i t y o f the onset  Compare e q u a t i o n s (6)  W(Z)  a t  p  and ( 7 ) ,  v o r t e x d e n s i t y t h e r e , and t h i s (7)  we g e t  (8)  =. Y ( Z ) .  T h e r e f o r e the complex v e l o c i t y W(Z)  *  flow.  a t the a i r f o i l  is only true i f  s u r f a c e s i s equal to  *  the  the i n t e g r a t i o n i n e q u a t i o n  i s done i n the c o u n t e r c l o c k w i s e d i r e c t i o n .  - 40 APPENDIX  2  EVALUATION OF THE INTEGRAL IN EQUATION  (5)  The most s t r a i g h t f o r w a r d method o f e v a l u a t i n g t h e i n t e g r a l i n equation The  (5) i s t o do so u s i n g a numerical  i n t e g r a t i o n procedure.  s u r f a c e element i s chosen t o be curved  d i s t r i b u t i o n can v a r y over t h e element.  and t h e v e l o c i t y  The i n f l u e n c e o f t h i s  d i s t r i b u t i o n on one element on t h e c o n t r o l p o i n t o f another element i s c a l c u l a t e d by t h e i n t e g r a l S2 V fc^ >r- (I nC , C\ = 2T / Y(S)£n S) Ac dS. SI  1  A coordinate  system  **  ^  ±  ( n ) i s s e t up w i t h o r i g i n a t t h e  c o n t r o l p o i n t o f t h e i n f l u e n c i n g element as shown i n F i g . (11). The  influenced c o n t r o l point i s located a t (b,a)in t h i s  system.  The s u r f a c e element i s d e f i n e d by n=n_(£) .  coordinate  In the neighbour-  hood o f t h e o r i g i n a power s e r i e s expansion i s used,  K  2  n =  c  + e C  +  The  i n t e g r a l i s taken over t h e s u r f a c e d i s t a n c e and i t i s  3  ...  (2f*  convenient to use: dS  / , , d l ^ -.^  on expanding  **  **  (3) as a s e r i e s about £ = 0 ,  || = 1 + 2 c The  •  2  C  2  +  6 ce ?  + . . .  3  (4)**  v o r t e x d e n s i t y can a l s o be w r i t t e n as a s e r i e s d e f i n e d  by, T(S)  - Y  ( 0 ) +  Y  ( 1 )  S  + Y  ( 2 )  S  2 +  Y  ( 3 )  S  3 +  ...  (5)  **  Applying  (4)  Y  and (5) , = <0)  m  T  +  Y  <D  e  +  y  (2) 2 5  ,2^(1,  +  +  Y  (3), 3 £  _  +  (6) The d i s t a n c e r(C^,S) from the c o n t r o l p o i n t t o the s u r f a c e i s , r  (c S) =  {(a-n) +  (b - 5 ) }  At  t h i s p o i n t i t i s n e c e s s a r y t o employ the t e c h n i q u e  2  i f  used by Hess [14].  2  (7)**  %  I n s t e a d o f expanding t h i s term d i r e c t l y a  m o d i f i c a t i o n i s used which p e r m i t s the b a s i c , f l a t to  appear as the f i r s t 2  the  =  2  a*  +  term o f the s e r i e s .  (b -£)  r  term  By w r i t i n g  2  d i s t a n c e t o the f l a t  expanded  element,  ** (8) s u r f a c e n=0,  the r e m a i n i n g terms are  and, 2  = r  2 f  - 2ac r  Substituting  2  - 2ae T  3  .  + ...  ** (9)  (9) i n the l o g a r i t h m term and expanding a l l  2 but  the r .  In  f  term about £=0 y i e l d s , r = - \ in 2 |_'  The i n t e g r a l of  /  (r  2* f  2cic  )  w2  r  a c c u r a c y as i s d e s i r e d .  ^ £  f  r  ** can now (1)  ** (10) .  2s.s »_3 ,  ^ £  +  f  be e v a l u a t e d t o as h i g h a degree  In t h i s case o n l y the f i r s t  are  r e t a i n e d , h i g h e r o r d e r terms b e i n g o f d i m i n i s h i n g ** E q u a t i o n (1) then becomes, 1=  hf^  i0)  *»  +  T  U  )  r  2 f  5  -^| <°> Y  terms  importance.  2 5  few  +  ... , d  5  (11)** the  first  term i s t h e s t r a i g h t element, c o n s t a n t v e l o c i t y case,  second term i n t r o d u c e s a l i n e a r v e l o c i t y d i s t r i b u t i o n and the third  i n t r o d u c e s the s u r f a c e c u r v a t u r e .  - 42 -  **  Each term i n (11) can be i n t e g r a t e d s e p a r a t e l y ,  as f o l l o w s :  A IS  J/  £n r  d£ =  2 f  (b+A) In r^  - (b-A) Hn r  2  2 2  - 4A  -A  +  An r  2 f  2  a tan" ( ^ ) a + b -A 1  2  5 d K = a ^ ,** 2  2  2  i n (^)  2  - 2bA  ~ , , ~1 . 2 a A . tan (—x ^—2~) a + b -^A  •r 2 a b  2  A  d E. = ( b - a ) t a n " ( —5-) " ; uaii v 2 2 2 a +b -A  5  2  2  1  2 a A  + a b  i n (—) l  2 " s  -A  --.e f  **  (12)  a  2  (13)  + 2aA  (14)  r  For t h e s t r a i g h t element  'j',  which makes an a n g l e 6.. w i t h  the X-wind a x i s , t h e 'a' and 'b' a r e g i v e n by, a = (Xj-x^  s i n 9 j + (y^-y^)  cose.,  b = (x.-x.) cos0. + (y.-y.) s i n 9 .  Also r  1  and  r  l  **  (15)  can be w r i t t e n a s ,  =A  2  + (b + A )  2  ** and r  2  A  (16) 2  + (b - A )  2  - 43 In the above e q u a t i o n s o f the c o n t r o l  ( x . , y.) and ( x . , y . )  p o i n t s on the elements  ' i ' and ' j '  are the  coordinates  respectively.  In case o f s t r a i g h t elements w i t h c o n s t a n t v o r t e x d e n s i t i e s y^ the i n t e g r a l  (l)  g i v e n by e q u a t i o n  i s c a l l e d the BASIC i n f l u e n c e c o e f f i c i e n t s (12)  it it  In t h i s case t h i s was done by f i t t i n g  through s e t s o f t h r e e a d j a c e n t element end p o i n t s . determined was assumed to be the a i r f o i l o f the t h r e e p o i n t s . by i n t e r p o l a t i o n .  and i t  is  .  For h i g h e r o r d e r terms i n v o l v i n g s u r f a c e c u r v a t u r e must be d e t e r m i n e d .  K-.  a t the  thus centre  p o i n t s were then found  The v a l u e s a t the elements a d j a c e n t  edge were found by e x t r a p o l a t i o n .  parabolas  The c u r v a t u r e  surface curvature  The c u r v a t u r e at the c o n t r o l  the c o n s t a n t C  to the  trailing  Having determined the c u r v a t u r e  element the l o c a t i o n o f the c o n t r o l  of  p o i n t can be c a l c u l a t e d as t h i s  the  point  i s no l o n g e r on the s t r a i g h t l i n e j o i n i n g the element end p o i n t s .  In employing v a r i a t i o n s  i n the s i n g u l a r i t y s t r e n g t h the term  i s an unknown and must be r e l a t e d  to the y ^ .  a b l e to do t h i s and the t e c h n i q u e used by Hess derivatives  o f the d i s t r i b u t i o n on the j  t  h  V a r i o u s schemes are [14]  y ^ avail-  i s followed here.  The  element are determined by a s -  suming a p a r a b o l i c d i s t r i b u t i o n through the t h r e e s u c c e s s i v e v a l u e s Y-j_-/°^ (0)  (0) j  Y  '  Y  j+l•  The l i n e a r  v o r t e x d e n s i t y t e r m , u n l i k e the o t h e r  t e r m s , i s t h e r e f o r e comprised o f terms t h a t i n v o l v e the v o r t e x of  two  densities  adjacent-elements. The a p p l i c a t i o n of the h i g h e r o r d e r methods to the s o l u t i o n  the c a l c u l a t i o n o f the e x t r a terms  ( 1 3 ) * * and ( 1 4 ) * * .  The c u r v a t u r e  involves terms  -44 K-. c a l c u l a t e d f o r  are simply added to the i n f l u e n c e c o e f f i c i e n t basic case.  The l i n e a r v e l o c i t y  K.  K.  • -,, K..,  calculations  In  . ,.  terms must be added to the  Although t h i s  the  coefficients  i s not d i f f i c u l t to d o , the e x t r a  i n v o l v e d do take c o n s i d e r a b l e amounts o f time to p e r f o r m .  summary, e q u a t i o n s  (1)  and (12.)  give,  K.. = 1_ {(b+A) Jin ( r ) - (b-A) Jin ( r ) ^^ • i' ' • 2  2  L  + 2a t a n "  1  (  ^ ) aSb'-A ?  2  a  2  - 4A>  ** while  K.. 11  = — {Jin TT  A - 1}  (18)**  APPENDIX 3 CALCULATION OF THE VELOCITY COMPONENT INDUCED AT A POINT IN THE FIELD OF VORTEX DISTRIBUTION: The element  stream f u n c t i o n a t a p o i n t P^ due t o a s t r a i g h t  ' j ' , with constant  vortex  d e n s i t y Yj d i s t r i b u t e d over t h a t  element,  i s g i v e n by _  Y -;  P  * * *  -i  where r i s t h e d i s t a n c e  from t h e p o i n t P^ t o t h e p o i n t  Q on t h e  s u r f a c e , as i n F i g . (10a), and i t can be w r i t t e n as r.  + yp  = {(x^-O  L  (2)  Thus —Y .  *ij  =  The  ± j  v..  o  f  A  ~2%  *  n { ( x  j-5  o i«  +y- > ?  }  (3)  d  v e l o c i t y components i n d i r e c t i o n s p a r a l l e l and normal  t o t h e element u  A  'j'  r e s p e c t i v e l y , induced a t P^ a r e : -Y • t *1 r = 2TT I/ ( x l - ^ )i + y ! dC ,  *** (4a)  j  =  - _ =  3x.  Y. A ^ = - ^ - l /r 2TT  /  -A  3  2  T2  (x!-£) L_  _  (x^-^ +y' 2  d  ?  (  4  b  )  ***  2  Therefore Y. U  ij  = 2?  , (x'-A) {  t  a  y T -"  n  , t  a  n  (x'+A)  - ^ T —  < >  }  5a  ***  and v, . = - J - ' 13 4TT  *n{ y ! ^ + ( x ! - A ) ^ / y ^ + t x l + A ) ^ 3 3 -*3 3 z  Where x^ and y^ a r e t h e c o - o r d i n a t e s respect  }  of the point P  (5b)  i  with  ( r e f e r r i n g ) t o a system o f a x i s i t s o r i g i n i s the c o n t r o l  - 46 -  p o i n t on t h e element  'j'  which has t h e l e n g t h 2A and makes an  angle 6.. w i t h t h e wind a x i s , and they a r e g i v e n by  ***  x^ =  (x -x .) c o s 6 j +  ( y ^ y - j ) sine..  (6a)  y! =  (x^-x^  ( y ^ Y j ) cos6..  (6b)  i  ;  sine.. +  ***  To e x p r e s s t h e v e l o c i t i e s t a n g e n t i a l and normal c o n t r o l s u r f a c e we  to the i  consider the f o l l o w i n g :  From F i g . (7b) and from the vector analysis of the element •j',  t h e v e l o c i t y v e c t o r V.. can be w r i t t e n as 3  V.. = u.. t . + v.. n. 3-D  1 3  also  D  I D  (7)  D  ***  V. . = X. . I + y. . 3 I D  I D  * * *  '(8)  I D  thus X. . = V. . I 3 I D  *** (9)  x  t h e n from e q u a t i o n s X  ±j- ij U  C  O  s  (7)*** and 9  j  (9)***, we  get  ^ne.  (10)***  similarly  *** .(11)  Y. .' = u. . s i n e . + v. . c o s e . I D  I D  D  I D  _ D  A l s o t h e v e l o c i t y v e c t o r V\ ^, r e f e r r i n g t o F i g . (7b) and from t h e v e c t o r a n a l y s i s o f the element V.. = V I D  T m  ± j  t. + V i N  i  j  'i',  n. x  can be w r i t t e n as  ***  (12)  thus  -V i  then from e q u a t i o n s V_ = X.. T j I D similarly  (8)  and  cos6. + Y.. 1  ±  —  *** (13>  I D  (13) sine. i  x  get  ***  (14)  *** (15)  —  V.. = V. . n. i j 3 = y.. cose. - x.. I D i I D  , we  1  sine. i  - 47 -  ***  substitute hence v  (10)  =  T  V V  T  and V  u  i ;  .  = -u  N  respectively,  and  ***  (11)  c o s t e . - e . . )  i 3  s i n f e . - e . . )  i n t o equations  +  v  i ;  .  + v  are the t a n g e n t i a l  (14)  ***  and  ***  (15)  ,  s i n t e ^ O j )  ± j  (  cos ( 9 ^ )  tangential  6  a  ***  y  (16b)***  and t h e normal v e l o c i t i e s ,  induced a t element * i ' ,  1  and normal t o  i t , due t o a s t r a i g h t v o r t e x element ' j ' w i t h c o n s t a n t d e n s i t y  - 48 -  APPENDIX 4 AN ANALYTICAL APPROACH TO THE FLOW IN THE WIND TUNNEL PLENUM  A.  Introduction An extension of the current analysis i s to make the geometry of the  flow representation more l i k e that which actually occurs experimentally i n the test-section, with the plenum surrounding the slotted walls.  Figure  (6) compares the flow representation of the current analysis with the phys i c a l flow which actually occurs i n the test-section.  As shown i n F i g . (6)a, two streamlines AB and CD are used to simulate the actual shear layers shown i n F i g . 6(b).  AB leaves the  test-section upstream of the test a i r f o i l and opposite to i t s negative pressure side, while CD enters the test-section upstream of the test a i r f o i l and opposite to i t s p o s i t i v e pressure side.  In t h i s section the flows  in the upper and lower plenums are represented by considering these  two  streamlines t© be constant pressure streamlines, i n the current analysis.  In order to define the geometry of the constant pressure streaml i n e s and investigate i t s e f f e c t s , the current theory should be  developed  to provide the induced tangential and normal v e l o c i t i e s of the flow at any point i n the f i e l d , since they are necessary terms for tracking any streamline.  B.  The Induced Tangential arid Normal V e l o c i t i e s due to V o r t i c i t y Distribution  The stream function ¥.. at a control point P. whose coordinates i] with respect to the j straight element are P^= CX! • y')., i F i g . (7a), 1  a s  n  - 49 due to v o r t i c i t y having a constant density y  -Y  ij  ¥  =  j"I  2^  over that element i s given  In r .(P_ ,1 ^ '£>  l  n  r  ( P  (1)****  d  L  A  Then the v e l o c i t y components can be calculated from equation (1) . Using the notation of F i g . (7) the v e l o c i t y components induced at the element 'i  1  due to a vortex element ' j ' , are, u. . in  Y-i = -=J 2TT  .  Y-; = -L  1  v  -  i]  -,  ,  {Tan  —1 X  x'.-A n  f-^-)  y!  •,  - Tan  —1  X  D  z  n  4n  x'.+A 1  (^-)} y,  ,  ****  (2a)  y  . 9 { (y'/+ :  (x!-A) D  2 . 2 )/ (y! + D  (x-  3  +  . . 2. -i A) )}  (2  b)  ****  The directions of u.. and v.. at P. are p a r a l l e l and normal to the ID ID i d i r e c t i o n of element ' j , 1  respectively.  With respect to cartesian 'wind  axes' X and Y (X i s the wind d i r e c t i o n ) , the j element ' i '  vortex element and the  are included at angles 6^ and 6^ to the X-axis respectively.  Thus, V  V  ij  = u.. cos ^  (9.-9.) i D  + v..  = v. . cos  (9.-9.)  - u. .  1  ID  l  D  sin(9.-e.)  (3a)****  sin(9.-9.)  (3b)****  i  ID  ID  1  D  ;  D  are the tangential and the normal v e l o c i t i e s induced at the element ' i '  due  to the vortex element ' j ' of density y . The l o c a l tangential v e l o c i t y . i V , . i s - — w h i l e the l o c a l normal v e l o c i t y V„.. i s zero. The d e t a i l s of Tn 2 Nn Y  m  D  1  **** **** the c a l c u l a t i o n of equations (2) and (3) are provided i n Appendix 3.  Therefore, the tangential and the normal v e l o c i t i e s induced at  - 50 -  the control point on the element * i '  due to a system of 'N' vortex elements  immersed i n an i n f i n i t e uniform flow U, p a r a l l e l to the X-direction, are,  V  T. l  V„  =  N Z  {u. . ID  cos(9.-9.)  + v. . s i n  (9.-6.)}  n  Z  (v..  cos(6.-9.)  - u.. s i n  (6 - 9  . D=l  =  l  D  13  Hence, at any free point  3  1  + U«,cos  )} _ n s i n  1  ****  9.  (4a )  9-  (4b)  i n the f i e l d of the above system the  tangential and the normal v e l o c i t i e s of the flow there are given by,  V  T  V N  = V ±  = V  i  Y  X  =  N Z  =  N Z  j=l  ****  {u., cos 6 . - v.. s i n 9.} + U ID 3 ID D °°  (5a)  {v.. cos 6.+ u.. s i n  <  1 3  3  3  9.}  5 b  )  ****  3  (i = 1,2,...., N ) and the pressure c o e f f i c i e n t of the flow at this point P ^  p  o  P  ±  = 1 - (V  C.  p  + V i  i s given by,  n  ****  ) ,  ( i = 1,2,...., N )  (6) .  i  Simulated Shear Layer Tracking The routine described i n the previous chapter must be carried out  to calculate N vortex densities, y , f o r the N elements which represent the test a i r f o i l , the wall s l a t s and the s o l i d walls.  The values of the  •k 4c "k It  vortex densities can then be inserted i n equations (5a)  it it * *  and (5b)  to get  the induced tangential and normal v e l o c i t i e s , respectively, at a specified point (x., y ) i n the flow f i e l d . i 1  Thus from a s t a r t i n g point (x., 1  v  i  ),  which i s supposed to be at the downstream edge of the s o l i d wall but, to  - 51 -  avoid the s i n g u l a r i t y there i s located instead very close to the edge, the flow d i r e c t i o n 0^ i s calculated, given by, _,  = tan"  6  where V  ****  and V  T  (7)  (V /V ) 1 1  1  1  are the induced tangential and normal v e l o c i t i e s 1  at the point (x^, y ^ ) , respectively, and they are given by equation (5a) **** and (5b)  Then 0.^ i s used to calculate the next point (x^, y ) where, 2  . **** x  2  Y  2  = x.^ +  = y  ±  (8a)  Ax  + Ax  * tan  6  ****  The flow d i r e c t i o n 6 ^ i 0^.  (8b)  1  s  then calculated there i n the same way as  The flow directions are averaged to give 0, and the y-co-ordinate i s  changed so that the next point i s now, ****  x  2  = x^^ +  A x  y  2  = y  A x  x  +  (9a) ****  * tan 0  (9b.)  Having defined the geometry of the streamlines  simulating the shear  layers, which i s assumed to remain constant i n the following analysis, one can proceed to include them i n the present t h e o r e t i c a l analysis as follows: F i r s t the pressure v a r i a t i o n along each of the streamlines i s calculated, **** using Equation (6) , and the large negative pressure excursions, as the flow accelerates when i n the v i c i n i t y of the wall s l a t s , i s excluded and an average pressure c o e f f i c i e n t , C , would be obtained for each p  - 52 -  streamline.  Then each streamline i s divided into a number of elements  n, which adds n vortex elements to the previous N elements.  Thus we should  once more solve (N'+M) questions, (N'=N+n) elements and M components, prescribed by the following set of equations,  N + E  ¥  K. . Y. = R. ,  S  ID 'D  (i=l,2,  N)  (8)  i  and ¥_  +  s  m  m  N E K . Y- = tp ,] D J—Jm  tp, m  (m=s=l,2,  M)  10)  The additional n equations required f o r a solution to this problem, are the prescribed - tangential - v e l o c i t y equations for streamlines, i N V  {u. . c o s ( 0 . - 6  = E  l  ) + vm  sin(6.-9.)}  + U cos9, ,  **** (4a)  D=1  where V T T  i  = ± ^d-^ ) / i  (i=l,2,  n)  (10)  ****  P  The sign of the squareroot i s p o s i t i v e i f the vortex elements are d i s t r i b u t e d sequentially f o r calculations i n the same sense as the flow directions and C  i s the prescribed pressure c o e f f i c i e n t along either of the i two constant pressure streamlines (C^ could be the average t h e o r e t i c a l -i pressure c o e f f i c i e n t or any other experimental value). p  Thus the problem of including the flow i n the wind tunnel plenum into the present t h e o r e t i c a l analysis has therefore been reduced to that of solving the above (N+n+M) equations simultaneously to get (N+n) vortex densities, Y»  a  n  d M  stream functions ¥ .  - 53 -  APPENDIX 5 LIST OF EQUIPMENT USED IN THE EXPERIMENTAL INVESTIGATION Instrument 1.  Barocel  Description  Pressure  Sensor  Type 511 (10-mm Hg=10 v o l t s ) .  Manometer  Pressure transducer tube measurements.  for  pitot-static  2.  Barocel E l e c t r o n i c Type 1018B.  3.  Barocel S i g n a l Type 1015.  4.  Betz Water Micromanometer.  For p i t o t - s t a t i c measurements.  5.  Setra Pressure Model 237.  For S c a n i v a l v e a i r f o i l p r e s s u r e measurements.  6.  NEFF Data A c q u i s i t i o n System.  For scanning and d i g i t i z i n g pressure transducer s i g n a l .  7.  PDP-11 Computer hooked up to NEFF.  For e v a l u a t i n g , d i s p l a y i n g and s t o r i n g the s u r f a c e p r e s s u r e coefficients.  Conditioner  Transducer  4-1/2 d i g i t v o l t m e t e r f o r i n d i c a t i n g the t o t a l p r e s s u r e head from p i t o t s t a t i c tube.  A m p l i f i e r f o r Barocel transducer.  pressure  tube wind speed  surface  the  - 54 -  FIGURE  (1)  COMPARISON OF AIRFOIL PRESSURE COEFFICIENTS: REF.  [9]  THEORY  - 55 -  FIGURE  (2)  VORTEX REPRESENTATION OF TWO COMPONENT AIRFOIL:  REF.  - 56 -  FIGURE  (3)  NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS:  REF.  [10]  - 57 -  — G r o w i n g of the i t a r t i n g vortex.  FIGURE (4)  STREAMLINES CONTOURS AROUND AN AIRFOIL:  REF.  [12]  - 58  -  FIGURE ( 5 ) LOCATION OF T R A I L I N G CONTROL POINT:  REF. [10]  (a)  THEORY  FIGURE (6)  AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL  -  FIGURE  (7)  NOTATION  60 ^  U S E D TO C A L C U L A T E  INDUCED  VELOCITIES  - 61 -  FIGURE (8)  LOCATION OF ELEMENTS ON AIRFOIL SURFACE:  REF.  [10]  - 62 -  i  i  i  i  1  1  1  r  T  c>  V  KarmanTrefftz Aerofoil  a = 0°  Element Shape  Vortex Density  O  Straight Line  Constant  •  Curved  Constant  V  Straight Line  Linear  A  Curved  Linear  —= Exact Analytic  FIGURE (9)  COMPARISON OF HIGHER ORDER METHODS REF.  [10]  - 63 -  x/c FIGURE (10)  COMPARISON OF AIRFOIL VELOCITY DISTRIBUTIONS REF.  flQ]  - 64 -  FIGURE ( 1 1 )  GEOMETRY FOR CALCULATION OF HIGHER ORDER TERMS:  REF. [ 1 0 ]  FIGURE (12)  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL SECTION  IN FREE AIR AND BETWEEN SOLID WALLS TEST  FIGURE (13)  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION  - 67 -  FIGURE (14)  COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL  IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION  - 68 -  FIGURE  (15)  COMPARISON OF PRESSURE  COEFFICIENTS FOR NACA-0015 AIRFOIL  IN FREE AIR AND BETWEEN SINGLE UNIFORMLY SLOTTED WALL TEST SECTION  - 69 -  FIGURE  (16)  COMPARISON OF PRESSURE  COEFFICIENTS FOR NACA-0015 AIRFOIL  IN FREE AIR AND BETWEEN DOUBLE UNIFORMLY SLOTTED WALL TEST SECTION  FIGURE (17)  COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN LIFT COEFFICIENTS  - 71 -  NACA-0015 a = 10°  / NACA-23012 WITH 25.66% SLOTTED FLAP  ' FIGURE  (18)  5  """S  r —  /  5— ~~S—'—I r  COMPARISON OF THE EFFECT OF TEST AIRFOIL  C/H  SIZE ON THE  RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS  -72-  -8.  0  -7.  0  FREE AIR DOUBLE NON-UNIFORM SLOTTED WALLS WITH A0AR=59% NACA-0015 C/H=. 8 ALPHA= 10 DEGREES  CP -6.  0  E =0.62 %  E  L  -5.  0L ©  -4.  0  -3.  0  M  = _ 1  -  2 1  %  2 s  - 2 . 0 i  a  -1. 0  ^ * * A 6  t 0 . 01  . ^' $  £  6  _ — — » — , — * — '© £> © & ©  A** 0-0  FIGURE  0 . 5  (19)  X/H  1b 0  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NONUN I FORMLY SLOTTED WALL TEST SECTION  -73-  -14.  0  -13.  0  12. 0  •  CP  •1 1. 0  D  -i0.0D -Q.  0  FREE AIR DOUBLE NON-UNIFORM SLOTTED WALLS WITH A0AR=59% NACA-23012 C/H=. 8 ALPHA= 8 GAMA= 20 E =-4.03 *  I*  E =-0.91 %  L  M  -8. 0 -7.  0  -6. 0 -5. 0 -4. 0  •  -3. 0 -2. 0 -1. 0 O  -0. 0 g $ d  £  1. 0.0 0. FIGURE  £  ^  ®  ©  © © ©  0.5 (20)  1.0  COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL  IN FREE AIR AND BETWEEN DOUBLE NON-  UNIFORMLY SLOTTED HALL TEST SECTION  -74-  -14.  0  -13.  0  |© ^ FREE AIR 12. 01 ! DOUBLE NON-UNIFORM SLOTTED CP WALLS WITH A0AR=59% 11. 01A CLARK-Y 14% C/H=. 8 -iB. 4 ALPHA= 10 DEGREES -S. 0 E =-0.79 %  E  L  M ~ ' =  1  2 4  %  -S. 0 -7.  0  -e.  0 ©  -5.  0 A  -4. 0  © •  s  -3. 0  ffi  -2. 0  6  6 6  -1. 0  d  d  * 6  -0. 0  , i » * 0.0 FIGURE  M  • • • • • 0.5  (21)  X/H  1 , 0  COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NONUNIFORMLY SLOTTED WALL TEST SECTION  - 75 -  FIGURE  (22)  COMPARISON OF-THE EFFECT OF TWO DIFFERENT ANGLES OF ATTACK ON THE RELATIVE -ERROR IN LIFT COEFFICIENTS FOR NACA-0015 AIRFOIL  C/H = .8  FIGURE  (23)  c/C = .15  a = 10  GEOMETRY OF THE IDEALIZED FLOW IN THE PLENUM CHAMBER FOR NACA-0015 AIRFOIL  - 77 -  B.  0  7.  0  r  FREE AIR SLOTTED WALLS A0AR=59% WITH THE FLOW IN THE PLENUM C/H=. 8 ALPHA= 10 DEGREES  CP 6.  0  C.  5. 4  -  3.  0 0  L  ©  PA  0  T  =1.2137  C  E. =-0.98%  .  ©  •  A  i  M  M  c/4  =-0.0412  E =-1.82% M u  ft  i 2 .  ©  0 L  •  1  *  • O  •  l"*r  *  ©  A ©  ©  0.  0  '  A A  A © *A © i . 0[2©  £  ~©  o  A *  °  i  *  " ©  4"  • $  • $  t A  * © ~ $ $A'  ©  0 . 0  FIGURE (24)  0 . 5  X/H  COMPARISON OF NACA-0015 AIRFOIL PRESSURE FLOW IN WIND TUNNEL PLENUM IS ANALYSIS  1  "  0  COEFFICIENTS:  INCLUDED IN THEORETICAL  - 78 -  MM OF WATER (TEST-SECTION) 413. 0  r  FIGURE  (25) CALIBRATION OF NOZZLE AND TEST SECTION DYNAMIC PRESSURES  - 79 -  a. 0  r  SOLID WALLS SLOTTED WALLS A0AR=59% RE= 5 0 0 , 0 0 0 C/H=. 34 ALPHA= 10 DEGREES  e  7. 0  A  CP 6. 0  5. 0  4. 0  A©  3. 0  A®  A  2. 0  s  © A©  *©©© i . 0t  A  S  © *  »  0. 0  1. 0} 0.0  FIGURE  0-5  (26)  9  44-  X/H  1 - 0  WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34:  EXPERIMENT  - 80 -  B. 0  © SOLID WALLS A SLOTTED WALLS A0AR=59% RE= 5 0 0 , 0 0 0 C/H=. 67 ALPHA= 10 DEGREES  7. 0  CP 6. 0  5. 0  4. 0  ©  © ©  3. 0  • A©  A  ©  ©O  2. 0  A  ©, A  A  '©,© A  ~  © A  s  a e a a g ta ft-  0. 0 *  A©  6  *  1. 0{4fe 0-0  FIGURE (27)  0-5  X/H  1 , 0  WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67:  EXPERIMENT  - 81 -  —8. 0  o SLOTTED WALLS AOAR-50% A SLOTTED WALLS A0AR=55% + SLOTTED WALLS A0AR=59% C/H=. 34 RE=500, 000 ALPHA = 10 DEGREES  -7. 0  CP -6. 0  -5.  0  -4. 0 © 04  -3. 0  ©  -2. 0  4© 4  -1. 01 9 •  0. 0  1. 0 0 0  •  FIGURE (28)  ' • ^•• • • •  A  * .• • • •  •  0 . 5  »* t 1 t '  X/H  1 , 0  EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34:  EXPERIMENT  - 82 -  — B. 0  -7.  ©  0  CP -6.  0  -5.  0  •4.  0  •3.  0  SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 RE=500, 000 ALPHA = 10 DEGREES  I I  + -2.  0  -1.  0+  • * » 0.  1.  0  0 ( _ 0 . 0  FIGURE (29)  0  .  5  x/ H  1  -  0  EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL,  C/H=.67:  EXPERIMENT  - 83 -  S. 0  7. 0  CP  o C A R 34 C/H=. 67 SLOTTED WALLS A0AR=59% RE= 5 0 0 , 0 0 0 ALPHA= 10 DEGREES A  6. 0  5. 0  4. 0 3-  0&  2. 0  1 .  0 •  V  C #  •  •  A  0 [  0.0  FIGURE  • ft ®  0. 0  1 .  66  0.5  (30)  X/H  COMPARISON OF NACA-0015 AIRFOIL EXPERIMENT  m  1 - 0  PRESSURE COEFFICIENTS  2.  0  CL 1.  5  1.  0  0.  5  © SOLID WALLS A* SLOTTED WALLS AOAR=50% + SLOTTED WALLS AOAR-55% SLOTTED WALLS A0AR=59% C/H=. 34 RE=500,000 x  ALPHA - 4 .  FIGURE  16.  (31)  EFFECT OF WALL CONFIGURATION ON LIFT COEFFICIENT FOR NACA-0015 AIRFOIL, C/H=.34:  EXPERIMENT  0  - 85 -  2.  0  ©  CL 1.  + X  5  SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 © © A RE=500,000 o  o  1.  0  A  X  ©  © 0.  5  0.  0|_  X  ALPHA -0.  50  -4.  FIGURE  0  (32)  0.  0  4.  0  e. 0  12.  0  16.  EFFECT OF WALL CONFIGURATION ON LIFT COEFFICIENT FOR NACA-0015 AIRFOIL, C / F K 6 7 :  EXPERIMENT  0  - 86 -  0.  10 ©  CMC/4  A  + X  0.  SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 34 RE=500, 000  05  ©  o 6 +  A  0  0.  A  * - 4  00?-  X  ALPHA •0.  05 -4.  00  FIGURE (33)  0.  00  4.  00  s.  00  12. 00  16.  00  EFFECT OF WALL CONFIGURATION ON PITCHING MOMENT COEFFICIENT FOR NACA-0015 AIRFOIL OF SIZE C/H=.34:  EXPERIMENT  - 87  0.  10  ©  CM  4-  C/4  X  0.  SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 RE=500, 000  05  + 0.  0l£  ¥ 4^ X  O O ALPHA •0. 05 -4.  00  FIGURE (34)  0 .  0  0  4.  00  B.  00  12. 00  16.  00  EFFECT OF WALL CONFIGURATION ON PITCHING MOMENT COEFFICIENT FOR NACA-0015 AIRFOIL OF SIZE C/H=.67:  EXPERIMENT  - 88 -  FIGURE (35) EFFECT OF AIRFOIL SIZE ON E,_FOR NACA-0015:. COMPARISON OF EXPERIMENT AND THEORY  -89-  N . A. C. A.tlrton oooo..  0012..  0015.  Re= 500,000 m 0018..  F  = 0.0915  (millions) 8.470 8.280 6.100 8.410 1.760 .882 .446 .223 .112 8.870 R450 6.280 (.640 1.740 .871 .449 8.610 8.990 8.350 1.730 • 874 .222 .113 7.840 6.240 1.300 1.730 .866 .430 .214 .100  (dec)  m  0 0  aoee .067 .097 .066 .096 .105 .117 .104 .099 .100 .097 .067 .096 .064 .098  0 0  0  0  0 0 0  0 0 0 0  0 0  0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  3412..  8.240 6.100 8.420 1.730 .870 .438 .218 .110  -3.0 -2.1 - 2. 0 -2.1 -2.1 - 2. 0 -2.2 -1.3  33012..  8.370 8.160 6.070 8.400 1.760 .884 .446 .221 .112 8.000 6.360 3.380 1.760 .600 .4M 8.370 6.810 8.540 1.770 .884 .464  -1.2 -1.2 -1.2 -1.2 -1.2 -1.3 -1.6 -1.4 -1.2 -1.2 -1.2 -1.2 -1.2 -1.4 —.6 -.7 -.7 -.8 -.8 -.9  33012-33..  2Ril2  4406.  4412..  4418.  6412..  8.080 4.S70 8.340 1.700 .866 .438 .218 .110 7.920  e. ioo  S.270 1.680 .874 .433 .216 .111 7.920 6.280 3.340 1.730 .682 .431 .219 .110 8.210 6.020 8.350 1.700 .882 .441 .219 .110  (percent c)  .067 .066 .064 .063 ..092 •*ro»i .101 .134 .096 .096 .066 .065 .060 .086 .062 .114 .068 .097 .068 .066 .066 .066 .102  -3.9 -3.6 -4.0 -4.0 -4.1 -4.1 -3.7 -Z8 -4.0 —4.1 —4.1 -4.2 -4.3 -4.3 -4.3 -36 -4.0 -4.0 —4.1 -4.2 -4.3 -4.4 -4.4 -3.1 -8.9 -8.9 -61 -6.2 —6.3 -6.2 -8.9 -8.4  .100 .098 .068 .097 .066 .096 .109 .067 .096 .096 .066 .094 .066 .098 .067 .067 .065 .066 .100 .066 .066 .095 .068 .066 .097 .105 .115 .068 .096 .098 .097 .096 .094 .100 .113  41.89 *1.38 D.94 D.86 D.86 D.gi O.B3 ».7B 41.66 *1.65 41.62 '1.49 41.18 D.B1 C. 89 41.66 41.60 C1.48 C1.28 ci.oe o.98 ».89 O.tO 41.63 41.83 C1.42 oi.26 C1.15 41.03 o.96 D. 86  • 0.0061 .0064 .0064 .0062 .0060 .0049 .0065 .0131 .0138 .0066 .0069 .0073 .0077 .0075 .0065 .0105  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  41.72 41.68 C1.83 01.S3 »1.16 »1.08 01.08 01.03  .14 .14 .15 .30 .22 .42 .26  41.72 *1.67 41.83 »1.41 »1.28 »1.19 »1.15 Oi.OO •1.49 41.42 »1.26 »1.12 01.07 Ol.Ol 41.61 "1.85 "1.44 »1.28 »1.14 01.06  .06 .06 .05 .16 .28 .12 .37 .20 .20 .10 .23 .28 .10 .40 .10 .02 .11 .23 .28 .85 .26 .26 .84 .41 .40 .65 .87 .77  41.77 »1.70 C1.80 01.29 01.26 • »1.23 »1.21 »1.09  1.0 1.0 1.8 1.7  0 0 0 0 0 0 0 0 0 0 0  .6 .6 .8 1.0 1.1  O  0  1.3 1.1 1.3 2.4 1.5  .0077 .0082 .0086 .0088 .0084 .0076 .0148 .0158 .0088 .0092 .0068 .0100 .0102 .0127 .0176 .0267  .0071 .0080 .0076 .0069 .0085 .0067 .0156 .0227  ».0071 .0070 .0078 .0080 .0060 .0084 .0066 .0179 .0182 .0071 .0075 .0076 .0071 .0084 .0096 .0073 .0078 .0077 .0077 .0073 .0118 .0073 .0080 .0077 .0084 .0080 .0097 .0066 .0189  (percent c)  1.7 1.6 33 3.2 3.4 1.8 -.043 -.043 -.045 -.045 -.054  .8 1.1 1.1 .9 1.8  .006 .007 .007 .012 .010  1.2 1.3 1.3 1.4 30  .010 .010 .011 .014 .011 .014 .005 .006 .005 .002 -.001  .6 .6 1.0 .9 .9 .4 1.0 1.1 1.0 .6 1.1  .066 .068 .090 .092 .098  .6 .7 1.0 1.1 1.4  3 1 -1 -1 -4  3 1 -1 -5  01.74 01.70 Ol.61 »t.4« B1.86 Dl.Sl 01.82 O1.20  .82 .22 .30 .37 .86 .61 .87  .0082 .0085 .0067 .0065 .0061 .0106 .0164 .0276  .088 .088 .061 .065 .067  .8 .9 1.0 1.3 1.1  .097 .065 .096 .065 .094 .066 .086  C1.72 Ol.66 01.86 01.48 01.41 D1.35 01.84 01.84  .0090 .0093 .0064 .0069 .0103 .0123 .0168 .0366  .085 .086 .085 .090 .092  1.0 1.4 1.4 1.7 1.4  .066 .096 .067 .087 .097 .067 .106  01.82 01.75 D1.64 Ol.B4 01.48 »1.47 »1.46 Ol.4S  .32 .20 .33 .31 .84 .89 .46 .68 .87 .35 .38 .82 .60 .85 .70  .0061 .0096 .0099 .0104 .0066 .0129 .0205 .0160  -.133 -.130 -.131 -.135  .9 1.1 .8 1.0  -2  FIGURE (36) AIRFOIL SECTION CHARACTERISTICS AS AFFECTED BY VARIATIONS OF THE REYNOLDS NUMER:  REF. [17j  FIGURE  (37)  U.B.C. MECHANICAL ENGINEERING LOW-SPEED CLOSED-CIRCUIT WIND TUNNEL  - 91 -  TABLE  I  AERODYNAMIC CHARACTERISTICS OF AIRFOILS EXAMINED THEORETICALLY 1.1  NACA-0015  Free A i r  Values  a = 10° C, L  Wall  Configuration  S o l i d Walls  „ „  „ „  "  "  H  „  S l o t t e d Walls A0AR=59% II  H  .  = 1.2257  C  F  M  M  = 0.2829  0  C M  %  M  = -.0189  c/4  C/H  \  0.1  1.2335  0.2841  -0.0196  0.2  1.2562  0.2841  -0.0217  0.3  1.2939  0.2936  -0.0250  0.4  1.3458  0.3019  -0.0294  0.5  1.4115  0.3127  -0.0348  .9-6  1.4906  0.3260  -0.0410  0.7  1.5825  0.3420  -0.0476  0.8  1.6873  0.3607  -0.0547  0.9  1.8050  0.3825  -0.0619  1.0  1.9361  0.4075  -0.0692  0.1  1.2341  0.2841  -0.0198  0.2  1.2438  0.2863  -0.0199  C  M c/4  - 92 -  1.1  Wall  (Continued)  Configuration  Slotted Walls . A0AR=59%  c  C/H  o  M M  C M  C/4  Q.3  1.2474  0.2872  -0.0199  -0.0206  n  n  0.4  1.2443  0.2857  ii  ii  0.5  1.2385  0.2825  n  H  0.6  1.2336  0.2784  -0.0253  H  H  0.7  1.2320  0.2741  -0.0293  n  H  0.8  1.2333  0.2699  j  -0.0338  n  n  0.9  1.2380 ;  0.2659  '  -0.0389  1.0  1.2452  0.2623  1.2  NACA-23012  Free A i r Values  a = 8  Wall  Configuration  Solid  Walls  II  H  n  n  -0.0442  6 = 20°  C. = 2.7959 L  -0.0224  .  C  F  M  M  = 0.3890  C  0  M  M  = -0.3031  c/4  C/H  \  0.1  2.8099  0.3904  -0.3052  2,8499  0.3990  -0.3065  2.9170  0.4107  -0.3115  0.3  M  0  M  c/4  - 93 1.2  (Continued)  Kali  Configuration  S o l i d Walls  C/H  ...  v  K  c/4  0.4  3.0106  0.5  3.1299  0.4441  -0.3308  ,  0.4256  -0.3197  n  n  II  II  0.6  3.2752  0.4665  -0.3444  n  II  0.7  3.4465  0.4932  -0.3601  n  II  0.8  3.6448  0.5248  -0.3775  II  n  0.9  3.8711  0.5620  n  n  1.0  4.1276  0.6057  -0.4162  0.1  2.8092  0.3902  -0.3052  0.2  2.8260  0.3928  -0.3068  S l o t t e d Walls A0AR=59%  :  -0.3963  II  n  H  II  0.3  2.8237  0.3922  -0.3068  II  II  0.4  2.8008  0.3862  -0.3072  n  II  0.5  2.7704  0.3762  -0.3097  0.6  2.7364  0.3635  -0.3140  n  II  0.7  2.7073  0.3497  -0.3205  ti  II  0.8  2.6831  0,3358  -0.3285  •  - 94 1.2  (Continued)  Wall  Configuration  1  S l o t t e d Walls A0AR=59%  1.3  C/H  L L  c  T  M  M  Configuration  c/4  2.6655  0.3219  -0.3380  1.0  2.6527  0.3083  . -0.3484  a = 10° C,  = 2.1353  C  M  = 0.3926  C  Lp  Wall  C M  0  0.9  CLARK-Y 14%  Free A i r Values  C  C/H  = - 0.1331  H  "c/4  C  i T  M  0  c' M  c/4  Walls  0.1  2.1490  0.3946  -0.1345  M  II  0.2  2.1883  0.4006  :-0.1382  n  n  0.3  2.2546  0.4108  -0.1443  n  n  0.4  2.3478  0.4254  -0.1526  0.5  2.4687  0.4448  -0.1630  0.6  2.6184  0.4696  -0.1751  Solid  II  II  II  n  0.7  2.7989  0.5004  -0.1887  II  n  0.8  3.0131  0,5383  -0.2035  II  H  3.2652  0,5845  -0,2194  3.5611  0,6406  -0.2361  °-9 H  n  1.0  - 95 1.3  (Continued)  S l o t t e d Walls A0AR=59% ii  n  n  n  C/H  V  0.1  2.1504  0.3946  -0.1348  0.2  2.1656  0.3978  -0.1354  0.3  2.1692  0.3982  -0.1358  0.4  2.1595  0.3945  -0.1372  0.5  2.1446  0.3878  -0.1402  n  Q  F 1  c/4  n  n  II  H  n  n  II  II  0.6  2.1312  0.3795  -0.1452  II  M  0.7  2.1220  0.3708  -0.1517  II  II  0.8  2.1185  0.3620  0.9  2.1206  0.3536  -0.1685  1.0  2.1270  0.3454  -0.1782  n  H  n  n  ;  -0.1596  - 96 -  TABLE  II  EFFECT OF THE PRESSURE ALONG.THE SHEAR LAYERS ON THE NEW WIND TUNNEL THEORETICAL DATA  NACA-0015 AIRFOIL Free A i r Values  C. L  V  a = 10°  C/H = .8  =1 .2257  C  =-0.0189.  M  F  c/4  \  .  E% L  C M  c/4  -0.4  0.11  1.4038  14.53  -0.0555  -2.99  -0.3  0.11  1.3094  6.83  -0.0484  -2.42  -0.2  0.11  1.2137  -0.98  -0.0412  -1.82  -0.1  0.11  1.1164  -8.91  -0.0338  -1.22  -0.1  0.2  1.1163  -8.93  -0.041  -1.8  -0.1  0.3  1.1137  -9.14  -0.0488  -2.44  -0.1  0.4  .1,1079  -9.61  -0.0561  -3.04  - 97 -  TABLE  III  LOCATIONS OF THE PRESSURE TAPS  III.1  For C/H = .34 S e c t i o n  Lower  Upper S u r f a c e x/H  Surface y/H  x/H  y/H  0.00346  0.01224  0.0  0.0  0.01060  0.02188  0.01800  0.02797  0.01750  0.02778  0.03190  0.03633  0.02930  0.03493  0.05410  0.04587  0.04980  0.04426  0.07880  0.05338  0.07520  0.05241  0.10400  0.05917  0.10200  0.05876  0.15400  0.06701  0.12700  0.06331  0.20500  0.07176  0.15200  0.06677  0.2550  0.07406  0.17800  0.06959  0.30400  0.07458  0.20200  0.07155  0.35400  0.07379  0.22800  0.07309  0.40400  0.07179  0.25200  0.073977  0.45300  0.06899  0.27600  0.07443  0.50300  0.06518  0.30300  0.07459  0.55300  0.06100  0.351000  0.07387  0.65200  0.05066  0.40100  0.07194  0.75100  0.03855  0.45300  0.06899  0.85100  0.02462  .  - 98 III.l  (continued)  Lower S u r f a c e  • Upper S u r f a c e  III.2  x/H  y/H  x/H  y/H  0.50200  0.06526  0.90000  0. 01714  0.55200  0.06526  0.95100  0. 00894  0.60200  0.05612  0.65200  0.05066  0.70100  0.04488  0.75100  0.03855  0.80000  0.03195  0.84900  0.02491  0.90000  0.01714  For C/H = .67 S e c t i o n  Lower S u r f a c e  Upper S u r f a c e x/H  y/H  x/H  y/H  0.00664  0.06969  0.0  0.0  0.01808  0.00705  0.00667  0.01894  0.02847  0.06378  0.01874  0.02852  0.05345  0.05929  0.02887  0.03469  0.07914  0.05347  0.05401  0.04584  0.10457  0.04564  0.07984  0.05365  - 99 -  III.2  (Continued)  Lower S u r f a c e  Upper S u r f a c e x/H  y/H  x/H  y/H  0.13003  0.03466  0.10467  0.05931  0.15426  0.02803  0.13031  0.06382  0.17915  0.01890  0.15423  0.06704  0.20921  0.07204  0.17990  0.06974  0.22971  0.07317  0.20922  0.07204  0.25405  0.07403  0.22986  0.07318  0.27904  0.07447  0.25452  0.07404  0.30338  0.07459  0.27949  0.07448  0.35456  0.07377  0.30406  0.07458  0.40460  0.07177  0.35351  0.07380  0.45178  0.06907  0.40306  0.07184  0.50045  0.06538  0.45223  0.06904  0.59984  0.05635  0.50149  0.06530  0.69811  0.04523  0.55123  0.06117  0.79565  0.03255  0.60081  0.05625  0.88765  0.01907  0.69823  0.04522  0.96950  0.00567  0.79608  0.03249  0.89199  0.01839  0.97060  0.00547  - 100 -  TABLE  IV  THE EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF NACA-0015 AIRFOIL  IV.1  Wall  C/H = .34  Configuration  Re = 500,000  0  C  a  L L  T .  %  C  M n  c/4  Walls  -4  -0.4349  -0.1054  0.00302  II  n  -2  -0.1815  -0.02573  0.01960  n  n  0  -0.0051  0.0086  0.0098  II  n  2  0.1891  0.0485  0.0013  n  II  4  0.4348  0.1189  0.0105  n  n  6  0.6944  0.1988  0.0262  II  II  8  0.9.061  0.2646  0.0403  n  II  10  1.0176  0.2766  0.0260  H  n  12  1.1326  0.3033  0.0253  n  H  14  1.1915  0.3322  0.0431  -4  -0.3568  -0.0786  0.0104  -2  -0.1626  -0.0237  0.0169  0.0109  0.0120  0.0093  Solid  Slotted  Walls  AOAR = bU/o n  II  II  n  0.  .. .  - 101 -  IV.1  Wall  (Continued)  Configuration  S l o t t e d Walls AOAR  = 50%  0  or  M  c/4  -0.000  2  0.1686  0.0416  4  0.3751  0.0974  0.0038  6  0.5746  0.1542  0.0114  8  0.7979  0.2100  0.0124  II  n  n  II  n  n  H  II  10  0.9203  0.2341  0.0075  II  II  12  1.1096  0.2995  0.0282  H  II  14  1.1425  0.3016  0.0243  -4  -0.3597  -0.0780  0.0117  S l o t t e d Walls AOAR  = 55%  H  n  -2  -0.1825  -0.0284  0.0172  ti  H  0  -0.0144  0.0071  0.0107  n  H  2  0.1549  0.0400  0.0013  n  II  4  0.3541  0.0920  0.0037  n  n  6  0.5570  0.1460  0.0075  n  II  8  0.7786  0.2073  0.0146  n  n  10  0.8677  0.2144  0.0008  - 102 -  IV.1  Wall  (Continued)  Configuration  S l o t t e d Walls AOAR  II  = 55% n  S l o t t e d Walls AOAR  II  = 59%  II  a  0  \  \  M  c/4  12  1.0574  0.2786  0.0201  14  1.1122  0.2910  0.0212  -4  -0.3370  -0.0705  0.0136  -2  -0.1561  -0.0208  0.0182  0  -0.0135  0.0067  0.0101  II  II  2  0.1540  0.0386  0.0001  II  H  4  0.3337  0.0828  -0.0004  H  II  6  0.5360  0.13710  0.0038  H  II  8  0.7583  0.20160  0.0139  II  H  10  0.8611  0.2124  0.0004  12  0.9510  0.2222  -0.0103  14  1.0900  0.2841  0.0197  n  n  - 103 IV.2  C/H = .67  Wall  Configuration  S o l i d Walls  Re = 500,000  0  a  C M  c/4  -4  -0.4747  -0.1139  0.0044  -2  -0.2650  -0.0487  0.0175  0  -0.0171  0.0125  0.0167  n  II  2  0.1927  0.0643  0.0161  n  II  4  0.4229  0.1223  0.0169  6  0.7078  0.2066  0.0306  0.0137  II  n  8  0.9233  0.2423  II  n  10  1.1053  0.2784  II  II  12  1.2643  0.29884  -0.0103  14  1.4256  0.32251  -0.0233  16  1.4373  0.3325  -0.0130  -4  -0.3299  -0.0670  0.0153  n  II  n  n  S l o t t e d Walls AOAR = 5 0 %  s  0.0063  II  H  -2  -0.1594  -0.0233  0.0166  II  II  0  0.0252  0.0233  0.0170  '  II  II  2  0.1903  0.0641  0.0166  I  n  II  4  0.3794  0.1093  0.0146  - 104 IV.2  (Continued)  Wall C o n f i g u r a t i o n Slotted AOAR  Walls =  50%  II  n  II  n  a  0 L  T  .  . .  M  0  0.1675  C  M  C/4  0.0238  6  0.5781  8  0.83.63  0.2245  0.0174  10  0.9658  0.2512  0.0134  12  1.1193  0.2812  0.0075  II  H  14  1.2235  0.2928  -0.0040  II  II  16  1.3391  0.3172  -0.0046  -4  -0.3514  -0.0772  0.0105  -2  -0.2039  -0.0305  0.0204  0  0.0107  0.0291  0.0264  2  0.1629  0.0566  0.0159  4  0.3562  0.1037  0.0149  6  0.5990  0.1728  0.0239  Slotted AOAR n  Walls =  55%  II  H  n  n  n  n  H  8  0.7998  0.2163  0.0183  n  II  10  0.9404  0.2459  0.0144  n  II  12  1.0720  0.2698  0.0077  II  n  14  1.1924  0.2846  -0.0047  n  n  16  1.2886  0.3043  -0.0054  - 105 IV.2  Wall  (Continued)  Configuration  S l o t t e d Walls AOAR = 59% .  0  a  c  4  \  C M  c/4  -4  -0.3186  -0.0612-  0,0183  II  H  -2  -0.1616  -0.0222  0.0182  H  n  0  -0.0041  0.0170  0.0181  n  H  2  0.1757  0.0594  0.0155  II  n  4  0.3506  0.0996  0.0122  n  H  6  0.5286  0.1539  0.0224  n  II  8  0.7719  0.2088  0.0177  n  n  10  0.8982  0.2334  0.0122  n  II  12  1.0287  0.2601  0.0086  n  n  14  1.1657  0.2791  -0.0037  n  n  16  1.2753  0.2980  -0.0085  - 106 -  TABLE V THE LIFT-CURVE SLOPES OBTAINED .... EXPERIMENTALLY FOR NACA-0015 AIRFOIL  m ( 0 ° ^ a<10° Wall  Configuration  )  For C/H = .34  C/H = .67  0.10750  0.11560  Slotted Walls A0AR=50%  0.09478  0.09771  Slotted Walls A0AR=55%  0.09264  0.09717  S l o t t e d Walls A0AR=59%  0.09126  0.09254  Solid  Walls  

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