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

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A T H E O R E T I C A L I N V E S T I G A T I O N O F A L O W - C O R R E C T I O N W I N D T U N N E L W A L L C O N F I G U R A T I O N F O R A I R F O I L T E S T I N G b y AHMED FOUAD MALEK B . S c . ( E n g . ) , A l e x a n d r i a U n i v e r s i t y , 1 9 7 0 B . S c . ( M a t h . ) , A l e x a n d r i a U n i v e r s i t y , 1 9 7 4 -A THESIS SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A u g u s t 1 9 7 9 ^ Ahmed F o u a d M a l e k , 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f 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 i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 c o p y i n g 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 DE-6 BP 75-51 1 E A B S T R A C T T h i s t h e s i s d e a l s w i t h a n e w a p p r o a c h t o r e d u c e w a l l c o r r e c t i o n s i n h i g h - l i f t a i r f o i l t e s t i n g , b y e m p l o y i n g s y m m e t r i c -a l l y t r a n s v e r s e l y s l o t t e d w a l l s . T h e s o l i d e l e m e n t s o f t h e s l o t t e d w a l l a r e s y m m e t r i c a l a i r f o i l s a t z e r o i n c i d e n c e , t h e s p a c e s b e t w e e n t h e s l a t s a r e n o n u n i f o r m , i n c r e a s i n g l i n e a r l y t o w a r d s t h e r e a r . T h i s w a l l c o n f i g u r a t i o n p r o v i d e s f l o w c o n d i t i o n s c l o s e t o t h e f r e e a i r t e s t e n v i r o n m e n t w h i c h l e a d s t o n e g l i g i b l e o r . s m a l l w a l l c o r r e c t i o n s . T h e t h e o r y u s e s t h e p o t e n t i a l f l o w s u r f a c e v o r t e x - e l e m e n t m e t h o d , w i t h " F u l l L o a d " K u t t a C o n d i t i o n s s a t i s f i e d o n t h e t e s t a i r f o i l a n d w a l l s l a t s . T h i s m e t h o d i s v e r y w e l l s u p p o r t e d b y p h y s i c a l e v i d e n c e a n d i t i s s i m p l e t o u s e . T h e s u r f a c e v e l o c i t i e s c a n b e c a l c u l a t e d d i r e c t l y a n d t h e a e r o d y n a m i c l i f t a n d p i t c h i n g moment a r e d e t e r m i n e d b y n u m e r i c a l i n t e g r a t i o n o f t h e c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n s a r o u n d t h e a i r f o i l c o n t o u r . T h i s m e t h o d c a n b e d e v e l o p e d i n o r d e r t o i n c l u d e t h e f l o w i n t h e p l e n u m c h a m b e r s i n t h e a n a l y s i s . - i i -TABLE OF CONTENTS Page Abstract i Table of Contents i i L i s t of Figures i i i Symbols v Acknowledgement v i I - ..INTRODUCTION 1 II - POTENTIAL FLOW ANALYSIS 4 I I - l Introduction 4 II-2 Surface Singularity Theory 5 II-3 The Kutta Condition 8 III - METHOD OF SOLUTION AND NUMERICAL ANALYSIS 10 IV - THE FLOW IN THE WINDTUNNEL PLENUM 15 IV-1 Introduction 15 IV-2 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 15 IV-3 Free Streamline Tracking 17 V - RESULTS AND DISCUSSION 19 VI - CONCLUSIONS 23 References 25 APPENDIX 1 - Evaluation of the Integral in Equation (5) 2 6 APPENDIX 2 - Calculation of the Velocity Component Induced at a Point i n the F i e l d of Vortex D i s t r i b u t i o n 30 Figures 33 - i i i -LIST OF FIGURES Page Figure 1 Comparison of A i r f o i l Pressure C o e f f i c i e n t s : Theory, Ref. (3) 33 Figure 2 Vortex Representation of Two Component A i r f o i l 34 Figure 3 Notation Used to Calculate Influence C o e f f i c i e n t s 35 Figure 4 Streamline Contours Around an A i r f o i l 3 6 Figure 5 Location of T r a i l i n g Control Point 37 Figure 6 Location of Elements on A i r f o i l Surface 38 Figure 7 Comparison of Higher Order Methods, Ref. (6) 3 9 Figure 8 Comparison of A i r f o i l V elocity D i s t r i b u t i o n s , Ref. (6) 40 Figure 9 An A i r f o i l Inside Tunnel Test Section with Double Slotted Wall 41 Figure 10 Notation Used to Calculate Induced V e l o c i t i e s 4 2 Figure 11 Geometry of Higher Order Terms 4 3 Figure 12 Comparison of Pressure C o e f f i c i e n t s for NACA-0015 A i r f o i l i n Free A i r and Between Solid Walls Test Section 44 Figure 13 Comparison of Pressure C o e f f i c i e n t s for NACA-23012 A i r f o i l i n Free A i r and Between Solid Walls Test Section 4 5 Figure 14 Comparison of Pressure C o e f f i c i e n t s for CLARK-Y 14% A i r f o i l i n Free A i r and Between Solid Walls Test Section 4 6 Figure 15 Comparison of Pressure C o e f f i c i e n t s for NACA-0015 A i r f o i l i n Free A i r and Between Single Uniformly Slotted Wall Test Section 47 Figure 16 Comparison of Pressure C o e f f i c i e n t s for NACA-0015 A i r f o i l i n Free A i r and Between Double Uniformly Slotted Wall Test Section 48 - i v -Page Figure 17 Comparison of the E f f e c t of Test A i r f o i l Size on the Relative Error i n L i f t C o e f f i c i e n t s 4 9 Figure 18 Comparison of the E f f e c t of Test A i r f o i l Size on the Relative Error i n Pitching Moment Coe f f i c i e n t s 50 Figure 19 Comparison of Pressure C o e f f i c i e n t s for NACA-0015 A i r f o i l i n Free A i r and Between Double Non-uniformly Slotted Walls Test Section with the Flow i n the Plenum 51 Figure 20 Comparison of Pressure C o e f f i c i e n t s for NACA-23012 A i r f o i l i n Free A i r and Between Double Non-uniformly Slotted Walls Test Section with the Flow i n the Plenum 52 Figure 21 Comparison of Pressure C o e f f i c i e n t s for CLARK-Y 14% A i r f o i l i n Free A i r and Between Double Non-uniformly Slotted Walls Test Section with the Flow i n the Plenum 53 Figure 22 Comparison of the E f f e c t of Two Different Angles of Attack on the Relative Error in L i f t C o e f f i c i e n t s for NACA-0015 A i r f o i l 54 Figure 23 Geometry of the Idealized Flow i n the Plenum Chamber for NACA-0015 A i r f o i l 55 -v-SYMBOLS Symbol Y K. 11 N M M Mc/4 < >P ( ) T UOAR AOAR c/C C/h De f i n i t i o n 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 . Influence c o e f f i c i e n t of the element " j " on the control point " i " . Number of vortex elements. Number of l i f t i n g bodies or components. Relative error i n l i f t c o e f f i c i e n t E_ = (CT - C T )/CT * 100 T L F L F Relative error i n pitching moment c o e f f i c i e n t . EM " ( CMC/4 T " CMC/4 F'/ CL F * 1 0 0 Pressure c o e f f i c i e n t . L i f t 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 . Free a i r data. Windtunnel data. 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 . - v i -ACKNOWLEDGEMENT This research was carried out under the supervision of Dr. G.V. Parkinson, whose expert advice and guidance i s gr a t e f u l l y acknowledged. A l l the computing was done at the U.B.C. Computing Center. This research was supported by the Department of Mechanical Engineering at U.B.C. - 1 -CHAPTER I - INTRODUCTION T r a d i t i o n a l l y , the subsonic measurements of the aerodynamic c h a r a c t e r i s t i c s of a i r f o i l s i n windtunnels with s o l i d walls re-quired but small corrections by the standard methods, provided that the test a i r f o i l s were small r e l a t i v e to the tes t section cross-sections and developed r e l a t i v e l y small l i f t c o e f f i c i e n t s . Now, however, with the frequent testing of very high l i f t a i r f o i l sections, using large models which produce r e a l i s t i c a l l y high Reynolds' numbers, the standard wall-correction theory i s no longer accurate unless windtunnels with very large test cross-sections are used. Since the cost of construction of windtunnels of s u f f i c i e n t size i s not economical, the alternative i s therefore to develop a windtunnel wall correction theory for subsonic testing which can answer to the c a l l for r e a l i s t i c measurements and accurate corrections. Contributions to such a theory are presented i n t h i s thesis. It i s well known that most corrections to data i n windtunnels with open jets are opposite i n sign to those i n windtunnels whose walls are s o l i d , Ref. (1). These opposing effects 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 can-c e l l i n g the corrective e f f e c t s of the two types of wall. Recently, two such designs for t h i s purpose have been considered. One has walls with narrow longitudinal s l a t s . The other has walls patterned with small holes. Using the l i n e a r theory to investigate those two types of . wall configurations, Parkinson and Lim (2) and others have found that there i s a lack of agreement between the experimental r e s u l t s - 2 -a n d t h o s e w h i c h a r e p r e d i c t e d b y t h e t h e o r y f o r t h e l o n g i t u d i n a l s l o t t e d w a l l . A l s o t h e y h a v e f o u n d t h a t t h e " p o r o s i t y p a r a m e t e r " i s n o t s i m p l y a n e m p i r i c a l f u n c t i o n o f t h e o p e n a r e a r a t i o b u t i t m u s t be d e t e r m i n e d e m p i r i c a l l y f o r e a c h a i r f o i l u n d e r t e s t , a n i m p o s s i b l e s i t u a t i o n f o r t h e p r a c t i c a l u s e o f p o r o u s w a l l c o n f i g -u r a t i o n . P a r k i n s o n a n d L i m (2) h a v e a t t r i b u t e d t h e l a c k o f s u c c e s s o f t h e l o n g i t u d i n a l s l a t s a n d p o r o u s w a l l t h e o r i e s t o t h e o c c u r r e n c e e x p e r i m e n t a l l y o f s e p a r a t e d f l o w s i n t h e s l a t s a n d h o l e s . S u c h f l o w s a r e n o t a c c o u n t e d f o r i n t h e t h e o r i e s , p r i m a r i l y as t h e y a d d u n d e s i r a b l e n o n l i n e a r i t i e s t o t h e m . I n a d d i t i o n , t h o s e f l o w s e p a r a t i o n s s e r i o u s l y d e g r a d e t h e m a i n f l o w i n t h e v i n c i n i t y o f t h e w a l l s . W i l l i a m s (3) , u s i n g a s u r f a c e s o u r c e m e t h o d , h a s i n v e s t i g a t e d a n o t h e r t y p e o f w i n d t u n n e l , one w h i c h h a s an u p p e r t r a n s v e r s e l y s l o t t e d w a l l , w i t h u n i f o r m g a p s , a n d a l o w e r s l o t t e d w a l l . H i s t h e o r e t i c a l a n a l y s i s shows t h a t u s i n g t h i s t y p e o f w a l l c o n f i g u r -a t i o n w i l l i m p r o v e t h e p e r f o r m a n c e o f t h e t u n n e l t e s t s e c t i o n . The a n a l y s i s p r e d i c t s t h a t s u c h a t e s t s e c t i o n w i l l p r o d u c e l i f t d a t a w i t h i n a s m a l l p e r c e n t a g e e r r o r o f t h e f r e e - a i r v a l u e s , w h i l e t h e p r e s s u r e d i s t r i b u t i o n a n d t h e p i t c h i n g moment d a t a a r e o f .. l o w e r a c c u r a c y , F i g . (1). A l s o h i s e x p e r i m e n t a l i n v e s t i g a t i o n shows t h a t s u c h . a w a l l c o n f i g u r a t i o n p r o v i d e s a f l o w f r e e o f s e p a r a t i o n o n t h e t r a n s v e r s e s l a t s . I n t h e p r e s e n t t h e s i s , a s u r f a c e s i n g u l a r i t y a n a l y s i s , u s i n g a s u r f a c e v o r t i c i t y m e t h o d , i s u s e d t o i n v e s t i g a t e a w i n d t u n n e l w i t h b o t h w a l l s t r a n s v e r s e l y s l o t t e d a s a n o t h e r a p p r o a c h t o m o d i f y t h e w i n d t u n n e l w a l l c o n f i g u r a t i o n i n o r d e r t o r e c r e a t e t h e f r e e -- 3 -a i r streamline patterns about the test a i r f o i l , which would then experience the corresponding f r e e - a i r loading. 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 incidence. 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 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 Fig.(16)) the upper surface of the a i r f o i l section i n the presence of the slotted 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 e f f e c t s tend to cancel for l i f t but lead to appreciable errors i n pitching moment. A solution to t h i s 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 , as shown i n F i g . (9a). Also the surface s i n g u l a r i t y analysis has been used here to track the free streamlines which enter or leave the test section from or to the upper and the lower plenum, respectively, i n order to represent the flows i n there. - 4 -CHAPTER I I - POTENTIAL FLOW ANALYSIS I I - l - I n t r o d u c t i o n An e f f i c i e n t , r e l i a b l e m e t h o d f o r c a l c u l a t i n g , t h e v e l o c i t y d i s t r i b u t i o n o n t h e s u r f a c e o f a i r f o i l s e c t i o n s i s r e q u i r e d . C o n -f o r m a l t r a n s f o r m a t i o n m e t h o d s s u c h as t h a t o f T h e o d o r s e n (4) c a n a n a l y z e s e c t i o n s o f a r b i t r a r y s h a p e . T h e s e m e t h o d s a r e b a s e d o n t h e t h e o r e m w h i c h s t a t e s t h a t i t i s a l w a y s p o s s i b l e t o t r a n s f o r m t h e p o t e n t i a l f i e l d a r o u n d a n y c l o s e d c o n t o u r i n t o t h e p o t e n t i a l f i e l d a r o u n d a c i r c l e . S u c h m e t h o d s a r e n o t s i m p l e a n d , as t h e r e i s no s u c h t h e o r e m f o r t r a n s f o r m i n g t h e p o t e n t i a l f i e l d a r o u n d m u l t i - c o m p o n e n t s e c t i o n s , o n e l o o k s t o s u r f a c e s i n g u l a r i t y m e t h o d s o f a n a l y s i s . T h e s e m e t h o d s r e p l a c e t h e p o t e n t i a l f l o w f i e l d o u t -s i d e t h e a i r f o i l c o n t o u r w i t h t h a t a b o u t a s e t o f s i n g u l a r i t i e s , s o u r c e s o r v o r t i c e s , w h i c h s a t i s f y t h e same b o u n d a r y c o n d i t i o n s . The s u r f a c e s i n g u l a r i t y m e t h o d s c a n d e a l e a s i l y w i t h m u l t i -c o m p o n e n t s e c t i o n s a n d a r e no l e s s a c c u r a t e t h a n t h e c o n f o r m a l t r a n s f o r m a t i o n m e t h o d s o n s i n g l e - c o m p o n e n t c a s e s . The m o s t w i d e l y u s e d s u r f a c e s i n g u l a r i t y m e t h o d i s t h a t o f Hess a n d S m i t h ( 5 ) . T h i s m e t h o d u s e s a d i s t r i b u t i o n o f s o u r c e s a n d s i n k s o n t h e s u r f a c e o f t h e a i r f o i l s e c t i o n c o m b i n e d w i t h a v o r t i c i t y d i s t r i b u t i o n t o g e n e r a t e c i r c u l a t i o n . W i l l i a m s ( 3 ) , among o t h e r s , h a s e m p l o y e d t h i s t o i n v e s t i g a t e h i s w i n d t u n n e l m o d e l . The t e c h n i q u e w h i c h t h e y h a v e u s e d h a s , h o w e v e r , some d r a w b a c k s . One p a r t i c u l a r p r o b l e m a r i s e s f r o m t h e i r a p p l i c a t i o n o f t h e K u t t a C o n d i t i o n , i n t h e f o r m o f e q u a l v e l o c i t y m a g n i t u d e s a t t h e c o n t r o l p o i n t s o f t h e u p p e r a n d l o w e r t r a i l i n g edge e l e m e n t s J (U = - U ) . T h i s c a n t n t 1 u _ 5 _ be c a l l e d the NO LOAD Kutta Condition since 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 (9). 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 . As i n the a i r f o i l analysis the boundary condition which i s applied here i s that the s o l i d surfaces of the a i r f o i l section are streamlines and the stream functions are required to be constant. II-2 - Surface Singularity Theory In two dimensional, incompressible, i r r o t a t i o n a l flow the stream function must s a t i s f y Laplace's equation, i f i + i J L - n (i) 2 2 dx dy 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 if) (K=l,2,....,M) on the is. surfaces of M components on a multi-component section are constants, the boundary condition for equation (1) can be written as, if> = \bv , on the surface (2) The stream function for a uniform stream incident to the pos i t i v e X axis at an angle a i s given by if) = y cos a - x sin a (3) 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. - 6 -T h e d i s t a n c e s a r e d i m e n s i o n l e s s w i t h r e s p e c t t o t h e c h o r d l e n g t h C , t h e v e l o c i t i e s w i t h r e s p e c t t o t h e f r e e s y s t e m v e l o c i t y U o o a n d t h e s t r e a m f u n c t i o n s w i t h r e s p e c t t o t h e p r o d u c t U o o C . The p o i n t v o r t e x o f s t r e n g t h r , l o c a t e d a t ( X q , y ) h a s t h e s t r e a m f u n c t i o n ^ = " 2T £ n ( r ) ' ( 4 ) 1 /2 2 2 ' w h e r e r = I(x-x ) + ( y - y ) l o o e q u a t i o n (4 ) a l s o s a t i s f i e s ( 1 ) , e x c e p t a t r = 0 . B e c a u s e o f t h e l i n e a r i t y o f e q u a t i o n (1 ) a n y c o l l e c t i o n o f p o i n t v o r t i c e s o r a n y c o n t i n u o u s d i s t r i b u t i o n o f t h e m a s i n F i g . ( 2 ) , t h a t l i e s o n t h e a i r f o i l s u r f a c e , S , w i l l s a t i s f y e q u a t i o n (1) i n t h e r e g i o n o u t -s i d e o f S. T h e n t h e s t r e a m f u n c t i o n a t a g e n e r a l p o i n t P d u e t o v o r t i c i t y h a v i n g a d e n s i t y Y ( S ' ) a t S ' a n d c o n t i n u o u s l y d i s t r i b -u t e d o v e r t h e a i r f o i l s u r f a c e S, i s g i v e n b y ^P = 2T f'V(s') £ n r ( p ' s ' > d s ' (5 ) S A p p l y i n g t h e b o u n d a r y c o n d i t i o n , e q u a t i o n ( 2 ) , t h e c o m b i n e d f l o w d u e t o a u n i f o r m s t r e a m p l u s t h e a b o v e d i s t r i b u t i o n o f t h e v o r t i c i t y , o n e o b t a i n s % = y S ° O S a ~ X S S i n a ~ ITF ^ ^ ( S ' ) £n r ( S , S ' ) d S ' (6 ) S ' The a i r f o i l s u r f a c e i s d i v i d e d u p i n some m a n n e r i n t o N s m a l l s u r f a c e e l e m e n t s . On e a c h o f t h e s e t h e r e i s a c o n t r o l p o i n t , C ^ , l o c a t e d a t ( x ^ , y ^ ) , a t w h i c h t h e b o u n d a r y c o n d i t i o n , e q u a t i o n ( 6 ) , i s made t o a p p l y . E a c h e l e m e n t j h a s v o r t i c i t y o f d e n s i t y Y ( S j ) d i s t r i b u t e d o n i t s s u r f a c e . The i n t e g r a l i n e q u a t i o n ( 6 ) , o v e r t h e w h o l e s u r f a c e S , i s t h e n r e p l a c e d b y a s u m m a t i o n o f N i n t e g r a l s - 7 -o v e r t h e N s u r f a c e e l e m e n t s . A p p l y i n g e q u a t i o n (6) a t t h e c o n t r o l p o i n t , C^, one o b t a i n s , •N iK + E ^ — / y (S !) I n r (C. , S ! ) d S ! = y . c o s a - x . s i n a r k . , 2ir / ] 1 3 ] ^ i l 3 S j (7 ) The r e s u l t s r e q u i r e d o f an a i r f o i l a n a l y s i s m e t h o d a r e t h e s u r f a c e v e l o c i t i e s . K e n n e d y (6) shows t h a t t h e t a n g e n t i a l v e l o c i t y a t t h e i n t e r i o r o f t h e s o l i d s u r f a c e s h a s t o be z e r o so t h a t t h e s e s u r f a c e s become s t r e a m l i n e s , w h i c h a l s o r e s u l t s i n t h e d i s c o n t i n -u i t y i n t h e t a n g e n t i a l v e l o c i t y a c r o s s a v o r t e x s h e e t b e i n g e q u a l t o t h e d e n s i t y o f t h e v o r t e x s h e e t . T h u s Y (S^) i s e q u a l t o t h e s u r f a c e v e l o c i t y . I n s o l v i n g t h e e q u a t i o n (7 ) o n e t h e r e f o r e s o l v e s d i r e c t l y f o r t h e v e l o c i t i e s o n t h e a i r f o i l s u r f a c e s . A t t h i s p o i n t i t i s n e c e s s a r y t o make some a s s u m p t i o n a b o u t t h e s e c t i o n g e o m e t r y , t h e l o c a t i o n o f • t h e c o n t r o l p o i n t s a n d t h e f o r m o f y ( S j ) o v e r e a c h e l e m e n t j . The s i m p l e s t a p p r o x i m a t i o n i s t o assume t h a t t h e e l e m e n t s a r e s t r a i g h t l i n e s w i t h c o n t r o l p o i n t s a t t h e e l e m e n t m i d p o i n t s a n d Y ( S j ) i s a c o n s t a n t o v e r e a c h e l e m e n t . U s i n g t h e a b o v e a p p r o x i m a t i o n a n d a p p l y i n g e q u a t i o n (7) a t e a c h c o n t r o l p o i n t y i e l d s t h e s y s t e m o f e q u a t i o n s , N E j = l \ + . E , K i j Y j = R i ' ( i = 1 , . . . N) (8) w h e r e K ^ j i s t h e i n f l u e n c e c o e f f i c i e n t o f t h e e l e m e n t j o n t h e c o n t r o l p o i n t i , R^ i s t h e r i g h t h a n d s i d e o f e q u a t i o n (7) e v a l -u a t e d a t c o n t r o l p o i n t i a n d ip i s t h e s t r e a m f u n c t i o n f o r t h e K a i r f o i l c o m p o n e n t K. U s i n g t h e n o t a t i o n o f F i g . ( 3 ) , t h e BASIC i n f l u e n c e c o -e f f i c i e n t s c a n be w r i t t e n , - 8 -K.. = j- { (b+A) in (A) - (b-A) £n • (A + 2a tan 1 (-~^ ~) - 4A J 1 a +b -A (9) The d e t a i l s of the c a l c u l a t i o n of t h i s equation are provided i n Appendix 1 . The K^j and are purely functions of the geometry of the surface elements and the angle of attack. The system of equation (8) i s a set of N equations for the N unknown y. and M unknown IJJ, , where there are M components. 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 wall, 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 geometries 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, 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, i n f i n i t e . 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. It has been shown by Milne-Thompson (7) that a consequence of t h i s assumption i s that - 9 -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 (8) show t h i s 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. Such a Kutta Condition was used successfully by Bhateley and Bradley (9). The 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). It i s then assumed that the streamlines through the other control points of that component also pass through t h i s control point. Equation (8) then applies to these t r a i l i n g control points, C t , and the Kutta Condition can be written as, N + Z K t D i Y V = RtD (m = 1 2 M) <10> m j = l t pm , : i D t pm ' ( m 1 ^ " " ' M ' There are M such t r a i l i n g control points, one for each component, and hence M Kutta Condition equations. Thus the problem of potential flow over an a i r f o i l section has therefore been reduced to that of solving (N+M) equations, prescribed by equations (8) and (10), simultaneously to get N vortex densities, y • , and M stream functions i k . - 10 -CHAPTER I I I - METHOD OF SOLUTION AND NUMERICAL ANALYSIS The f i r s t s t e p i n t h e s o l u t i o n i s t o d e f i n e t h e e l e m e n t s w h i c h d e s c r i b e t h e a i r f o i l s u r f a c e . One o b v i o u s m e t h o d o f d o i n g t h i s i s t o l e t t h e s u p p l i e d c o - o r d i n a t e s be t h e e n d p o i n t s o f t h e s u r f a c e e l e m e n t s . T h i s h a s t h e d i s a d v a n t a g e t h a t t h e r e may be i n s u f f i c i e n t c o - o r d i n a t e s a v a i l a b l e o r t h a t t h e y may be i r r e g u l a r l y s p a c e d . To o v e r c o m e t h e s e p r o b l e m s t h e a i r f o i l i s d i v i d e d u p , f r o m i t s l e a d i n g edge a t x=0 t o i t s t r a i l i n g e d g e a t x = l . The e n d p o i n t s o f t h e s u r f a c e e l e m e n t s a r e l o c a t e d , as shown i n F i g . (5) , a t x - c o - o r d i n a t e s g i v e n b y , x £ = | ( l -cos<J> £ ) , (£ = 0,1,2, ,N) (11) w h e r e " 2 T T £ h = I T H e r e N m u s t b e a n e v e n n u m b e r i n o r d e r t h a t t h e e n d p o i n t b e l o c a t e d a t t h e a i r f o i l t r a i l i n g e d g e . T h i s d i s t r i b u t i o n o f p o i n t s p r o v i d e s , i n g e n e r a l , a m o r e a c c u r a t e s o l u t i o n b e c a u s e i t c o n c e n t r a t e s t h e c o n t r o l p o i n t s n e a r t h e l e a d i n g e d g e a n d t r a i l i n g ed g e w h e r e t h e l a r g e s t v e l o c i t y g r a d i e n t s g e n e r a l l y o c c u r . The c o r r e s p o n d i n g c o - o r d i n a t e s y £ o f t h e e l e m e n t e n d p o i n t s a r e d e t e r m i n e d b y i n t e r p o l a t i o n o n t h e g i v e n a i r f o i l d a t a . The u s e o f a c u b i c s p l i n e f u n c t i o n h a s b e e n f o u n d t o be t h e m o s t r e l i a b l e m e t h o d , s i n c e i t g i v e s s m o o t h c u r v e s t h r o u g h t h e g i v e n p o i n t s a n d c a n b e e a s i l y a n d e f f i c i e n t l y c o m p u t e d . H e r e t h e U . B . C . c o m p u t e r s u b r o u t i n e SAINT h a s b e e n u s e d f o r i n t e r p o l a t i o n . The c o n t r o l p o i n t s a r e t a k e n as t h e m i d - p o i n t s o f e a c h s u r f a c e e l e m e n t , as shown i n F i g . (6), t h e n t h e b i s e c t o r o f t h e t r a i l i n g e d g e i s e x t e n d e d , as shown i n F i g . (5) , a n d t h e c o n t r o l - 11 -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 esults 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 a, 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. cosa + y. sina 1 1 1 1 (12) Y. = y. cosa - x. sina l 2 1 l 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^j, 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. l l (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, i = l i=N i=N+l K 1,1 K1,N 1 K N,l * • ' Kutta Condition KN,N 1 Y-Y N R, R tp (14) The a b o v e s y s t e m o f e q u a t i o n s i s t h e n s o l v e d f o r t h e u n k n o w n N v o r t e x d e n s i t i e s y. a n d t h e s t r e a m f u n c t i o n tf>, . When t h i s t e c h n i q u e i s e x t e n d e d t o m u l t i - c o m p o n e n t a i r f o i l s e c t i o n s t h e p o i n t d i s t r i b u t i o n , g i v e n b y e q u a t i o n (11 ) i s f i r s t s c a l e d t o t h e c h o r d o f e a c h i n d i v i d u a l c o m p o n e n t b e f o r e b e i n g a p p l i e d . I t i s t h e n n e c e s s a r y t o move e a c h c o m p o n e n t t o i t s c o r r e c t l o c a t i o n . T h i s i s d o n e b y s p e c i f y i n g t h e a m o u n t s by w h i c h t h e l e a d i n g e d g e o f t h e c o m p o n e n t i s t r a n s l a t e d a n d t h e a n g l e t h r o u g h w h i c h t h e c o m p o n e n t i s r o t a t e d . W i t h t h e ;. g e o m e t r y t h u s d e f i n e d , o n e c a n c a l c u l a t e t h e K . . a n d R. f r o m e q u a t i o n s (9 ) a n d l j 1 • ( 1 3 ) . The m u l t i - c o m p o n e n t c a s e g i v e s r i s e t o a d i f f e r e n t s t r e a m f u n c t i o n f o r e a c h c o m p o n e n t a n d e a c h c o m p o n e n t h a s i t s own K u t t a C o n d i t i o n . A t w o - c o m p o n e n t a i r f o i l w i t h N e l e m e n t s o n e a c h c o m -p o n e n t g i v e s r i s e t o a s y s t e m o f e q u a t i o n s w h i c h c a n b e w r i t t e n : i = l K l , l * * ' K 1 , 2 N 1 0 Y l R l i = N K N , 1 K N , 2 N 1 0 Y N * N i = N + l K N + 1 , 1 • K N + 1 , 2 N ° 1 Y N + 1 = * N + 1 i = 2 N K 2 N , 1 ' * ' K 2 N , 2 N ° i Y 2 N R 2 N K u t t a c o n d i t i o n , c o m p o n e n t 1 * 1 l = 2 N + 2 K u t t a c o n d i t i o n , c o m p o n e n t 2 * 2 t p 2 (15 ) - 13 -The U . B . C . c o m p u t e r s u b r o u t i n e FSLE h a s b e e n u s e d h e r e t o s o l v e t h e s y s t e m o f e q u a t i o n s (14 ) a n d ( 1 5 ) . The s o l u t i o n s a r e t h e d i m e n s i o n l e s s s u r f a c e v e l o c i t i e s a t t h e c o n t r o l p o i n t s a n d t h e d i m e n s i o n l e s s s t r e a m f u n c t i o n s o f e a c h c o m p o n e n t . . The p r e s s u r e d i s t r i b u t i o n , l i f t c o e f f i c i e n t a n d t h e l e a d i n g - e d g e p i t c h i n g moment c a n be c a l c u l a t e d f r o m t h e v e l o c i t i e s as f o l l o w s 2 C = 1 - Y • l N C l = . \ C P . A X 1 = 1 l N C = - £ C n ( x . Ax + y . Ay) (16 ) % - i = l P i 1 1 a n d s u m m a t i o n s a r e p e r f o r m e d c o u n t e r ' c l o c k w i s e a r o u n d t h e p o l y -g o n a l c o n t o u r s . The t e c h n i q u e d e s c r i b e d so f a r makes t h e s i m p l i f y i n g assump-t i o n s o f s t r a i g h t l i n e e l e m e n t s a n d c o n s t a n t v o r t e x d e n s i t y o n e a c h e l e m e n t , w h i c h i s r e f e r r e d t o h e r e as t h e BASIC m e t h o d . K e n n e d y (6) h o w e v e r , h a s s t u d i e d t h e e f f e c t s o f i n c l u d i n g t h e h i g h e r o r d e r t e r m s due t o s u r f a c e c u r v a t u r e and a l i n e a r l y v a r y i n g v o r t e x d e n s i t y o n e a c h e l e m e n t , t h e r e s u l t s a r e shown i n F i g . ( 7 ) . T h e s e r e s u l t s show t h a t t h e i n c l u s i o n o f e l e m e n t c u r v a t u r e r a i s e s t h e v e l o c i t i e s w h i l e i n c l u d i n g t h e l i n e a r v o r t e x d e n s i t y d e c r e a s e s t h e v e l o c i t i e s . A l s o f r o m a p p e n d i x 1 , o n e c a n n o t i c e t h a t t h e t w o t e r m s w h i c h i n t r o d u c e t h e 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 s u r f a c e c u r v a t u r e i n t o t h e i n f l u e n c e c o e f f i c i e n t s a r e o f t h e same m a g n i t u d e b u t o f o p p o s i t e s i g n s , t h u s t h e i r e f f e c t s t e n d t o c a n c e l when t h e y a r e c o m b i n e d . I t i s t h e r e f o r e r e c o m m e n d e d t h a t o n l y t h e BASIC m e t h o d w i t h s t r a i g h t l i n e e l e m e n t s a n d c o n s t a n t v o r t e x - 14 -d e n s i t y b e u s e d . K e n n e d y ' s i n v e s t i g a t i o n ( 6 ) , h a s shown t h a t t h e BASIC m e t h o d w i t h " F u l l L o a d " K u t t a C o n d i t i o n g i v e s a c c u r a t e r e s u l t s f o r m o s t a i r f o i l s . An e x a m p l e o f a s e c t i o n w i t h a f a i r l y s h a r p p e a k i n t h e v e l o c i t y d i s t r i b u t i o n i s g i v e n i n F i g . ( 8 ) . T h i s i s a J o u k o w s k i a i r f o i l w i t h a c u s p e d t r a i l i n g e d g e f o r w h i c h 4 0 e l e m e n t s w e r e u s e d t o d e s c r i b e t h e s e c t i o n , a n d t h e r e i s e x c e l l e n t a g r e e m e n t w i t h t h e e x a c t s o l u t i o n . I n t h e c u r r e n t i n v e s t i g a t i o n t h e s i n g l e - c o m p o n e n t a i r f o i l s e c t i o n s , NACA-0015 a n d CLARK-Y 14% h a v e b e e n r e p r e s e n t e d b y 4 0 e l e m e n t s e a c h w h i l e . t h e t w o - c o m p o n e n t a i r f o i l , NACA-23012 w i t h 25% s l o t t e d f l a p d e f l e c t e d 20 d e g r e e s , h a s b e e n r e p r e s e n t e d b y 7 0 e l e m e n t s (40 e l e m e n t s f o r t h e m a i n a i r f o i l a n d 30 e l e m e n t s f o r t h e f l a p ) . The w i n d t u n n e l w i t h s o l i d w a l l s i s r e p r e s e n t e d b y 128 e l e m e n t s w h i l e t h e w i n d t u n n e l w i t h s l o t t e d w a l l s i s r e p r e s e n t e d b y 64 e l e m e n t s f o r t h e s o l i d p a r t s o f t h e w a l l a n d 8 e l e m e n t s f o r e a c h o f t h e 16 s l a t s (8 s l a t s f o r e a c h w a l l , w i t h c / C = . 1 5 ) . The t o t a l e x t e n t o f t h e w i n d t u n n e l w a l l h a s b e e n f o u r t i m e s t h e t e s t a i r f o i l c h o r d , C. - 15 -CHAPTER T V - THE FLOW I N THE WINDTUNNEL PLENUM I V - 1 - I n t r o d u c t i o n The e x t e n s i o n o f t h e c u r r e n t a n a l y s i s i s t o make t h e g e o m e t r y o f t h e f l o w r e p r e s e n t a t i o n m o r e l i k e t h a t w h i c h a c t u a l l y o c c u r s e x p e r i m e n t a l l y i n t h e t e s t - s e c t i o n , w i t h t h e p l e n u m s u r r o u n d i n g t h e s l o t t e d w a l l s . F i g u r e (9 ) c o m p a r e s t h e f l o w r e p r e s e n t a t i o n o f t h e c u r r e n t a n a l y s i s w i t h t h e p h y s i c a l f l o w w h i c h a c t u a l l y o c c u r s i n t h e t e s t - s e c t i o n . As i t i s shown i n F i g . (9a) t h a t t h e r e a r e t w o f r e e s t r e a m -l i n e s AB a n d CD, AB l e a v e s t h e t e s t - s e c t i o n u p s t r e a m o f t h e t e s t a i r f o i l a n d o p p o s i t e t o i t s n e g a t i v e p r e s s u r e s i d e , w h i l e CD e n t e r s t h e t e s t - s e c t i o n u p s t r e a m o f t h e t e s t a i r f o i l a n d o p p o s i t e t o i t s p o s i t i v e p r e s s u r e s i d e . I n t h i s s e c t i o n t h e f l o w s i n t h e u p p e r a n d l o w e r p l e n u m s a r e r e p r e s e n t e d b y c o n s i d e r i n g t h e s e t w o f r e e s t r e a m l i n e s i n t h e c u r r e n t a n a l y s i s . I n o r d e r t o d e f i n e t h e g e o m e t r y o f t h e f r e e s t r e a m l i n e s t h e c u r r e n t t h e o r y s h o u l d be d e v e l o p e d t o p r o v i d e t h e i n d u c e d t a n g e n t i a l a n d n o r m a l v e l o c i t i e s o f t h e f l o w a t a n y p o i n t i n t h e f i e l d , s i n c e t h e y a r e n e c e s s a r y t e r m s f o r t r a c k i n g a n y f r e e s t r e a m l i n e . I V - 2 - The I n d u c e d T a n g e n t i a l a n d N o r m a l V e l o c i t i e s Due t o V o r t i c i t y The s t r e a m f u n c t i o n \1). . a t a c o n t r o l p o i n t P. whose c o -xj c 1 t h o r d i n a t e s w i t h r e s p e c t t o t h e j s t r a i g h t e l e m e n t a r e P^={X\,Y\), as i n F i g . (10a), d u e t o v o r t i c i t y h a v i n g a c o n s t a n t d e n s i t y y o v e r t h a t e l e m e n t i s g i v e n b y , * i j = 2 ? 1 / £ n r ( P i ' ^ d ^ (17) -A - 16 -A Then the v e l o c i t y components can be calculated from equation (17). Using the notation of F i g . (10) the v e l o c i t y components induced at the element Vi' due to a vortex element ' j ' , are Y- -I x!-A x'.+A u. . = -1 {Tan - 1 (-J-T-) - Tan 1 (—L—) } (18a) 1 *1 v i j =.4? £ n { {y'y+ ( x j ~ A ) 2 ) / ( y - 2 + + A)2>} (i8b) The dir e c t i o n s of u.. and v.. at P. are p a r a l l e l and ID i l i normal to the d i r e c t i o n of element 1 j ' , respectively. With respect to cartesian 'wind axes' X and Y (X i s the wind d i r e c t i o n ) , t h the j vortex element and the element ' i ' are i n c l i n e d at angles 0j and 0^ to the X-axis respectively. Thus, V m = u.. cos (9.-0.) + v.. sin(0.-0.) (19a) T i j 11 i l i l I D VN. . = V i j C O S ( 9 i ~ e j ) " u i j sin(e i-6 ;.) (19b) 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 tan-Y i 3 gential v e l o c i t y V_ . . i s -=— while the l o c a l normal v e l o c i t y V\T. . ^ J T n 2 J N n i s zero. The d e t a i l s of the c a l c u l a t i o n of equations (18) and (19) are provided i n Appendix 2. Therefore, the tangential and the normal v e l o c i t i e s induced at the control point on the element ' i ' due to a system of 1N' 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 - 17 -N V = I {u, . cos (9 -0 ) + v s i n (0.-0.)} + Uoocos 0. i j = l X J 1 D 1 J 1 1 3 (20a) N VN. = E { v i i cos(8.-8.) - u s i n (9.-9.)} -LU^sin 0. l j=l J J X J 1 1 i (20b) Hence, at any free point 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, N VT. = V X = ._ { U i j C O S V i i s i n 9 i > + U o o l 1-1 J J J N (21a) V M - V v = E {v, _. co j = 1,2, . . . , N N ± ~ VY = l± i v i j c o s 9 j + u i j s i n 9j> (21b) and the pressure c o e f f i c i e n t of the flow at t h i s point p^, i s i s given by, C P ± = 1 " (V^ + ) ( 2 2 ) i i IV-3 - Free Streamline Tracking The routine described i n the previous chapter must be carried out to calculate N vortex d e n s i t i e s , yj» f ° 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 vortex densities can then be inserted i n equations (21a) and (21b) 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 . Thus from a s t a r t i n g point (x^, y^), which i s supposed to be at the downstream edge of the s o l i d wall but, to avoid the - 18 -s i n g u l a r i t y t h e r e i s l o c a t e d i n s t e a d v e r y c l o s e t o t h e e d g e , t h e f l o w d i r e c t i o n 0^ i s c a l c u l a t e d , g i v e n b y , 9 1 = T A N _ 1 ^ N / ^ (23) w h e r e V m a n d V > T a r e t h e i n d u c e d t a n g e n t i a l a n d n o r m a l v e l o c i t i e s 1 1 a t t h e p o i n t (x^, Yj_) / r e s p e c t i v e l y , and" t h e y a r e g i v e n b y e q u a t i o n s ' ' ( 21a ) a n d ( 2 1 b ) . T h e n 0^ i s u s e d t o c a l c u l a t e t h e n e x t p o i n t ( x 2 , y 2 ) w h e r e x 2 = x 1 + Ax (24a ) y 2 = y 1 + Ax * t a n 6 1 ( 24b ) The f l o w d i r e c t i o n 0 2 i s t h e n c a l c u l a t e d t h e r e i n t h e same way a s 0^ .. The f l o w d i r e c t i o n s a r e a v e r a g e d t o g i v e 0, a n d t h e y -c o - o r d i n a t e i s c h a n g e d so t h a t t h e n e x t p o i n t i s now x 2 = x1 + Ax (25a ) y 2 = y 1 + Ax * t a n 0 ( 25b ) The f l o w d i r e c t i o n 0^ i s c a l c u l a t e d t h e r e , a n d so o n . H a v i n g d e f i n e d t h e g e o m e t r y o f t h e f r e e s t r e a m l i n e s o n e c a n p r o c e e d t o c a l c u l a t e t h e s t r e a m f u n c t i o n s o n e a c h , u s i n g e q u a t i o n ( 8 ) . T h e n e a c h s t r e a m l i n e i s d i v i d e d i n t o 8 e l e m e n t s , w h i c h a d d s 16 v o r t e x e l e m e n t s t o t h e p r e v i o u s N e l e m e n t s . T h u s t o i n c l u d e t h e e f f e c t o f t h e s e t w o f r e e s t r e a m l i n e s i n t h e p r e s e n t a n a l y s i s we s h o u l d o n c e m o r e s o l v e (N 1 + M) e q u a t i o n s , (N 1 = N+16) e l e m e n t s a n d M c o m p o n e n t s , p r e s c r i b e d b y e q u a t i o n s (8) a n d ( 1 0 ) , s i m u l t a n e o u s l y t o g e t N ' v o r t e x , d e n s i t i e s , , a n d M s t r e a m f u n c t i o n s . - 19 -CHAPTER V - RESULTS AND DISCUSSION The t h e o r e t i c a l c u r v e s p r e s e n t e d h e r e w e r e c a l c u l a t e d b y t h e m e t h o d o f R e f . ( 6 ) . F i g u r e s ( 1 2 ) , (13 ) a n d (14) show c o m p a r -i s o n s o f t h e o r e t i c a l p r e s s u r e d i s t r i b u t i o n s f o r t h e a i r f o i l s , o f c h o r d C, NACA-0015 a t a = 1 0 ° NACA-23012 a t . a = 8 w i t h 2 5 . 66% s l o t t e d f l a p a t 6 = 2 0 ° a n d CLARK-Y 14% a t a = 1 0 ° r e s p e c t i v e l y , i n f r e e a i r a n d b e t w e e n t h e s o l i d w a l l s o f a c o n v e n t i o n a l w i n d t u n n e l t e s t s e c t i o n o f h e i g h t h , w i t h C / h = . 8 . The t u n n e l l i f t c o -e f f i c i e n t s C T a r e 1 . 6 8 7 3 , 3 . 6 4 4 8 a n d 3 . 0 1 3 1 w h i c h a r e 3 7 . 6 6 % , T 3 0 . 3 6 % a n d 4 1 . 1 1 % h i g h e r t h a n t h e f r e e a i r l i f t c o e f f i c i e n t s C T L F o f 1 . 2 2 5 7 , 2 . 7 9 5 9 a n d 2 . 1 3 5 3 r e s p e c t i v e l y . The v e r y l a r g e e r r o r s i n t h e a i r f o i l l i f t c o e f f i c i e n t s d e v e l o p e d i n t h e w i n d t u n n e l w i t h s o l i d w a l l s shown b y t h e a b o v e r e s u l t s , p r o m p t e d a s e a r c h f o r a w a l l c o n f i g u r a t i o n t h a t w o u l d e x h i b i t t h e known c a n c e l l i n g e f f e c t s o f p a r t l y o p e n , p a r t l y c l o s e d w a l l s , a n d w h i c h w o u l d t h e r e f o r e p r o v i d e n e g l i g i b l e o r s m a l l e r r o r s . F r o m t h e a b o v e r e s u l t s i t c a n be s e e n t h a t n e a r l y a l l o f t h e i n c r e a s e d v a l u e s i n t h e t u n n e l l i f t c o e f f i c i e n t s a r e d u e t o t h e g r e a t e r s u c t i o n o v e r t h e t o p s u r f a c e o f t h e a i r f o i l . T h e r e f o r e , t h e f i r s t s t e p t o w a r d s t h e m o d i f i c a t i o n o f t h e c o n v e n t i o n a l w i n d -t u n n e l was t o c h a n g e t h e w a l l c o n f i g u r a t i o n o f t h e u p p e r s o l i d w a l l , a n d t h i s was d o n e h e r e b y u s i n g a u n i f o r m l y t r a n s v e r s e l y -s l o t t e d u p p e r w a l l . The p r e s s u r e d i s t r i b u t i o n f o r t h e a i r f o i l NACA-0015 a t a = l Q ? f o r s u c h a w a l l c o n f i g u r a t i o n o f 59% UOAR, w i t h s o l i d l o w e r w a l l s , a p p e a r s i n F i g . (15) a l o n g w i t h t h e c o r r e s p o n d i n g p r e s s u r e d i s t r i b u t i o n f o r f r e e a i r . T h i s f i g u r e shows some i m p r o v e m e n t i n t h e p r e s s u r e d i s t r i b u t i o n o p p o s i t e t o t h e u p p e r u n i f o r m l y s l o t t e d w a l l , b u t i t a l s o shows t h a t t h e r e a r e d i f f e r e n c e s i n t h e l o w e r s u r f a c e p r e s s u r e d i s t r i b u t i o n s , a n d i t s u g g e s t s t h a t t h e f l o w t h e r e i s e x p e r i e n c i n g l o w e r i n d u c e d v e l o c i t i e s i n t h e p r e s e n c e o f t h e l o w e r s o l i d w a l l t h a n i n f r e e a i r , a n d a p o s s i b l e s o l u t i o n t o t h i s p r o b l e m i s t o u s e a s l o t t e d l o w e r w a l l as w e l l . A c c o r d i n g l y , t h e w i n d t u n n e l w i t h d o u b l e u n i f o r m l y t r a n s v e r s e -l y s l o t t e d w a l l h a s b e e n i n v e s t i g a t e d h e r e a n d t h e r e s u l t s , as i n F i g . ( 1 6 ) , show t h a t t h e p r e s s u r e d i s t r i b u t i o n s f o r NACA-0015 a t a = 10 a n d o f C / h = . 8 , o p p o s i t e t o t h e l o w e r s l o t t e d w a l l i s c l o s e r t o t h e f r e e a i r v a l u e s t h a n b e f o r e , w h i l e t h e u p p e r s u r f a c e s u c t i o n s s t i l l t e n d t o be l o w n e a r t h e l e a d i n g edge a n d s l i g h t l y h i g h n e a r t h e t r a i l i n g e d g e , f o r t h e same UOAR 59%. A s o l u t i o n t o t h i s p r o b l e m w o u l d a p p e a r t o be t o u s e g r a d e d , w i d e r g a p s r e a r w a r d a n d n a r r o w o n e s f o r w a r d , r a t h e r t h a n u n i f o r m s p a c i n g b e t w e e n t h e w a l l s l a t s . A l l t h e a b o v e s u p p o r t t h e m o t i v a t i o n f o r u s i n g t h e d o u b l e s l o t t e d w a l l t e s t s e c t i o n w i t h n o n u n i f o r m s p a c i n g , F i g . ( 9 b ) . F o r t h e a c t u a l r e p r e s e n t a t i o n o f t h e f l o w i n t h i s d o u b l e s l o t t e d w a l l t e s t s e c t i o n t h e l i m i t i n g u p p e r a n d l o w e r s t r e a m l i n e s AB a n d CD, as shown i n F i g . ( 9 a ) , s h o u l d be c o n s i d e r e d . P h y s i c a l l y , t h e s t r e a m l i n e s . a r e s h e a r l a y e r s a n d t h e y c o u l d be i d e a l i z e d as f r e e s t r e a m l i n e s . a t c o n s t a n t p r e s s u r e . T h i s h a s n o t b e e n a t t e m p t e d , b u t , u s i n g t h e m e t h o d w h i c h h a s b e e n d e v e l o p e d h e r e f o r t r a c k i n g f r e e s t r e a m l i n e s , t h e i r g e o m e t r i e s c a n be d e s c r i b e d a n d c o n s e q u e n t l y t h e y c a n b e i n c l u d e d i n t h e r e p r e s e n t a t i o n . S e v e r a l a v e r a g e OAR h a v e b e e n e x a m i n e d t o l o o k f o r t h e m o s t s u i t a b l e w a l l c o n f i g u r a t i o n w h i c h w o u l d d e v e l o p t h e l e a s t e r r o r s - 2 1 -i n t h e t u n n e l l i f t a n d p i t c h i n g moment c o e f f i c i e n t s f o r t h e t h r e e d i f f e r e n t a i r f o i l s , m e n t i o n e d b e f o r e , a n d o f d i f f e r e n t s i z e s . I t h a s b e e n f o u n d t h a t AOAR=59% w i t h 2% i n c r e m e n t , t h e g a p s a r e l i n e a r l y i n c r e a s e d d o w n s t r e a m , i s t h e m o s t s u i t a b l e w a l l c o n f i g u r -a t i o n f o r t h i s w i d e r a n g e o f a i r f o i l s h a p e s a n d s i z e s . F i g u r e ( 1 7 ) shows c o m p a r i s o n s o f t h e e f f e c t o n t h e t h e o r e t -C i c a l l i f t c o e f f i c i e n t s o f t h e 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 a n d t e s t s e c t i o n w a l l c o n f i g u r a t i o n . The a i r f o i l s a n d t h e i r a n g l e s o f a t t a c k a r e t h o s e u s e d b e f o r e , a n d t h e t e s t s e c t i o n w a l l s a r e e i t h e r s o l i d o r d o u b l e u n i f o r m l y s l o t t e d w i t h AOAR=59%. I t c a n be s e e n t h a t w i t h t h e s o l i d w a l l s , f o r t h e c a l c u l a t e d r a n g e o f C ^ , t h e c o r r e c t i o n s c a n e x c e e d 4 0% o f t h e f r e e - a i r v a l u e s , w h i l e w i t h a s u i t a b l e AOAR o f t h e s l o t t e d t e s t s e c t i o n t h e p r e d i c t e d e r r o r s c a n b e k e p t w i t h i n 4% f o r t h e t h r e e a i r f o i l s , a n d f o r p.s? . 8 . A l s o F i g . ( 1 8 ) s h o w s c o m p a r i s o n s o f t h e e f f e c t o n t h e t h e o r e t i c a l C q u a r t e r c h o r d p i t c h i n g moment c o e f f i c i e n t s o f t h e r a t i o ^ f o r t h e same a i r f o i l s a n d t e s t s e c t i o n w a l l c o n f i g u r a t i o n s . I t c a n b e s e e n t h a t w i t h t h e s o l i d w a l l s t h e c o r r e c t i o n s c a n e x c e e d 3% o f t h e f r e e - a i r v a l u e s , w h i l e w i t h t h e s l o t t e d t e s t s e c t i o n o f AOAR= 59% t h e p r e d i c t e d e r r o r s c a n b e k e p t w i t h i n 1.8% f o r t h e t h r e e a i r f o i l s , a n d f o r ^ £ . 8 . F i g u r e s ( 1 9 ) , ( 2 0 ) a n d (21 ) show c o m p a r i s o n s o f t h e t h e o r e t i c a l p r e s s u r e d i s t r i b u t i o n s , f o r t h e a b o v e a i r f o i l s a n d a n g l e s o f a t t a c k , i n f r e e a i r a n d i n t h e d o u b l e C s l o t t e d t e s t s e c t i o n s o f AOAR=59%, a g a i n w i t h ^ = . 8 . I t c a n b e s e e n t h a t t h e r e i s q u i t e g o o d a g r e e m e n t b e t w e e n t h e t w o d i s t r i b u -t i o n s f o r t h e t h r e e d i f f e r e n t a i r f o i l s . - 22 -Figure (22) shows comparisons of the e f f e c t on the theoret-C i c a l l i f t c o e f f i c i e n t of the r a t i o t - for NACA-0015 a i r f o i l at two h d i f f e r e n t angles of attack, a=10° and a=20°, and with three d i f f e r e n t wall conditions; s o l i d walls, double nonuniformly slotted of AOAR= 59% without and with the flow i n the plenums, which i s i d e a l i z e d here by free streamlines. The f i r s t two wall conditions exhibit the known blockage eff 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 t h i r d wall: condition shows the opposite. To explain t h i s trend one should study the l a s t two wall conditions separately. The above figure shows that the error i n l i f t co-e f f i c i e n t s increases as we include the flow i n the plenum i n the analysis for the same angle of attack and for a l l r a t i o s of ^. Also the analysis of streamline geometry shows that an increase i n angle of attack causes more curvature i n the upper free streamline which tends to reduce the e f f e c t of t h i s streamline i n increasing the negative pressure c o e f f i c i e n t at the a i r f o i l leading edge. So we have two opposite e f f e c t s . The above figure shows that the block-age e f f e c t i s weaker than the e f f e c t of the i n c l u s i o n of the flow i n the plenum, and because of that the error i n l i f t c o e f f i c i e n t s improves when the two e f f e c t s are combined. Also, F i g . (22) shows that the slotted walls with AOAR=59% s t i l l i s the most suitable wall configuration for a wide range of angles of attack. Figure (23) shows the geometry of the idealized flow i n the plenum chambers, the two free streamlines, of a test section with the new wall configuration of AOAR=59%, and the test a i r f o i l NACA-C 0015 at a=10°, It also shows the values of the stream function, ips, and the average pressure c o e f f i c i e n t C , for each free stream-P l i n e . - 23 -CHAPTER V I - CONCLUSIONS A t w o - d i m e n s i o n a l t h e o r y w h i c h p r e d i c t s a s a t i s f a c t o r i l y c o r r e c t i o n - f r e e w i n d t u n n e l t e s t c o n f i g u r a t i o n h a s b e e n d e v e l o p e d . The t h e o r y i s a n e x t e n s i o n o f t h e t w o - d i m e n s i o n a l p o t e n t i a l f l o w t h e o r y b a s e d o n t h e m e t h o d o f d i s t r i b u t e d s u r f a c e s i n g u l a r i t i e s . The e x t e n d e d t h e o r y t a k e s i n t o c o n s i d e r a t i o n n o t o n l y a w i d e r a n g e o f a i r f o i l s i z e s a n d s h a p e s , b u t a l s o t h e e f f e c t o n t h e a i r f o i l l o a d i n g s o f d i f f e r e n t w i n d t u n n e l w a l l c o n f i g u r a t i o n s . The a b o v e p o t e n t i a l f l o w t h e o r y was t h e n d e v e l o p e d t o a c c o u n t f o r t h e f l o w i n t h e p l e n u m c h a m b e r s , i n o r d e r t o r e p r e s e n t t h e f l o w i n t h e t e s t s e c t i o n as c l o s e as p o s s i b l e t o t h e p h y s i c a l s i t u a t i o n . The r e s u l t s o f t h e t h e o r e t i c a l s t u d y i n d i c a t e t h a t f o r a i r -f o i l t e s t i n g , a w i n d t u n n e l c o n s i s t i n g o f t w o s y m m e t r i c a l l y n o n -u n i f o r m t r a n s v e r s e l y - s l o t t e d w a l l s , w i t h t h e g a p s b e t w e e n t h e s l a t s l i n e a r l y i n c r e a s e d d o w n s t r e a m , a n d t h e s l a t s s y m m e t r i c a l a i r f o i l - s h a p e d a t z e r o i n c i d e n c e w i t h a v e r a g e o p e n a r e a r a t i o 59%, w i l l y i e l d u n c o r r e c t e d p r e s s u r e d i s t r i b u t i o n s , l i f t c o e f f i c i e n t s a n d p i t c h i n g moment c o e f f i c i e n t s w h i c h a r e w i t h i n a f e w p e r c e n t o f t h e f r e e a i r v a l u e s . The t h e o r y p r e d i c t s t h a t t h i s l o w c o r r e c t i o n w a l l c o n f i g u r a t i o n w i l l r e m a i n r e l a t i v e l y c o r r e c t i o n - f r e e f o r a w i d e r a n g e o f a i r f o i l s i z e s a n d shapes. . I n t h e p r e s e n t a n a l y s i s , t h e f l o w i n t h e p l e n u m c h a m b e r s i s i d e a l i z e d h e r e b y t w o b o u n d i n g s t r e a m l i n e s , b u t a c t u a l l y t h e r e a r e t w o s h e a r l a y e r s . E a c h o n e d i v i d e s t w o f l o w s , t h e h i g h - e n e r g y f l o w w h i c h e x i s t s i n t h e t e s t s e c t i o n , ar id t h e l o w - e n e r g y s t a g n a n t f l o w o u t s i d e i n t h e p l e n u m . So t h e s i n g l e f r e e - s t r e a m l i n e r e p r e s e n t -a t i o n f o r e a c h s h e a r l a y e r d o e s n o t m o d e l e x a c t l y t h e d i v i s i o n o f t h e t w o f l o w s o f d i f f e r e n t e n e r g y l e v e l a n d m o r e w o r k s h o u l d be d o n e t o r e p r e s e n t t h e s e t w o s h e a r l a y e r s m o r e a c c u r a t e l y . A l s o t h e p r e s e n t p o t e n t i a l f l o w a n a l y s i s s h o u l d b e d e v e l o p e d so t h a t i t w o u l d a c c o u n t f o r t h e v i s c o u s e f f e c t s t h a t o c c u r e x p e r -i m e n t a l l y . F i n a l l y t h e p e r f o r m a n c e o f t h e t e s t . s e c t i o n w i t h t h e new w a l l c o n f i g u r a t i o n n e e d s t o b e e x a m i n e d e x p e r i m e n t a l l y . - 25 -REFERENCES (1 ) i Pope a n d H a r p e r , "Low Speed W i n d t u n n e l T e s t i n g , " W i l e y , 1 9 6 6 . i i P a n k h u r s t a n d H o l d e r , " W i n d t u n n e l T e c h n i q u e , " P i t m a n , 1 9 5 2 . (2 ) P a r k i n s o n , G . V . a n d L i m , A . K . , "On The Use o f S l o t t e d W a l l s i n T w o - D i m e n s i o n a l T e s t i n g o f L o w - S p e e d A i r f o i l s , " CASI T r a n s . 4 , S e p t . 1 9 7 1 . (3 ) W i l l i a m s , C D . , "A New S l o t t e d ^ - W a l l M e t h o d f o r P r o d u c i n g Low B o u n d a r y C o r r e c t i o n s i n T w o - D i m e n s i o n a l A i r f o i l T e s t i n g , " P h . D . T h e s i s , O c t . 1 9 7 5 , U n i v e r s i t y o f B r i t i s h C o l u m b i a . (4 ) T h e o d o r s e n , T . , " T h e o r y o f W i n g S e c t i o n s o f A r b i t r a r y S h a p e , " NACA R e p o r t N o . 4 1 1 , 1 9 3 1 . (5 ) H e s s , J . L . a n d S m i t h , A . M . O . , " C a l c u l a t i o n o f P o t e n t i a l F l o w A b o u t A r b i t r a r y B o d i e s , " P r o g , i n A e r o . S c i . , 8 , P e r g a m o n , 1 9 6 6 . (6 ) K e n n e d y , J . L . , " T h e D e s i g n a n d A n a l y s i s o f A i r f o i l S e c t i o n s , " P h . D . T h e s i s , 1 9 7 7 , U n i v e r s i t y o f A l b e r t a . (7 ) M i l n e - T h o m s o n , L ' ^M. , " T h e o r e t i c a l A e r o d y n a m i c s , " D o v e r , 1 9 7 3 . (8 ) P r a n d t l , L . a n d T i e t j e n s , O . G . , " A p p l i e d H y d r o - a n d A e r o -d y n a m i c s , " D o v e r , 1 9 5 7 . (9 ) B h a t e l e y , T . C . a n d B r a d l e y , R . G . , "A S i m p l i f i e d M a t h e m a t i c a l M o d e l f o r t h e A n a l y s i s o f M u l t i E l e m e n t A i r f o i l s N e a r t h e S t a l l , " AGARD-CP-102 , 1 9 7 2 . (10 ) H e s s , J . L . , " H i g h e r O r d e r N u m e r i c a l S o l u t i o n s o f t h e I n t e g r a l E q u a t i o n f o r t h e T w o - d i m e n s i o n a l Neumann P r o b l e m , " C o m p u t e r M e t h o d s i n A p p l i e d M e c h a n i c s , V o l . 2 , p p . 1 - 1 5 , 1 9 7 3 . - 26 -APPENDIX .1 EVALUATION OF THE INTEGRAL I N EQUATION (5) The m o s t s t r a i g h t f o r w a r d m e t h o d 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 e q u a t i o n (5) i s t o d o so u s i n g a n u m e r i c a l i n t e g r a t i o n p r o c e d u r e . The s u r f a c e e l e m e n t i s c h o s e n t o be c u r v e d a n d t h e v e l o c i t y d i s t r i b u t i o n c a n v a r y o v e r t h e e l e m e n t . 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 o n o n e e l e m e n t o n t h e c o n t r o l p o i n t o f a n o t h e r e l e m e n t i s c a l c u l a t e d b y t h e i n t e g r a l 1 S2 I = 27 / Y ( S H n r ( C i , S) d S . S I A c o o r d i n a t e s y s t e m (E,, n) i s s e t u p 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 e l e m e n t as shown i n F i g . ( 1 1 ) . The i n f l u e n c e d c o n t r o l p o i n t i s l o c a t e d a t ( b , a ) i n t h i s c o o r d i n a t e s y s t e m . The s u r f a c e e l e m e n t i s d e f i n e d b y n=n . (£ ) . I n t h e n e i g h b o u r -h o o d o f t h e o r i g i n a p o w e r s e r i e s e x p a n s i o n i s u s e d , n = c C 2 + e + . . . ( 2 ) * The i n t e g r a l i s t a k e n o v e r t h e s u r f a c e d i s t a n c e a n d i t i s c o n v e n i e n t t o u s e : 2 k d s - i i j. d n - , * d T " { 1 + ~ ^ " } (3) . , d r -o n e x p a n d i n g ( 3 ) * a s a s e r i e s a b o u t 5 = 0 , ^ • = l + 2 c ? + 6 c e 5 + . . . (4 ) The v o r t e x d e n s i t y c a n 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 b y , Y ( S ) = Y<°» + y( 1 » S + Y( 2 ) S 2 + Y ( 3 ) S 3 + . . . | * A p p l y i n g ( 4 f t o (5)* , T I E ) = Y < 0 ) + YU ) 5 + Y( 2 » C 2 + ( | c2 Y < 1 ) + Y( 3 ) ) C 3 + * (6) The d i s t a n c e r ( C . ^ , S ) f r o m t h e c o n t r o l p o i n t t o t h e s u r f a c e i s , r ( c i f S ) = { ( a - n )2 + (b -02}H ( 7 ) * A t t h i s p o i n t i t i s n e c e s s a r y t o e m p l o y t h e t e c h n i q u e u s e d b y Hess ( 5 ) . I n s t e a d o f e x p a n d i n g t h i s t e r m d i r e c t l y a m o d i f i c a t i o n i s u s e d w h i c h p e r m i t s t h e b a s i c , f l a t e l e m e n t , t e r m t o a p p e a r as t h e f i r s t t e r m o f t h e s e r i e s . By w r i t i n g T O O * r fz = a + (b -K) (8 ) t h e d i s t a n c e t o t h e f l a t s u r f a c e n = 0 , t h e r e m a i n i n g t e r m s a r e e x p a n d e d a n d , 2 2 2 3 * r = i - 2ac E, - 2ae £ + . . . (9) S u b s t i t u t i n g ( 9 ) * i n t h e l o g a r i t h m t e r m a n d e x p a n d i n g a l l 2 b u t t h e r f t e r m a b o u t 5=0 y i e l d s , in r = \ In ( r f 2 ) - ^ § ? 2 - 5 3 + . . . ( 10 ) r f r f The i n t e g r a l ( l ) * c a n now be e v a l u a t e d t o as h i g h a d e g r e e o f a c c u r a c y a s i s d e s i r e d . I n t h i s c a s e o n l y t h e f i r s t f e w t e r m s a r e r e t a i n e d , h i g h e r o r d e r t e r m s b e i n g o f d i m i n i s h i n g i m p o r t a n c e . E q u a t i o n (1) t h e n b e c o m e s , I = ^ f u W * n r / . Y ' 1 ' i n r f2 K - 2 ^ W 5 2 + . . . . , « -A f t h e f i r s t t e r m i s t h e s t r a i g h t e l e m e n t , c o n s t a n t v e l o c i t y c a s e , s e c o n d t e r m 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 a n d t h e t h i r d i n t r o d u c e s t h e s u r f a c e c u r v a t u r e . - 28 -E a c h t e r m i n ( 1 1 ) * c a n b e 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 j Zrx r f 2 de; = (b+A) In r±2 - ( b - A ) £n r 2 2 - 4A —A + 2 a t a n " 1 ( 2 2 a A , 2 ) ( 1 2 ) * a + b -A Jin r f2 K & K = ^ £n ( ^ - ) - 2bA ^ 2ab t a n " 1 ( 2 a A 2 — 2 ) ( 1 3 ) a + b t - A * A .2 * / * - I s - d C = ( b 2 - a 2 ) t a n 1 ( 2 a 2 2 ) + 2aA / r , a +b -A •A f + a b Jin ( — ) ( 1 4 ) r l F o r t h e s t r a i g h t e l e m e n t ' j ' , w h i c h makes a n a n g l e 9_. w i t h t h e X - w i n d a x i s , t h e ' a ' a n d ' b ' a r e g i v e n b y , a = ( x . - x . ) s i n 8 . + ( y . - y . ) c o s 9 . ( 1 5 ) b = ( x . - x . ) c o s 9 . + ( y . - y . ) s i n 9 . I D D I D D w h e r e ( x . , y . ) a n d ( x . , y . ) a r e t h e c o - o r d i n a t e s o f t h e c o n t r o l 1 1 3 3 p o i n t s o n t h e e l e m e n t s ' i ' a n d ' j ' r e s p e c t i v e l y . I n c a s e o f s t r a i g h t e l e m e n t s 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 j t h e i n t e g r a l ( l ) - * , i s c a l l e d t h e BASIC i n f l u e n c e c o e f f i c i e n t s K ^ j a n d i t i s g i v e n b y e q u a t i o n ( 1 2 ) * . - 29 -For higher order terms involving surface curvature the constant C must be determined. In t h i s case t h i s was done by f i t t i n g parabolas through sets of three adjacent element end points. The curvature thus determined was assumed to be the a i r f o i l surface curvature at the centre of the three points. The curvature at the control points were then found by interpolation. The values at the elements adjacent to the t r a i l i n g edge were found by extrapolation. Having determined the curvature of the element the location of the control point can be calculated as t h i s point i s no longer on the straight l i n e j oining the element end points. In employing variations i n the s i n g u l a r i t y strength the term i s a n unknown and must be related to the y ^  • Various schemes are available to do t h i s and the technique used by Hess (10) i s followed here. The derivatives of the d i s t r i b u t i o n on the t h j element are determined by assuming a parabolic d i s t r i b u t i o n through the three successive values y ^ ^ j _ 1 , Y ^ j , Y ^ j + i * T h e l i n e a r vortex density term, unlike the other two terms, i s therefore comprised of terms that involve the vortex densities of adjacent elements. The application of the higher order methods to the solution involves the c a l c u l a t i o n of the extra terms (13)* and (14)*. The curvature terms are simply added to the influence c o e f f i c i e n t K^ ^ calculated for the basic case. The l i n e a r v e l o c i t y terms must be added to the c o e f f i c i e n t s K. . ,, K.., K. Although t h i s i s not d i f f i c u l t to do, the extra calculations involved do take considerable amounts of time to perform. - 30 APPENDIX 2 CALCULATION OF THE VELOCITY COMPONENT INDUCED AT A POINT I N THE F I E L D OF VORTEX D I S T R I B U T I O N : T h e s t r e a m f u n c t i o n a t a p o i n t P^ d u e t o a s t r a i g h t v o r t e x e l e m e n t ' j ' , w i t h c o n s t a n t d e n s i t y y. d i s t r i b u t e d o v e r t h a t e l e m e n t , i s g i v e n b y - i w h e r e r i s t h e d i s t a n c e f r o m t h e p o i n t P^ t o t h e p o i n t Q o n t h e s u r f a c e , a s i n F i g . ( 1 0 a ) , a n d i t c a n b e w r i t t e n a s r . . = i(x\-V 2 + Ypk ( 2 ) * T h u s -A T h e v e l o c i t y c o m p o n e n t s i n d i r e c t i o n s p a r a l l e l a n d n o r m a l ^1 y i n { ( x ! - a 2 + y ! 2 } ^ ( 3 ) * t o t h e e l e m e n t ' j 1 r e s p e c t i v e l y , i n d u c e d a t P^ a r e : dty • . —y . A y ' . u ^ = - r d - = -*k f 5 ( 4 a ) i j 8y^ 2TT / , , r . 2 L , 2 v , , = - ^ - J - = 97 y : L 9 5" d S (4b) * * ^ 2 l T 7 ( x ' . - 5 ) 2 + y ! 2 -A 3 S ^ T h e r e f o r e YD f , " I ( X i " A ) , " I ( X i + A ) , u . . = TJ-*- { t a n . — - 4 t a n } ( 5 a ) i i 2TT y : v ! v~><=w 3 3 a n d v . . = £ n { y ' ^ + f x ' . - A ) 2 / y ' . 2 + ( x ' + A ) 2 } ( 5 b ? * i j 4TT j j j j \-> u> W h e r e x . a n d y . a r e t h e c o - o r d i n a t e s o f t h e p o i n t P. w i t h 3 3 1 r e s p e c t ( r e f e r r i n g ) t o a s y s t e m o f a x i s i t s o r i g i n i s t h e c o n t r o l - 31 -point on the element ' j ' which has the length 2A and makes an angle 0j with the wind axis, and they are given by k k x! = (x.-x.) cos0. + (y.-y.) sin8. (6a) 1 1 1 D i 1 1 k ic y! = (x.-x.) sin0. + (y.-y.) cos0. (6b) 1 l i l i l 1 J — - u To express the v e l o c i t i e s tangential and normal to the i control surface we consider the following: From F i g . (10b) and from the vector analysis of the element ' j ' , the v e l o c i t y vector V.. can be written as V. . = u. . t. + v. . n . (7) i l i l 1 i l 1 also thus V. . = X. . i + Y. . j (8** i l i l i l x. . = v. . i (9) :• i i i i . ••. then from equations (?)•** and (9)**, we get X. . = u. . cos0 . - v. . sin9 . (10) i l i l 1 i l 1 s i m i l a r l y ~ k k Y. . = u. . sin0. + v. . cos0 . (11) i l i l 1 i l _1 Also the v e l o c i t y vector ^, r e f e r r i n g to F i g . (10b) and from the vector analysis of the element ' i ' , can be written as V.. = V m t. + V„ n. (12)* l ] T. . l N. . l A ' 11 11 thus — • kk v m - v. . t i ,(13) T . . l ] l 11 J then from equations (8)** and (13)**, we get V m = X . . cos0. + Y.. sin0. (14** T i j i l i i l i s i m i l a r l y VN.. = V i j H i ( 1 5 " 11 J = Y.. cos0. - X.. sin0. - 32 -s u b s t i t u t e (10)** a n d (11)** i n t o e q u a t i o n s (14)** and (15)**, h e n c e V . = u i j eos(e.-e.) + v ± j s i n ( 6 i - e j ) ( 1 6 a ) ** V „ = - u . . s i n(6.-0 . ) + v . . c o s (6.-0.) - f i f t h ) j ij i 1 ID I D U b D ; V m a n d V „ a r e t h e t a n g e n t i a l a n d t h e n o r m a l v e l o c i t i e s , ID ID r e s p e c t i v e l y , i n d u c e d a t e l e m e n t ' i ' , t a n g e n t i a l a n d n o r m a l t o i t , d u e t o a s t r a i g h t v o r t e x e l e m e n t ' j ' w i t h c o n s t a n t d e n s i t y * * - 33 -F I G U R E 1 - COMPARISON OF A I R F O I L P R E S S U R E C O E F F I C I E N T S : THEORY Ref. (3) - 34 -VORTEX R E P R E S E N T A T I O N OF TWO COMPONENT A I R F M L - 35 -GURE 3- NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS FIGURE 4- STREAMLINES CONTOURS AROUND AN AIRFOIL Ref. (8) - 37 -LOCATION OF TRAILING CONTROL POINT - 38 -F I G U R E 6- L O C A T I O N OF ELEMENTS ON A I R F O I L S U R F A C E - 39 -3.0 2.5 2.0 1.5 1.0 Karman-Trefftz Aerofoil a = 0° Element Shape O Straight Line • Curved V Straight Line A Curved — « Exact Analytic Vortex Density Constant Constant Linear Linear F I G U R E 7- COMPARISON OF HIGHER ORDER METHODS Ref. (6) - 40 -0 0.2 " 0.4 0.6 0.8 1.0 x / c FIGURE 8- COMPARISON OF A I R F O I L V E L O C I T Y D I S T R I B U T I O N S Ref. (6) -'41 -(a) THEORY f ( (' f ( ( ( / / / / • / / / / / // / / i / f " y / ' v A ; —} > ) )—7—7—7—J—* i } t i (b) EXPERIMENT FIGURE 9- AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL - 42 -- 43 -F I G U R E 1 1 - GEOMETRY FOR C A L C U L A T I O N OF HIGHER ORDER TERMS - 44 -N A C A - 0 0 1 5 a =10 C'/h=. 8 — - F R E E A I R C. =1 . 2 2 5 7 I" S O L I D WALLS C, = 1 . 6 8 7 3 i E L = 3 7 . 6 6 % C N c / C - 0 1 8 9 E M = - 2 . 9 2 % F I G U R E 1 2 - C O M P A R I S O N OF P R E S S U R E C O E F F I C I E N T S FOR N A C A - 0 0 1 5 A I R F O I L IN F R E E A I R AND BETWEEN S O L I D WALLS T E S T S E C T I O N - 45 -i N A C A - 2 3 0 1 2 WITH 2 5 . 6 6 % S L O T T E D F L A P . 2 A .6 . 8 X/ c FIGURE 13- COMPARISON OF PRESSURE C O E F F I C I E N T S FOR N A C A - 2 3 0 1 2 A I R F O I L IN FREE A I R AND BETWEEN S O L I D WALLS T E S T S E C T I O N - 46 C L A R K - Y 1 4 % a =10 C / h = . 8 - - - F R E E A I R S O L I D WALLS C L p = 2.135.3 C M c / 4 = - . 1 3 3 1 C L _ = 3 . 0 1 3 1 C M c / 4 = ~ . 2 0 3 5 E L = 4 1 . 1 1 % ' E M = - 3 . 3 % FIGURE 15- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE A I R AND BETWEEN S I N G L E UNIFORMLY S L O T T E D WALL T E S T S E C T I O N - 48 -FIGURE 16- COMPARISON OF PRESSURE COEFFICIENTS IN FREE A I R AND BETWEEN DOUBLE WALL T E S T S E C T I O N FOR NACA-0015 AIRFOIL UNIFORMLY S L O T T E D - 49 -NACA-0015 a =10 NACA-23012 WITH 25.66% SLOTTED FLAP a =8 . 5 =20 CLARK-Y 14% a =10 A0AR=59% c/C=.15 / / / / / / / '/ / * / / / SOLID WALLS DOUBLE SLOTTED WALL WITH FLOW IN THE PLENUM I . 2 A C/h GURE 17- COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN LIFT COEFFICIENTS - 50 -C/h FIGURE 18- COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS FIGURE 19- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE A I R AND BETWEEN DOUBLE NON-UNIFORMLY S L O T T E D WALL T E S T S E C T I O N WITH THE FLOW IN THE PLENUM - 52 -NACA-23012 WITH 25.66% SLOTTED FLAP a =8 6 =20 C/h =.8 AOAR =59% — FREE AIR DOUBLE SLOTTED WALL WITH FLOW IN THE PLENUM V2- 7 4 1 7 w- 3 0 3 1 C, =2.7417 C M_ / / 1=-.3282 Ei — 1 . j t / o EM=-X/C FIGURE 20- COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE A I R AND BETWEEN DOUBLE NON-UN IFORMLY S L O T T E D WALL T E S T S E C T I O N WITH THE FLOW IN THE PLENUM - 53 -CLARK-Y 14% lj a =10 C/h=.8 A0AR=59% | - F R E E AIR " ^ = 2 . 1 3 5 3 ^ = - . 1 3 3 1 I DOUBLE SLOTTED WALL . 2 .4 .6 • *8 t. x/c FIGURE 21- COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE A I R AND BETWEEN DOUBLE NON-UNIFORMLY S L O T T E D WALL T E S T S E C T I O N WITH THE FLOW IN THE PLENUM - 54 -N A C A - 0 0 1 5 A 0 A R = 5 9 % — a =10 --- a =20 / / / / / / / / /y S O L I D WALLS DOUBLE S L O T T E D WALL WITH FLOW IN THE PLENUM DOUBLE S L O T T E D WALL WITHOUT FLOW IN THE PLENUM .6' 1. C/h F I G U R E 2.2- COMPARISON OF THE E F F E C T OF TWO D I F F E R E N T ANGLES OF A T T A C K ON THE R E L A T I V E ERROR IN L I F T C O E F F I C I E N T S FOR N A C A - 0 0 1 5 A I R F O I L C/h = .8 c/C = .15 a = 10 c p = - . i ^ s = . 6 8 2 2 *JJUJ*dJ~LLU Q> o o 'o o> o>, o- o -H h-c e^YjBj :fTT7T C > O O 0 > O 0 > 0 > O 5 1 % 5 3 % 5 5 % 57%. 5 9 % 6 1 % 6 3 % 6 5% 6 7% h T T T T FIGURE 2 3 - GEOMETRY OF THE I D E A L I Z E D FLOW IN THE PLENUM CHAMBER FOR N A C A - 0 0 1 5 A I R F O I L 

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