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

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A NEW SLOTTED-WALL METHOD FOR PRODUCING LOW BOUNDARY CORRECTIONS IN TWO-DIMENSIONAL AIRFOIL TESTING by CHRISTOPHER DWIGHT WILLIAMS B.A.Sc., University of British Columbia 1967 M.A.Sc., University of British Columbia 1973 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical E n g i n e e r i n g We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1975 In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis for s cho lar ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or p u b l i c a t i o n of th is thesis for f i n a n c i a l gain sha l l not be allowed without my writ ten pe rm i ss ion . Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e t.]> Qdi9kf 1 9 1 6 SUPERVISOR: Dr. G.V. P a r k i n s o n i i AB3 TRACT T h i s t h e s i s deals with a new approaca to r e d u c i n g wiadtunnai w a l l c o r r e c t i o n s i n a i r f o i l t e s t i n g , by employing a t r a n s v e r s e l y - s l o t t e d w a l l o p p o s i t e the s u c t i o n s i d e of the a i r f o i l , and a s o l i d w a l l o p p o s i t e the pressure s i d e . The s o l i d elements of the s l o t t e d w a i l are symmetrical a i r f o i l s at z e r o i n c i d e n c e . T h i s geometry permits the flow to assuaa c l o s e l y the s t r e a m l i n e p a t t e r n f o r unconfined flow, without degrading the flow q u a l i t y through shear l a y e r mixing near the t e s t a i r f o i l . The theory uses the p o t e n t i a l flow s u r f a c e source-element method, with Kutta c o n d i t i o n s s a t i s f i e d on the t e s t a i r f o i l and the w a l l s l a t s . In experiments using a range of s i z e s of a i r f o i l s of t h r e e d i f f e r e n t p r o f i l e s , good agreement with the p r e d i c t i o n s of the theory has been o b t a i n e d . I t appears t h a t u n c o r r e c t e d l i f t c o e f f i c i e n t s and p r e s s u r e d i s t r i b u t i o n s , a c c u r a t e to w i t h i n one percent, can be o b t a i n e d f o r a wide range of a i r f o i l shapes, s i z e s , and l i f t c o e f f i c i e n t s , u s ing a s l o t t e d w a l l of open-area r a t i o between 60 and 70 p e r c e n t . i i i C e t t e these d _ c r i t une n o u v e l l e inethode en vue de diminuer l e s c o r r e c t i o n s da p a r o i s en s o u f f l e r i e aux e s s a i s des a i l e s . C ette methods einploie un taur a. f e n t e t r a n s v e r s a l e , en f a c e du f l a n c de d e p r e s s i o n de l ' a i l e d ' e s s a i ; et un mux s o l i d e , en f a c e du f l a n c de p r e s s i o n . Les elements s o l i d e s du' mur a, f e n t e s o n t de p r o f i l s aerodynamiques et syraetriques a 1'angle d ' i n c i d e n c e z e r o . C e t t e f i g u r e geotuetrique permet l'ecoulement d ' a i r , s u i v i e de pres par l e s l i g n e s de courant. de l'ecoulement l i b r e ; ce r e s u l t a t e s t obtenu sans diiainuer l a g u a l i t e de l'ecoulement par l e melange de l a couche de d i s c o n t i n u i t e , pres da l ' a i l e d ' e s s a i - La t h e o r i e demande d ' u t i l i s e r l'ecoulement a p o t e n t i e l des elements de l a source de s u r f a c e , s a t i s f a i s a n t l e s c o n d i t i o n s Kutta a l ' a i l e d ' e s s a i et aux a i l e s murales. Les e x p e r i e n c e s ont ete f a r t s en eaployant. des a i l e s de corde de t r b i s p r o f i l s d i f f e r e n t s . Les t h e o r i e s obtenuas r e s p e c t e n t assez bien l e s hypotheses p r a e t a b l i e s . Les c o e f f i c i e n t s p o rtances non v e r i f i e s , et l e s d i s t r i b u t i o n s de p r e s s i o n s sont e x a c t s a un pourcent pres; e t i l s peuvant e t r e obtenus pour une qrande v a r i e t e de p r o f i l s , de dimensions at de c o e f f i c i e n t s p o r t a n c e s ; en u t i l i s a n t un mur a f e n t e , de q u o t i e n t e n t r e l a p a r o i e t l a s u r f a c e t o t a l e , de s o i x a n t e a s o i x a n t e - d i x pourcent. I V ACKNOWLEDGEMENT T h i s r e s e a r c h was c a r r i e d out under the s u p e r v i s i o n of Dr. G. V. Park i n s o n , whose expert a d v i c e and guidance i s g r a t e f u l l y acknowledged. In the design and c o n s t r u c t i o n of the v i a d t u n n e l models and equipment, the work done and the advice given by the t e c h n i c i a n s of the Mechanical E n g i n e e r i n g Department was e x t r e a e l y v a l u a b l e . A l l the computing was done at the U.B.C. Computing C e n t e r . T h i s r e s e a r c h was supported by the U n i v e r s i t y of B r i t i s h Columbia and the Defence Research Board of Canada. Encouragement and support were pr o v i d e d by F.M.W. , who c o n t r i b u t e d more than her share i n our j o i n t e f f o r t . Table of Contents A b s t r a c t Resume Acknowledgement L i s t o f F i g u r e s L i s t o f P l a t e s L i s t o f T a b l e s N o t a t i o n I n t r o d u c t i o n Survey o f Windtunnel W a l l C o r r e c t i o n Methods §2.1 C o n v e n t i o n a l L i n e a r T h e o r i e s §2.2 R e s u l t s o f C o n v e n t i o n a l L i n e a r T h e o r i e s §2.3 Low C o r r e c t i o n T e s t C o n f i g u r a t i o n s A New S l o t t e d - W a l l Theory §3.1 A P h y s i c a l B a s i s f o r the New Theory §3.2 F o r m u l a t i o n o f an Ex a c t Numerical Theory §3.3 Other A i r f o i l - W a l l C o n f i g u r a t i o n s Examined Methods o f Numerical S o l u t i o n §4.1 Assembling the Equations §4.2 S o l v i n g the Equations R e s u l t s o f the New Theory Experiments t o V e r i f y the New Theory §6.1 T e s t s e c t i o n Design §6.2 A i r f o i l Models T e s t e d §6.3 T e s t Procedures v i 7. E x p e r i m e n t a l R e s u l t s 5 4 8. E x t e n s i o n s t o the New Theory 60 §8.1 P o t e n t i a l Flow C o n s i d e r a t i o n s of V i s c o u s E f f e c t s 62 §8.2 The Flow i n the Plenum: The Bounding Shear L a y e r 69 §8.3 Summary 76 9. C o n c l u s i o n s 77 Appendix 1. The I n t e g r a t i o n of a Three-Dimensional P o i n t 79 Source to a Two-Dimensional F l a t D i s t r i b u t e d Source Element Appendix 2. A Procedure f o r Block Computation o f 84 M a t r i c e s A, B, and C. Appendix 3. Two Methods o f S o l v i n g the Systems o f 91 Simultaneous L i n e a r A l g e b r a i c E q u a t i o n s Appendix 4. A S t r e a m l i n e T r a c k i n g A l g o r i t h m 96 Appendix 5. Design of the Two-Dimensional N o z z l e I n s e r t 99 Appendix 6. An A n a l y t i c R e p r e s e n t a t i o n of a L i f t i n g 10 3 V o r t e x Between a S o l i d , a T r a n s v e r s e l y - S l o t t e d and a Constant P r e s s u r e Boundary Appendix 7. Standard S o l i d Wall C o r r e c t i o n s 111 Appendix 8. A Reduced A i r f o i l C i r c u l a t i o n Determined 113 from the Measured L i f t Appendix 9. A Reduced A i r f o i l C i r c u l a t i o n Determined 116 by M o d i f y i n g the P r o f i l e Appendix 10. The Computer Program f o r the E x a c t 118 Numerical Theory Appendix 11. L i s t o f Equipment Used 152 F i g u r e s 153 Plates Tables References v i i i L i s t of F i g u r e s F i g u r e 2.1 F i g u r e 2.2 F i g u r e 2.3 F i g u r e 3.1 F i g u r e 3.2 F i g u r e 3.3 F i g u r e 3.4 F i g u r e 5.1 F i g u r e 5.2 F i g u r e 5.3 F i g u r e 6.1 P o r o s i t y parameter as a f u n c t i o n of w a l l open-area r a t i o f o r l o n g i t u d i n a l s l o t s [12] R a t i o o f a i r f o i l l i f t - c u r v e s l o p e s f o r l o n g i t u d i n a l l y s l o t t e d w a l l s : Experiment[12] V a r i a t i o n o f p r e s s u r e c o e f f i c i e n t a l o n g a s t r a i g h t boundary: Theory An a i r f o i l between t r a n s v e r s e l y - s l o t t e d upper and s o l i d lower w a l l s : Theory W a l l e f f e c t on a i r f o i l p r e s s u r e c o e f f i c i e n t : Theory Geometry and n o t a t i o n of two-dimensional source elements f o r Smith's method S u r f a c e v e l o c i t y v a r i a t i o n s f o r a two- d i m e n s i o n a l source element E f f e c t of a i r f o i l s i z e on r a t i o of l i f t c o e f f i c i e n t s : Theory Comparison o f 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 V a r i a t i o n of p r e s s u r e c o e f f i c i e n t a l o n g a s t r a i g h t boundary f o r a two-dimensional a i r f o i l w i t h zero l i f t - c o r r e c t i o n : Theory U.B.C. Mechanical 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 windtunnel Page 153 154 155 156 157 158 159 160 161 162 163 F i g u r e 6.2 V a r i a t i o n o f mean windspeed i n two-dimensional t e s t s e c t i o n i n s e r t on v e r t i c a l p i t o t s t a t i c t r a v e r s e 164 F i g u r e 6.3 V e l o c i t y p r o f i l e i n f l o o r boundary l a y e r i n two-dimensional t e s t s e c t i o n i n s e r t 165 F i g u r e 6.4 E f f e c t o f endp l a t e l o a d i n g s on l i f t , d rag and p i t c h i n g moment c o e f f i c i e n t s f o r two-dimensional a i r f o i l t e s t s 166 F i g u r e 6.5 C a l i b r a t i o n o f n o z z l e and t e s t s e c t i o n dynamic p r e s s u r e s 167 F i g u r e 6.6 E r r o r b a r s f o r measured a i r f o i l l i f t c o e f f i c i e n t s 168 F i g u r e 6.7 V a r i a t i o n of measured a i r f o i l l i f t c o e f f i c i e n t s on t h r e e c o n s e c u t i v e runs 169 F i g u r e 7.1 V a r i a t i o n o f a i r f o i l l i f t c o e f f i c i e n t w i t h s l o t t e d - w a l l open-area r a t i o : Experiment 170 F i g u r e 7.2 E f f e c t o f s l o t t e d - w a l l open-area r a t i o on r a t i o o f l i f t - c u r v e s l o p e s f o r C l a r k - Y a i r f o i l 171 F i g u r e 7.3 E f f e c t o f s l o t t e d - w a l l open-area r a t i o on r a t i o o f l i f t - c u r v e s l o p e s f o r NACA-0015 a i r f o i l 172 F i g u r e 7.4 E f f e c t o f a i r f o i l s i z e on l i f t - c u r v e s l o p e f o r NACA-0015 a i r f o i l : Experiment 173 F i g u r e 7.5 E f f e c t o f a i r f o i l s i z e on l i f t - c u r v e s l o p e f o r C l a r k - Y a i r f o i l : Experiment 174 X F i g u r e 7.6 F i g u r e 7.7 F i g u r e 7.8 F i g u r e 7.9 F i g u r e 8.1 F i g u r e 8.2 F i g u r e 8.3 F i g u r e 8.4 F i g u r e 8.5 F i g u r e 8.6 F i g u r e 8.7 E f f e c t o f s l o t t e d w a l l on a i r f o i l p r e s s u r e c o e f f i c i e n t : Experiment 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 : Experiment V a r i a t i o n o f a i r f o i l midchord p i t c h i n g moment c o e f f i c i e n t w i t h s l o t t e d - w a l l open-area r a t i o : Experiment V a r i a t i o n o f a i r f o i l drag c o e f f i c i e n t w i t h s l o t t e d - w a l l open-area r a t i o : Experiment E f f e c t o f reduced c i r c u l a t i o n on a i r f o i l p r e s s u r e c o e f f i c i e n t : Theory (Appendix 8) M o d i f i c a t i o n of a i r f o i l p r o f i l e t o reduce t h e o r e t i c a l c i r c u l a t i o n t o measured v a l u e : Theory (Appendix 9) E f f e c t o f m o d i f i e d p r o f i l e on a i r f o i l p r e s s u r e c o e f f i c i e n t An a i r f o i l between a s l o t t e d upper and a s o l i d lower w a l l w i t h a plenum chamber The shear l a y e r i n the plenum chamber su r r o u n d i n g the s l o t t e d w a l l E f f e c t o f d i f f e r e n t types o f w a l l b o u n d a r i e s on r a t i o o f l i f t c o e f f i c i e n t s : Theory V a r i a t i o n o f p r e s s u r e c o e f f i c i e n t a l o n g a s t r e a m l i n e i n plenum chamber: Theory 175 176 177 178 179 180 181 182 183 184 185 x i F i g u r e 8.8 E f f e c t on a i r f o i l l i f t c o e f f i c i e n t s o f assumed p r e s s u r e c o e f f i c i e n t s on a s t r e a m l i n e r e p r e s e n t i n g the plenum shear l a y e r : Theory 186 F i g u r e A l . l Geometry f o r i n t e g r a t i o n o f a p o i n t source 187 F i g u r e A5.1 The two-dimensional n o z z l e i n s e r t 188 F i g u r e A6.1 A l i f t i n g v o r t e x between a s o l i d , a s l o t t e d , and a c o n s t a n t p r e s s u r e boundary: Theory 189 F i g u r e A6.2 The image system f o r a l i f t i n g v o r t e x between a s o l i d and a c o n s t a n t p r e s s u r e . boundary: Theory 190 F i g u r e A10.1 N o t a t i o n f o r the computer program o f Appendix 10 191 X X X LIST OF PLATES P l a t e l a . P l a t e l b . P l a t e 2. P l a t e 3. P l a t e 4. P l a t e 5. P l a t e 6. Page The U.B.C. Me c h a n i c a l 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 windtunnel The o c t a g o n a l t e s t s e c t i o n i n the windtunnel The a i r f o i l - s h a p e d w a l l s l a t s The w a l l s l a t s i n the s i d e w a l l frame The 616mm NACA-0015 a i r f o i l The 354mm C l a r k - Y a i r f o i l f i t t e d w i t h e n d p l a t e s 194 The 616mm NACA-0015 a i r f o i l i n the t e s t s e c t i o n 195 192 192 193 193 194 x i x i L i s t o f T a b l e s Page Tabl e 1. A i r f o i l p r o f i l e c o o r d i n a t e s 196 Table 2. Free a i r a i r f o i l c o e f f i c i e n t s : Theory 197 Table 3. A i r f o i l and w a l l c o n f i g u r a t i o n s examined 198 t h e o r e t i c a l l y T able 4. A i r f o i l and endp l a t e l o a d i n g s 203 Tabl e 5. Windtunnel balance r e s u l t s - C l a r k - Y a i r f o i l s 204 Table 6. Windtunnel balance r e s u l t s - NACA-0015 219 a i r f o i l s T a b l e 7. Windtunnel balance r e s u l t s - Joukowsky 234 a i r f o i l T a b l e 8. Q u a n t i t i e s d e r i v e d from b a l a n c e r e s u l t s 238 Ta b l e 9. P r e s s u r e c o e f f i c i e n t s f o r NACA-0015 a i r f o i l 243 Table 10. P r e s s u r e c o e f f i c i e n t s f o r Joukowsky a i r f o i l 246 Table A10 Equ a t i o n s f o r the computer program of 249 Appendix 10 x i v N o t a t i o n A.., B.., C.. m a t r i c e s o f d i s t u r b a n c e v e l o c i t i e s . D i D i D i c a i r f o i l chord c. . element of the m a t r i x C.. J i 31 C_ = D/qc drag c o e f f i c i e n t C = L/qc l i f t c o e f f i c i e n t C = Mo/qc 2 midchord p i t c h i n g moment Mo C M  = Mc/qc 2 q u a r t e r c h o r d p i t c h i n g moment t ' * C n = measured p r e s s u r e c o e f f i c i e n t v q C_ average p r e s s u r e c o e f f i c i e n t on s t r e a m l i n e C = 1- (v t / u ) 2 c a l c u l a t e d p r e s s u r e c o e f f i c i e n t i d. element o f column v e c t o r o f approach f l o w 1 boundary c o n d i t i o n s ds_.,dXj,dy^. s u r f a c e element l e n g t h d i f f e r e n t i a l s H windtunnel t e s t s e c t i o n h e i g h t (or t o t a l head) K(s) p o r o s i t y parameter f o r l o n g i t u d i n a l l y s l o t t e d w a l l s m = l i f t - c u r v e s l o p e it outward s u r f a c e normal OAR t r a n s v e r s e l y - s l o t t e d w a l l open a r e a r a t i o p l o c a l s t a t i c p r e s s u r e P(s) p o r o s i t y parameter f o r porous o r p e r f o r a t e d w a l l s q = -ipU 2 dynamic p r e s s u r e r . .,r(PQ) d i s t a n c e between s u r f a c e elements Re Reynolds number X V s d i s t a n c e along the s u r f a c e As. l e n g t h o f s u r f a c e element 3 U magnitude of approach flow v e l o c i t y V.,V ,V. ,V ,V : v e l o c i t y induced by a s u r f a c e element l n . ' t . x . y . 2 * . _̂  x x 1 1 approach f l o w v e l o c i t y , magnitude U x.,y. C a r t e s i a n axes f i x e d t o the j - t h surface, element 3 3 X,Y • wind axes, w i t h X - a x i s i n the flow d i r e c t i o n x aerodynamic c e n t e r d i s t a n c e ac x 0 midchord d i s t a n c e a a i r f o i l i n c i d e n c e r c i r c u l a t i o n Y ,v , ;•• s u r f a c e v o r t e x element s t r e n g t h d e n s i t y ° y V y s u r f a c e source element s t r e n g t h d e n s i t y <j> d i s t u r b a n c e v e l o c i t y p o t e n t i a l 8 i n c l i n a t i o n o f s u r f a c e element w.r.t. X - a x i s 3 p f l u i d d e n s i t y dC T = ^-Mq midchord p i t c h i n g moment curve s l o p e \\) stream f u n c t i o n S u b s c r i p t s : F f r e e a i r v a l u e S s o l i d w a l l v a l u e L lower s u r f a c e t t a n g e n t i a l d i r e c t i o n n normal d i r e c t i o n T windtunnel v a l u e s streamwise d i r e c t i o n U upper s u r f a c e oo upstream- undisturbed- flow c o n d i t i o n — • - 1 __. I n t r o d u c t i o n . I n t h e s u b s o n i c w i n d t u n n e l t a s t i n . g o i a i r f o i l s e c t i o n s , t h e e x i s t i n g t h e o r y f o r c o r r e c t i o n s t o t h e measured d a t a f o r t h e e f f e c t s o f w i n d t u n n e l w a l l c o n s t r a i n t s i s s a t i s f a c t o r y f o r w i n d t u n n e l s w i t h s o l i d w a l l s i n w h i c h t h e t e s t a i r f o i l s a r e s m a l l r e l a t i v e t o t h e t e s t s e c t i o n c r o s s s e c t i o n s , and d e v e l o p r e l a t i v e l y s m a l l l i f t c o e f f i c i e n t s . However, c u r r e n t r e s e a r c h on h i g h - l i f t a i r f o i l s e c t i o n s r e q u i r e s t e s t i n g a t v e r y h i g h l i f t c o e f f i c i e n t s , and t h e use o f r e l a t i v e l y l a r g e m o d e ls t o g i v e s u f f i c i e n t l y h i g h R e y n o l d s numbers. Under t h e s e c o n d i t i o n s , t h e w a l l c o r r e c t i o n s i n w i n d t u n n e l s w i t h s o l i d w a l l s may become u n a c c e p t a b l y l a r g e , u n l e s s w i n d t u n n e l s w i t h v e r y l a r g e t e s t c r o s s s e c t i o n s a r e a v a i l a b l e - Such w i n d t u n n e l s a r e e x p e n s i v e t o b u i l d and o p e r a t e , s o a method o f m o d i f i c a t i o n o f e x i s t i n g s m a l l e r w i n d t u n n e l s , t h a t would r e d u c e o r e l i m i n a t e t h e s e l a r g e w a l l c o r r e c t i o n s , would be most d e s i r a b l e . S i n c e t h e m a j o r c o r r e c t i o n s t o measured d a t a i n w i n d t u n n e l s w i t h s o l i d w a l l s a r e o f o p p o s i t e s i g n f r o m t h o s e f o r w i n d t u n n e l s w i t h open j e t s , an o b v i o u s p o s s i b i l i t y t o e x p l o r e i s t h e u s e o f p a r t l y s o l i d - p a r t l y open w a l l s , t o p r o d u c e c a n c e l l i n g e f f e c t s . Two s u c h f o r m s o f w i n d t u n n e l w a l l have been c o n s i d e r e d i n . r e c e n t y e a r s f o r t h i s p u r p o s e , w a l l s w i t h narrow l o n g i t u d i n a l s l o t s , and w a l l s w i t h a p a t t e r n o f s m a l l h o l e s . T h e o r i e s have b e e n p r e s e n t e d f o r p r e d i c t i o n of t h e c o r r e s p o n d i n g w a l l c o r r e c t i o n s . U n f o r t u n a t e l y , e x p e r i m e n t s h a v e shown t h a t t h e e x i s t i n g 2 t h e o r y f o r w a l l s w i t h l o n g i t u d i n a l s l o t s i s u s e l e s s f o r t h e p r e s e n t p u r p o s e s , and the t h e o r y f o r porous w a l l s i s i m p r a c t i c a l t o a p p l y . An e m p i r i c a l p o r o s i t y f a c t o r i s needed, which depends on the w a l l geometry and on the t e s t model. In t h e p r e s e n t t h e s i s , a 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 i s d e v e l o p e d f o r a d i f f e r e n t p a r t l y s o l i d - p a r t l y open w a l l s ystem, and the r e s u l t s o f e x p e r i m e n t s d e s i g n e d t o t e s t t h e t h e o r y a r e p r e s e n t e d . 3 2-. Survey, of Windtunnel Wall Correc t ion Met hod s.. 2̂ 1_ Conventional Linear Theor ies . In windtunnel t e s t ing of two-dimensional a i r f o i l s e c t i o n s at subsonic speeds , the windtunnel wal l cons tra in t i n f l u e n c e s the measured forces , moments, and pressure d i s t r i b u t i o n s . The current theories f or the correc t ions to be appl i ed to the measured values to account for wal l e f fec t s are s a t i s f a c t o r y only when the tes t a i r f o i l s are small r e l a t i v e to the windtunnel c ros s sec t i on , and develop smal l l i f t c o e f f i c i e n t s . Current prac t i ce for such cases i s wel l summarized in a report by Garner et a l [ 1 ]. The e s s e n t i a l feature of current wal l c o r r e c t i o n theor ie s i s the s e l ec t ion of an appropriate system of source , vortex and doublet s i n g u l a r i t i e s , together with images of the s i n g u l a r i t i e s i n the windtunnel boundaries, such that the flow c o n d i t i o n s at the boundaries are s a t i s f i e d . The l i f t - p r o d u c i n g c h a r a c t e r i s t i c s of the a i r f o i l ( incidence, camber), the a i r f o i l t h i c k n e s s , and the a i r f o i l wake, are associated with d i s t r i b u t i o n s of v o r t i c e s , doublets and sources r e s p e c t i v e l y . The c h a r a c t e r i s t i c s of the windtunnel walls are then simulated by a s s o c i a t i n g an appropriate set of images with each s i n g u l a r i t y i n .the f i e l d . For example, a s o l i d wall boundary cond i t ion requires zero disturbance v e l o c i t y normal to the boundary. This c o n d i t i o n may be simulated by an i n f i n i t e set of images i n the windtunnel wal l s ; sources, and doublets or iented i n the streamwise d i r e c t i o n , have images of the same s i g n ; v o r t i c e s have images of a l t e r n a t i n g s ign . For further d e t a i l s , see A l l e n and V i n c e n t i 4 [ 2 ] , and G o l d s t e i n [ 3 ] . When the net e f f e c t of the systems o f s i n g u l a r i t i e s and images has been c a l c u l a t e d , the v e l o c i t i e s t h e r e b y i n d u c e d a t the a i r f o i l may be d e t e r m i n e d . F o r s i m p l i c i t y both i n i n t e r p r e t a t i o n and a p p l i c a t i o n , i t i s u s e f u l i f the net e f f e c t o f a l l the systems i s the d i r e c t sum of the i n d u c e d e f f e c t s o f each of the i n d i v i d u a l sys tems . In terms of the c o r r e s p o n d i n g f i e l d e q u a t i o n s , a s u p e r p o s i t i o n i s p o s s i b l e i f the e x a c t boundary c o n d i t i o n s may be l i n e a r i z e d . The l i n e a r a p p r o x i m a t i o n s i n t u r n are v a l i d o n l y f o r s m a l l i n d u c e d v e l o c i t i e s , which i m p l i e s a s m a l l , t h i n , s l i g h t l y cambered a i r f o i l at low i n c i d e n c e . An a l t e r n a t i v e to the images t e c h n i q u e , which y i e l d s s i m i l a r r e s u l t s , i s t h e conforraa l mapping t e c h n i q u e deve loped by Woods [4 ] , The problem . o f f i n d i n g v e l o c i t y , p o t e n t i a l s i n a complex domain bounded by an a i r f o i l and w i n d t u n n e l w a l l s i s t r a n s f o r m e d , b y . c o n f o r m a l mappings , t o an e q u i v a l e n t , b u t g e o m e t r i c a l l y s i m p l e r , boundary v a l u e prob lem- The problem i s thus reduced t o the d e t e r m i n a t i o n of,, an a n a l y t i c f u n c t i o n on a t r a n s f o r m e d domain whose r e a l a n d / o r i m a g i n a r y p a r t s a r e p r e s c r i b e d on the boundary . Woods' t e c h n i q u e r e s u l t s i n i n t e g r a l e q u a t i o n s f o r the mapping f u n c t i o n s , and u s u a l l y n u m e r i c a l methods are r e q u i r e d f o r t h e i r s o l u t i o n . A l i n e a r i z e d form of h i s t h e o r y agrees wi th the r e s u l t s t o be found i n G a r n e r e t a l [ 1 ] . F o r the case of a t h i n f l a t p l a t e between p a r a l l e l b o u n d a r i e s , the s o l u t i o n s o f Havelock [ 5 ] and Tomot ika [ 6 ] a r e a v a i l a b l e u s e s t h a t Havelock uses the method of conformal t r a n s f o r m a t i o n s . o f images w h i l e The r e s u l t s are 5 Tomotika s i m i l a r . In r e c e n t p u b l i c a t i o n s , de Jager and van de Vooren [ 7 ] , and de V r i e s and S c h i p h o l t [ 8 ] , use image methods f o r t h i n a i r f o i l s with hinged f l a p s , between s o l i d windtunnel w a l l s . For porous or p e r f o r a t e d windtunnel w a l l s , Baldwin e t a l [ 9 ] propose an " e q u i v a l e n t homogeneous w a l l boundary c o n d i t i o n " . dary c o n d i t i o n i s a combination of the f o r s o l i d w a l l s and f o r an open j e t . ndary c o n d i t i o n s ar e expressed i n terms of a o c i ty p o t e n t i a l which f o r i n c o m p r e s s i b l e ace's e q u a t i o n . T he g e n e r a l l i n e a r w a l l ay be w r i t t e n : «(»>!!•+ »!.>!!• cowsS - o. F o r a s o l i d boundary where t h e r e i s z e r o d i s t u r b a n c e f l o w normal to the boundary, e q u a t i o n (2.1) has the form !*• = 0 • . (2.2) 8n . For an open j e t boundary i t i s assumed t h a t the d i s t u r b a n c e from the t e s t a i r f o i l at the j e t boundary i s s m a l l . The l i n e a r i z e d c o n d i t i o n of constant p r e s s u r e can be i n t e r p r e t e d , using B e r n o u i l l i ' s e q u a t i o n , as r e q u i r i n g zero streamwise d i s t u r b a n c e v e l o c i t y . Hence f o r an open j e t boundary, c o n d i t i o n (2.1) becomes ' 6 3* „ ' F o r p o r o u s o r p e r f o r a t e d w a l l s t h e p r e s s u r e d r o p a c r o s s t h e w a l l due t o t h e c r o s s - f l o w i s assumed t o be p r o p o r t i o n a l t o t h e n o r m a l d i s t u r b a n c e v e l o c i t y a t t h e w a l l . The r e s u l t i n g l i n e a r r e l a t i o n b e t w e e n t h e c r o s s - f l o w a n d s t r e a m w i s e d i s t u r b a n c e v e l o c i t y c o m p o n e n t s r e q u i r e s A s i m i l a r e x p r e s s i o n , b u t w i t h a d i f f e r e n t v a l u e o f P { s ) , i s u s e d f o r a w a l l w h e r e t h e p o r o s i t y c o n s i s t s o f t r a n s v e r s e s l a t s . F o r an i n f i n i t e l e n g t h t e s t s e c t i o n w a l l o f c l o s e l y s p a c e d t r a n s v e r s e s l a t s , M aeder a n d Hood [ 1 0 ] , a n d Woods [ 4 ] d e d u c e a c o n s t a n t v a l u e f o r P (s) : P = t a n g ) . (2.5) •2d I n e q u a t i o n ( 2 . 5 ) , »a' i s t h e s l o t w i d t h a n d 'd» i s t h e s l a t s p a c i n g ; a/d i s t h e o p e n a r e a r a t i o (OAB). F o r w a l l s w i t h l o n g i t u d i n a l s l o t s , a p o t e n t i a l f l o w m o d e l o f t h e c r o s s - f l o w t h r o u g h t h e s l o t s r e q u i r e s t h e d i s t u r b a n c e p o t e n t i a l t o v a n i s h and t h e p r e s s u r e t o be c o n s t a n t , i n t h e s l o t s . Hence 7 | i + K ( s ) ^ =• 0 . • •'• • (2.6). 8s 3s8n ' Maeder and Wood [ 10 ] g i v e , f o r an i n f i n i t e l e n g t h t e s t s e c t i o n w a l l o f u n i f o r m l y s p a c e d l o n g i t u d i n a l s l o t s , t h e v a l u e K = £ l o g csc(|§] , ' (2.7) where a/d i s a g a i n t h e OAS. 8 2.Z.Z. E ^ s u l t s of Convantional L i n e a r Tneories_. In the t h e o r e t i c a l determination of homogeneous boundary c o n d i t i o n s , the d e t a i l s of the s l o t or hole geometry can be, used to c o n s t r u c t an exact boundary c o n d i t i o n by, f o r example, the a p p l i c a t i o n of Kutta c o n d i t i o n s to the edges of s l o t s or h o l e s . Then by examining the flow through the w a l l from a poin t many s l o t (hole) widths away from the wa l l s the flow d e t a i l s due to the w a l l geometry are not f e l t , but only some "averaged" f l o w f i e l d i s detected. The exact boundary c o n d i t i o n i s thus re p l a c e d by a l i n e a r i z e d , averaged boundary c o n d i t i o n . In app l y i n g t h i s boundary c o n d i t i o n , the w a l l i s regarded as being g e o m e t r i c a l l y homogeneous. The advantage i s that a s i n g l e averaged boundary c o n d i t i o n can be a p p l i e d uniformly over.the plane of the w a l l so that i t i s not necessary to have separate boundary c o n d i t i o n s a p p l i e d i n s l o t s (holes) and on s o l i d s e c t i o n s . This averaging e f f e c t e x p l a i n s why the w a l l boundary c o n d i t i o n f o r porous, p e r f o r a t e d and trans v e r s e s l o t t e d w a l l s are s i m i l a r . For d e t a i l s see Maeder and Wood [ 10 ] . The same "averaging" e f f e c t , i f a p p l i e d to l o n g i t u d i n a l s l o t s , leads to erroneous p r e d i c t i o n s . In f a c t , the l o n g i t u d i n a l s l o t s render,the flow three-dimensional by imposing spanwise v a r i a t i o n s on the b a s i c two-dimensional flow c o n d i t i o n s . On the assumption t h a t , the flow i s quasi-plane, that i s the spanwise v a r i a t i o n s are only a small p e r t u r b a t i o n of the bas i c two- dimensional f l o w , an averaged.boundary c o n d i t i o n f o r the b a s i c two-dimensional flow can be deduced. For d e t a i l s see Woods [ 4 ] . In the use of such g e o m e t r i c a l l y homogeneous l i n e a r 9 boundary c o n d i t i o n s , a l l d e t a i l s of s l o t or p e r f o r a t i o n geometry, are l o s t , i n p a r t i c u l a r , t h e i r o r i e n t a t i o n ( l o n g i t u d i n a l or t r a n s v e r s e ) . Only the e f f e c t s of bulk p r o p e r t i e s such as -the p o r o s i t y or-CAE are r e t a i n e d . Wood [11] shows, f o r example t h a t the OAR f o r l o n g i t u d i n a l s l o t s would need to be l e s s than 1%, to achieve a boundary c o n d i t i o n a p p r e c i a b l y d i f f e r e n t from.-;, the open. j e t case. In p r a c t i c e , at such a s m a l l OAR, r e a l f l u i d e f f e c t s would be important, so a p o t e n t i a l flow model f o r the c r o s s - f l o w would not be v a l i d . Moreover, Wood's a n a l y s i s of t h i s boundary c o n d i t i o n i n d i c a t e s that only c r o s s - f l o w v e l o c i t i e s of l e s s than 0.5% of the mean flow would be i n keeping with the arguments f o r the l i n e a r i z a t i o n of terms i n v o l v e d i n the d e r i v a t i o n of t h i s boundary c o n d i t i o n . I n v e s t i g a t i o n s by Parkinson and Lim [ 1 2 ] , and Mokry [ 1 3 ] have found that the " p o r o s i t y parameter" P(s) i s not simply an e m p i r i c a l f u n c t i o n of w a l l OAR, but must be determined e m p i r i c a l l y f o r each a i r f o i l under t e s t . The usual procedure i s to choose a value of P(s) to match l i f t or pressure data taken at a p a r t i c u l a r i n c i d e n c e and f o r a p a r t i c u l a r s i z e of a i r f o i l and then to t r y to use t h i s same value of P(s) t o c a l c u l a t e the w a l l e f f e c t at other i n c i d e n c e s and f o r other s i z e s of a i r f o i l . G e n e r a l l y , the r e s u l t s are that P(s) depends on the w a l l OAR and the p a r t i c u l a r a i r f o i l under t e s t , an impos s i b l e s i t u a t i o n f o r the p r a c t i c a l use of such l i n e a r porous w a l l boundary c o n d i t i o n s . Figure 2.1 from [ 1 2 ] , f o r example, shows tha t f o r two d i f f e r e n t a i r f o i l p r o f i l e s t e s t e d , there are two completely d i f f e r e n t v a r i a t i o n s of " p o r o s i t y parameter" with OAR, and ne i t h e r agrees with the t h e o r e t i c a l v a r i a t i o n of r e l a t i o n (2-5). 10 Other r e s u l t s by Parkinson.and Lim [ 1 2 ] , Parker [ 1 4 ] , and Tsen [ 1 5 ] , have shown that the theory f o r the l o n g i t u d i n a l w a l l s l o t parameter K (s ) i s not useable. Figure 2.2, from [ 1 2 ] , f o r example, shows that f o r four a i r f o i l s of d i f f e r e n t s i z e , of the same p r o f i l e , the t h e o r e t i c a l w a l l i n t e r f e r e n c e curves corresponding to the values of w a l l OAR t e s t e d , are c l o s e l y grouped as though a l l of the w a l l c o n f i g u r a t i o n s were e f f e c t i v e l y open. C a t h e r a l l [16] s t a t e s t h a t "the usefulness of the method i s l i m i t e d by the doubts about t h i s ( l i n e a r homogenous w a l l boundary) c o n d i t i o n " . This i s c l e a r l y the case f o r measurements on high l i f t devices where the p r e d i c t e d w a l l c o r r e c t i o n s are of the same order as the measured values themselves. The great disadvantage of more p h y s i c a l l y a p p r o p r i a t e n o n l i n e a r w a l l boundary c o n d i t i o n s i s that the mathematical s o l u t i o n s f o r most boundary problems depend on such boundary c o n d i t i o n s being l i n e a r as, f o r example, i n complex v a r i a b l e theory. Wood [11] has developed a n o n l i n e a r boundary c o n d i t i o n fo r a two-dimensional Helmholtz j e t i s s u i n g from l o n g i t u d i n a l s l o t s , where the " p o r o s i t y " i s a f u n c t i o n of the c r o s s - f l o w v e l o c i t y . His a n a l y s i s i s f o r a n o n l i f t i n g a i r f o i l ; the extension to the case of a l i f t i n g a i r f o i l does not appear to have been made. Sears [17] comments t h a t : "Even i n those (flow) regimes where the flow p e r t u r b a t i o n s due to tunnel boundaries can be estimated, there i s a b a s i c flaw i n the idea of " c o r r e c t i n g " 11 measured aerodynamic data, because such c o r r e c t i o n r e q u i r e s t h a t the e f f e c t s of such p e r t u r b a t i o n s be known. I f the f i e l d of extraneous v e l o c i t i e s i s other than a uniform change of i n c i d e n c e , than i n some of the most important t e c h n i c a l cases these e f f e c t s are not known and cannot .be c a l c u l a t e d " . Figure 2.2, from [ 1 2 ] , a l s o shows that the theory f o r s o l i d w a l l s gives e x c e l l e n t agreement with the data. Therefore i n the absence of improvements to the theory f o r s l o t t e d - or p e r f o r a t e d - w a l l c o r r e c t i o n s , i t seems a d v i s a b l e to c a r r y out low-speed two-dimensional a i r f o i l t e s t s , even f o r l a r g e models developing high l i f t c o e f f i c i e n t s , i n c o n v e n t i o n a l windtunnels with s o l i d w a l l s . 12 2̂ .3 Low C o r r e c t i o n T e s t C o n f i g u r a t i o n s . An a l t e r n a t i v e a p p r o a c h i s ' t o m o d i f y t h e w a l l s o f t h e w i n d t u n n e l t o p r o v i d e f l o w c o n d i t i o n s a s c l o s e a s p o s s i b l e t o a f r e e - a i r ( u n c o n f i n e d ) t e s t e n v i r o n m e n t . Thus t h e w a l l ' , c o r r e c t i o n s w o u l d a u t o m a t i c a l l y be s m a l l . One a p p r o a c h i s t h a t o f t h e " s e l f - c o r r e c t i n g " w i n d t u n n e l [ 1 7 ] w h e r e b y an a r r a y o f s e n s o r s ( l o c a t e d on a c o n v e n i e n t " c o n t r o l s u r f a c e " i n s i d e t h e t u n n e l b u t n o t i n t h e w a l l b o u n d a r y l a y e r s ) m e a s u r e , s a y , t h e f l o w s p e e d a n d i n c l i n a t i o n t h e r e . A c a l c u l a t i o n i s p e r f o r m e d t o d e t e r m i n e i f t h e s e m e a s u r e d v a l u e s a r e c o m p a t i b l e w i t h p r e v i o u s l y c a l c u l a t e d v a l u e s o f t h e same v a r i a b l e s f o r an i m a g i n a r y i n f i n i t e i n v i s c i d f l o w f i e l d a b o u t t h e t e s t a i r f o i l . I f n o t , a d j u s t m e n t s a r e made, i t e r a t i v e l y , u n t i l s u c h c o n d i t i o n s a r e met. T h i s c o u l d be a c h i e v e d t h r o u g h t h e u s e o f f l e x i b l e w a l l s , a n d / o r w a l l s e c t i o n s o f v a r i a b l e p o r o s i t y . O b v i o u s d i s a d v a n t a g e s a r e t h e c o s t o f " o n - l i n e " c o m p u t i n g f a c i l i t i e s p l u s t h e l a r g e number o f p r e s s u r e t r a n s d u c e r s r e q u i r e d t o e x t r a c t and p r o c e s s t h e f l o w m e a s u r e m e n t s . W h a t e v e r t y p e o f t e s t s e c t i o n i s c h o s e n , t h e " p o r o s i t y " v a r i a t i o n must p r o d u c e r e s u l t s l i k e t h o s e o f F i g u r e 2 . 3 , i n o r d e r t o s i m u l a t e c o r r e c t l y t h e f r e e - a i r f l o w f i e l d . How t h i s i s a c c o m p l i s h e d m e c h a n i c a l l y i s up t o t h e w i n d t u n n e l d e s i g n e r . 13 Is. h New S o l t t e d - S a l l T h e o r y . 3.1 A P h y s i c a l B a s i s f o r t h e New T h e o r y . One r e a s o n f o r 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 o t and p o r o u s - w a l l t h e o r i e s i s 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 o t s a nd 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 a s 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 t h e o r i e s . I n a d d i t i o n , t h e 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 c i n i t y o f t h e w a l l s . The a p p r o a c h h e r e ( s e e F i g u r e 3 . 1 ) , u s e s t r a n s v e r s e w a l l s l o t s , w i t h s y m m e t r i c a l a i r f o i l - s h a p e d s o l i d s l a t s . The f l o w i n c l i n a t i o n s n e a r t h e w a l l w i l l be s m a l l e v e n f o r a n e a r l y u n c o n f i n e d f l o w f i e l d . Hence a l l t h e w a l l s l a t s w i l l o p e r a t e w i t h i n t h e i r u n s t a l l e d i n c i d e n c e r a n g e , s o t h a t f l o w s n e a r t h e w a l l w i l l be f r e e o f s e p a r a t e d wakes. M o r e o v e r , o n l y t h e w a l l o p p o s i t e t h e n e g a t i v e p r e s s u r e s i d e o f t h e t e s t a i r f o i l i s s l o t t e d . One r e a s o n f o r t h i s c h o i c e c a n be s e e n f r o m F i g u r e 3.2 w h i c h c o m p a r e s 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 ( c a l c u l a t e d by t h e m e t h o d s o f s e c t i o n 3.2) b e t w e e n s o l i d w a l l s and i n f r e e a i r , f o r a 14% C l a r k - Y a i r f o i l . : The r a t i o o f a i r f o i l c h o r d ' c ' t o w i n d t u n n e l t e s t s e c t i o n s i z e * H , - i s l a r g e a t 0.72, a n d t h e i n c i d e n c e * a* i s e x t r e m e a t 20 d e g r e e s , t o c r e a t e a l a r g e w a l l e f f e c t . T h e s e p r e s s u r e d i s t r i b u t i o n s show t h a t a l m o s t a l l o f t h e w a l l e f f e c t i s t o i n c r e a s e t h e m a g n i t u d e o f t h e n e g a t i v e p r e s s u r e on t h e u p p e r s u r f a c e o f t h e a i r f o i l . The e f f e c t on t h e u n d e r s i d e p r e s s u r e i s s o s m a l l a s t o be n e g l i g i b l e , e v e n i n t h i s r a t h e r e x t r e m e c a s e . 14 Another reason f o r using only one s l o t t e d w a l l i s to s i m p l i f y the flow f i e l d o p p o s i t e the pressure s i d e of the t e s t a i r f o i l . A s l o t t e d lower w a l l would allow an i n f l o w of low energy a i r from w i t h i n a plenum chamber (surrounding the s l o t t e d wall) to enter the t e s t s e c t i o n upstream of the a i r f o i l . T h i s i n f l o w would c o n s i s t of a shear l a y e r and i t s a s s o c i a t e d t u r b u l e n t mixing and would degrade the q u a l i t y of the, main flow, i n the v i c i n i t y of the a i r f o i l lower s u r f a c e . There w i l l be a c o r r e s p o n d i n g outflow from the t e s t s e c t i o n back i n t o the plenum downstraam of the t e s t a i r f o i l . On the other hand, on the w a l l o p p o s i t e the n e g a t i v e pressure s i d e , upstream of the t e s t a i r f o i l , there w i l l be an outflow from the t e s t s e c t i o n i n t o the plenum. The shear l a y e r so formed and i t s a s s o c i a t e d t u r b u l e n t mixing w i l l be s h i e l d e d from the t e s t a i r f o i l by the presence of the a i r f o i l - s h a p e d w a l l s l a t s with t h e i r boundary c o n d i t i o n s impressed on the f l o w . I f t h i s shear l a y e r were i d e a l i z e d as a c o n s t a n t - p r e s s u r e f r e e s t r e a m l i n e , any i n c o r r e c t n e s s i n pressure or l o c a t i o n i n such a r e p r e s e n t a t i o n of t h i s s t r e a m l i n e should have only secondary e f f e c t s on the t e s t a i r f o i l . The plenum a i r w i l l e n t e r the t e s t s e c t i o n downstream of the t e s t a i r f o i l ; however, i t s e f f e c t on the t e s t a i r f o i l by e n t e r i n g t h e r e w i l l be much s m a l l e r than f o r any a i r e n t e r i n g the t e s t s e c t i o n upstream of the t e s t a i r f o i l . As with i r r o t a t i o n a 1 f o r low-speed most windtunnel w a l l c o r r e c t i o n t h e o r i e s , flow i s assumed, and, s i n c e the method i s designed, h i g h - l i f t t e s t i n g , an i n c o m p r e s s i b l e p o t e n t i a l 15 f l o w method can be used. The t e s t a i r f o i l (and i t s component f l a p s ) and the a i r f o i l - s h a p e d w a l l s l a t s a r e a l l t r e a t e d as l i f t i n g a i r f o i l s . Hence the f l o w s a t i s f i e s the u s u a l t a n g e n t - v e l o c i t y and t r a i l i n g - e d g e K u t t a c o n d i t i o n s . The f l o w ' p a s t t h e s o l i d w a l l s e c t i o n s s a t i s f i e s the t a n g e n t - v e l o c i t y boundary c o n d i t i o n . 16 3.2.2 F o r m u l a t i o n o f an E x a c t N u m e r i c a l Theory... The f o r m u l a t i o n h e r e i s a 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 , based, on t h e s u r f a c e s i n g u l a r i t y d i s t r i b u t i o n method o f A.H.0. S m i t h and h i s c o l l e a g u e s [ 1 8 ] . I n t h i s method, t h e s u r f a c e s o f t h e s o l i d w a l l s , t h e a i r f o i l - s h a p e d s l a t s i n t h e s l o t t e d w a l l , and t h e t e s t a i r f o i l w i t h i t s component f l a p s , a r e r e p r e s e n t e d by a d i s t r i b u t i o n o f s o u r c e and v o r t e x e l e m e n t s . A n o r m a l - v e l o c i t y b o u n d a r y c o n d i t i o n w i l l p r e s c r i b e e i t h e r z e r o n o r m a l v e l o c i t y , on s o l i d s u r f a c e s , o r n o n - z e r o n o r m a l v e l o c i t y , f o r s u c t i o n o r b l o w i n g t h e r e . S o u r c e e l e m e n t s a r e t h e r e f o r e d i s t r i b u t e d o v e r any s u r f a c e on w h i c h a n o r m a l - v e l o c i t y b o u n d a r y c o n d i t i o n i s s p e c i f i e d . V o r t e x e l e m e n t s a r e used t o s e t t h e n e t c i r c u l a t i o n a b o u t a c l o s e d l i f t i n g body. T h e r e f o r e v o r t e x e l e m e n t s a r e d i s t r i b u t e d o v e r any s u r f a c e on which a t a n g e n t - v e l o c i t y b o u n d a r y c o n d i t i o n i s s p e c i f i e d . Hence s o u r c e e l e m e n t s o n l y a r e d i s t r i b u t e d o v e r t h e s o l i d w a l l s e c t i o n s , w h i l e b o t h s o u r c e and v o r t e x e l e m e n t s a r e d i s t r i b u t e d o v e r t h e s u r f a c e s o f t h e a i r f o i l - s h a p e d w a l l s l a t s a n d t h e t e s t a i r f o i l and i t s f l a p s . The v e l o c i t i e s a t any p o i n t i n t h e f l o w f i e l d due t o a l l s u c h s o u r c e s and v o r t i c e s a r e c a l c u l a t e d d i r e c t l y . The u s u a l f l o w b o u n d a r y c o n d i t i o n of~ z e r o n o r m a l v e l o c i t y i s a p p l i e d a t a l l s o l i d s u r f a c e s . In a d d i t i o n , a f i n i t e - v e l o c i t y K u t t a c o n d i t i o n i s a p p l i e d a t t h e t r a i l i n g e d g e s o f t h e w a l l s l a t s and t e s t a i r f o i l , i n c l u d i n g f l a p s . • A g a i n <j> i s t h e d i s t u r b a n c e v e l o c i t y p o t e n t i a l , w hich 17 .es Laplace's equation, vanishes at i n f i n i t y , and. s a t i s f i e s the above boundary c o n d i t i o n s . The p o t e n t i a l at a point P due to a s i n g l e t h r e e - dimensional, point source s i n g u l a r i t y at a point Q i s where m i s the volume flow r a t e of f l u i d emitted .by the source and r (PQ) i s the d i s t a n c e between the p o i n t s P and Q. The t o t a l p o t e n t i a l due to a l l such sources d i s t r i b u t e d over a s i n g l e surface S i s 4>(P) 0 ( Q ) d S , ( 3 .2 ) s r(PQ) where o(Q) i s the source s t r e n g t h d e n s i t y , i n c l u d i n g the f a c t o r 1/4IT , of the source element at Q. Since the disturbance v e l o c i t y i s the gradient of the v e l o c i t y p o t e n t i a l , the n o r m a l - v e l o c i t y boundary c o n d i t i o n a t a s u r f a c e can be expressed as 8© •> -> 9n = ~ V^-n + F, (3.3) where n i s the outward surface normal,, and » the undisturbed oo flow at upstream i n f i n i t y . The f u n c t i o n F denotes the value the normal v e l o c i t y must take at the a i r f o i l s u r f a c e . ? i s zero f o r a s o l i d (impermeable) s u r f a c e , but non-zero f o r s u c t i o n or blowing there. 1 8 A n a l y s i s ( H ess and S m i t h [ 1 8 ] ) shows t h a t t h e n o r m a l v e l o c i t y a t a p o i n t P on a s u r f a c e S, due t o a s o u r c e s t r e n g t h d e n s i t y d i s t r i b u t i o n a (Q) on S, c o n s i s t s o f two p a r t s . The " l o c a l " c o n t r i b u t i o n i s 2fro"(p) due t o t h e s o u r c e e l e m e n t a ( P ) a t P. The " f a r f i e l d " c o n t r i b u t i o n i s f _d_ dn 1 ) | r ( P Q ) J a ( Q ) d S (3.4) due t o t h e summation- o f t h e e f f e c t s o f a l l o t h e r s o u r c e e l e m e n t s a(Q) a t p o i n t s Q on S. The r e s u l t i n g e x p r e s s i o n o f t h i s b o u n d a r y c o n d i t i o n 2 i r a ( P ) - 9h i r ( P Q ) J a ( Q ) d S = - V - n + F oo ( 3 . 5 ) p r o d u c e s an i n t e g r a l e q u a t i o n f o r t h e unknown s o u r c e s t r e n g t h d e n s i t y d i s t r i b u t i o n f u n c t i o n c ( Q ) . T h i s e q u a t i o n i s a F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d . E x i s t e n c e a n d u n i q u e n e s s t h e o r e m s f o r s u c h e q u a t i o n s a r e w e l l known. The s u r f a c e S may be d i s j o i n t , b u t t h e o u t w a r d n o r m a l v e c t o r must be a c o n t i n u o u s f u n c t i o n o f p o s i t i o n . F o r a d i s c u s s i o n o f d i f f i c u l t i e s a s s o c i a t e d w i t h f u n c t i o n a l s i n g u l a r i t i e s i n s u c h s o u r c e d i s t r i b u t i o n s a t e d g e s o r c o r n e r s , s u c h a s a t an a i r f o i l t r a i l i n g - e d g e o r an u n f a i r e d w i n g - b o d y j u n c t i o n , s e e C r a g g s e t a l [ 1 9 ] . I n p r a c t i c e , t h e s u r f a c e s o f t h e s o l i d w a l l , t h e a i r f o i l - s h a p e d s l a t s , a n d t h e t e s t a i r f o i l and f l a p s , a r e r e p l a c e d by p o l y g o n a l e l e m e n t s . The c o n t i n u o u s d i s t r i b u t i o n o f s o u r c e s 19 thereby becomes a succession of f i n i t e d.istr ibuted-source each of these f i n i t e elements.was f l a t and of. constant uniform s t r e n g t h . S u c c e s s f u l refinements of the method have used h i g h e r - order polynomial curves f i t t e d to s e c t i o n s of the body s u r f a c e with the source s t r e n g t h d e n s i t y varying i n a l i n e a r or p a r a b o l i c way along these curved elements. For examples see Henshaw [20,21], or Hess [ 2 2 ] . The higher-order element shapes are needed f o r i n t e r n a l flow c a l c u l a t i o n s such as i n d u c t s , but f o r e x t e r n a l flow problems the f l a t elements give accurate r e s u l t s provided a l a r g e enough number of elements, i s used and t h e i r d i s p o s i t i o n on the body shape i s chosen c a r e f u l l y . Each v e l o c i t y boundary c o n d i t i o n i s a p p l i e d at a s i n g l e " c o n t r o l p o i n t " on each element. For f l a t elements a convenient choice i s the center of each element.. .Thus the exact i n t e g r a l equation f o r a continuous d i s t r i b u t i o n f u n c t i o n may be reduced to a set of N simultaneous l i n e a r a l g e b r a i c equations whose N unknowns are the s t r e n g t h s of the f i n i t e s u r f a c e elements. The above approximations become exact i n the l i m i t as N -»- °°. The method i s described as n u m e r i c a l l y exact i n the sense t h a t any degree of. accuracy may be obtained. By d e f i n i n g the l i n e a r operator elements. In the o r i g i n a l method of Smith and h i s c o l l e a g u e s . J JS. 3 _9_ dn • - as., 3^ X J J J (3.6) 20 the boundary condi t ion (3.5) appl ied at t h e . i - t h c o n t r o l point becomes N I A . . j = l a . = - V •n . +.F . . j 0 0 i 1 (3.7) This ind ica tes that A_^is the normal v e l o c i t y induced at a c o n t r o l point ' i ' by a unit strength source element . located at another point * j ' . Hence the " loca l" normal v e l o c i t y , ^ ± ± ' ^ s 2TT for a i l i= 1, 2, 3 , . . . N. . For the purposes of th i s problem, the three-dimensional point source of equation (3.1) must be integrated in to a f l a t two-dimensional d i s t r ibuted- sburce element; for the d e t a i l s see Appendix 1.. The uni t s of a (Q) are therefore: volume flow ra te per unit arc length along the contour per u n i t length i n the spanwise d i r e c t i o n . With respect to Cartes ian axes x and y f ixed to the j - t h j j element (Figure 3 .3 ) , the v e l o c i t y components induced at a point ' i ' by a source element at point ' j * are V = log K y % ) 2 + -yj) = 2 log R 2 (3.8) and V 2 j t a n -1 J 2" - tan -1 ( As, 1 y. = 2£2, (3.9) where x. and y. are the distances from the j - t h to the i - t h J J element; the j - t h element has length As . The v e l o c i t y f i e l d s j about a s i n g l e s o u r c e element a r e shown i n F i g u r e 3.'4. The d i r e c t i o n s of V v and 7„ a t ' i 1 a r e p a r a l l e l and normal t o t h e x. y. 3 3 d i r e c t i o n of the element a t ' j ' , r e s p e c t i v e l y . The i n v e r s e t a n g e n t s i n (3.9) a r e t o be e v a l u a t e d i n t h e range f - T r / 2 / +Tr/2) . The two i n v e r s e , t a n g e n t s may. be combined by means of the t a n g e n t law i n t o the a l t e r n a t i v e e x p r e s s i o n V = 2 t a n 1 Y3 Y. As. _ J i . l x j + y ? - ( ^ j ) 2 J (3.10) where t h i s s i n g l e i n v e r s e t a n g e n t i s t o be e v a l u a t e d i n t h e range (-1T, + Tr) by f a k i n g i n t o account t h e i n d i v i d u a l s i g n s o f t h e numerator and denominator of i t s argument. When c a l c u l a t i n g f l o w q u a n t i t i e s a t o f f - s u r f a c e p o i n t s which a r e c l o s e r t o t h e o r i g i n o f t h e element t h a n As/2, the f i r s t e x p r e s s i o n must be used. With r e s p e c t t o C a r t e s i a n "wind a x e s " X and Y, (X i s i n t h e wind d i r e c t i o n ) , t h e g-th s o u r c e element i s i n c l i n e d a t an a n g l e 8. t o t h e X - a x i s . Thus, A.. = V cos 9.-8. - V s i n ( 8 . - 8 . (3.11 i i y . i j x . i t v ' and B > ; L = V cos (6.-9.) + V s i n (9.-8.) .- (3.12) 3 3 1 2 Y3 1 :1 a r e the normal and t a n g e n t i a l v e l o c i t i e s r e s p e c t i v e l y i n d u c e d a t element ' i ' due to a u n i t s t r e n g t h d e n s i t y s o u r c e element a t a p o i n t ' j ' . -The " l o c a l " normal v e l o c i t y JL^ i s 2TT ; the " l o c a l " 22 t a n g e n t i a l v e l o c i t y 8 ^ i s z e r o . The d i r e c t i o n s o f a n d a t ' i» a r e n o r m a l ( p o s i t i v e o u t w a r d ) a n d p a r a l l e l ( p o s i t i v e c l o c k w i s e ) r e s p e c t i v e l y t o t h e d i r e c t i o n o f t h e e l e m e n t a t ' i ' . Hence f o r t h e e x t e r i o r f l o w a b o u t a s i n g l e c l o s e d c o n t o u r , t h e s o u r c e and v o r t e x e l e m e n t s a r e l a b e l l e d f o r c o m p u t a t i o n s i n a c l o c k w i s e o r d e r a b o u t t h e c o n t o u r . I n o r d e r t o f i x t h e c i r c u l a t i o n a b o u t a l i f t i n g b o d y , t h e u s u a l e q u a l - v e l o c i t y K u t t a c o n d i t i o n i s a p p l i e d . T h i s i s a c c o m p l i s h e d by a d d i n g f i n i t e d i s t r i b u t e d v o r t e x e l e m e n t s t o t h e body s u r f a c e , a l l o f t h e same v o r t e x s t r e n g t h d e n s i t y . The K u t t a c o n d i t i o n t h e n i m p l i e s t h a t t h e t a n g e n t i a l v e l o c i t i e s e s t a b l i s h e d a t t h e c o n t r o l p o i n t s on t h e u p p e r and l o w e r s u r f a c e s , a d j a c e n t t o t h e t r a i l i n g e d g e , must be. e q u a l i n m a g n i t u d e , and b o t h . d i r e c t e d t o w a r d t h e t r a i l i n g e d g e . S i n c e t h e v e l o c i t y due t o a v o r t e x i s s i m p l y t h a t due t o a s o u r c e , b u t r o t a t e d 90 d e g r e e s , e x p r e s s i o n s f o r t h e v e l o c i t y c o m p o n e n t s f o r d i s t r i b u t e d - v o r t e x e l e m e n t s o f c i r c u l a t i o n s t r e n g t h d e n s i t y Y(Q) c a n be w r i t t e n , c o r r e s p o n d i n g f o t h o s e o f (3. 8, 3 . 9) . F o r a d i s t r i b u t e d v o r t e x e l e m e n t o f u n i t s t r e n g t h d e n s i t y a t p o i n t ' j * t h e c o r r e s p o n d i n g n o r m a l a n d t a n g e n t i a l v e l o c i t i e s i n d u c e d a t e l e m e n t ' i ' a r e f o u n d t o be -B-. a n d A . , r e s p e c t i v e l y . The number and s i z e o f v o r t e x e l e m e n t s i s a r b i t r a r y , s i n c e t h e y a l l h a v e t h e same s t r e n g t h d e n s i t y . I t i s c o n v e n i e n t t o u s e t h e same number o f v o r t e x a s s o u r c e e l e m e n t s and t o h a v e s o u r c e and v o r t e x e l e m e n t s l o c a t e d t o c o i n c i d e e x a c t l y . The v e l o c i t i e s A . . and B.. c o m p u t e d f o r t h e s o u r c e 2 3 elements are then immediately useable f o r the vortex elements. Hence the normal and t a n g e n t i a l v e l o c i t i e s induced at the c o n t r o l point on element ' i ' due t o a system of H c o i n c i d e n t source and vortex elements immersed i n an i n f i n i t e uniform approach flow U ( p a r a l l e l to the X - d i r e c t i o n ) , are N N V n « . I N i 3=1 J J k = l and N N V t . = J V B j i a j + JnAkiYk + U c o s V (3.14) x j = l J J k = l S i n c e a l l the vortex elements on a s i n g l e c l o s e d l i f t i n g body are of equal s t r e n g t h Y , the d e s c r i p t i o n of the flow f i e l d about an N-sided p o l y g o n a l body i s complete when the N+1 q u a n t i t i e s O i , o 2 , . . . ,o , and Y are known. For zer o F^, the normal-flow boundary c o n d i t i o n s at each of the N elements p r o v i d e s N equations V n = 0. (3.15) i while the . f i n i t e - v e l o c i t y Kutta c o n d i t i o n a t the two c o n t r o l p o i n t s adjacent to the t r a i l i n g edge p r o v i d e s the s i n g l e (N+1)st equation (3. 16) 24 For the c o n f i g u r a t i o n of a i r f o i l - s h a p e d w a l l s l a t s , s o l i d w a l l s and t e s t a i r f o i l plus f l a p s of Figure 3.1, with a t o t a l of N source elements and a l i f t i n g bodies, there are N source and M vortex strength d e n s i t i e s to be determined. The zero normal- v e l o c i t y c o n d i t i o n a p p l i e d on each element on the a i r f o i l , f l a p s , s o l i d w a l l s and w a l l s l a t s , y i e l d s the N equations N M R(k) I A d . - I y I B = Usin9., i = l , 2 , . . . N . (3.17) j = l 3 i 3 k = 1 K m = 1 nu i A Kutta c o n d i t i o n a p p l i e d to each of the M bodies ( a i r f o i l , f l a p s , w a l l s l a t s ) y i e l d s the M equations N M R(k) •V B3U + B J L >°l\lv*l (Amu + AmL > = "U (cos9 D +cos6 L ),(3.18) j-1 J r J r . k=l m=l r r r r r=l,2,...M. The s u b s c r i p t s U and L i n d i c a t e the c o n t r o l p o i n t s adjacent to the t r a i l i n g edge"on the upper and lower s u r f a c e s r e s p e c t i v e l y of the r - t h l i f t i n g body; R(k) i s the number of source (and vortex) elements on t h i s body. In summary there are: - a t o t a l of N source elements d i s t r i b u t e d over the t e s t a i r f o i l , i t s f l a p s , the a i r f o i l - s h a p e d w a l l s l a t s , and the s o l i d w a l l s e c t i o n s . - a t o t a l of M bodies r e q u i r i n g Kutta c o n d i t i o n s . M a t o t a l of J R (k) vortex elements d i s t r i b u t e d over the k=l l i f t i n g bodies; there are R (k) source elements and R (k) e q u a l - s t r e n g t h density vortex elements d i s t r i b u t e d over the k-th body. - N unknown source stre n g t h d e n s i t i e s a j 25 - M unknown vortex strength d e n s i t i e s - M+N equations i n the M+N unknowns a i , a 2,...a , y i / y 2 , YM" 2 6 3^3 Other A i r f o i l - W a l l C o n f i g u r a t i o n s Examined.- . ' Obvious s i m p l i f i c a t i o n s of the above g e n e r a l e q u a t i o n s (.3.17,3.18) are f o r a t e s t a i r f o i l (a) i n an unbounded stream (fr e e a i r ) , (b) i n the proxi m i t y of a s i n g l e s o l i d lower s u r f a c e (ground e f f e c t ) , (c) between two s o l i d w a l l s , and. (d) between a s o l i d lower boundary and an upper boundary c o n s i s t i n g of s i n g l e - s i d e d t r a n s v e r s e s l a t s with no Kutta c o n d i t i o n s a p p l i e d . In each of -the cases (a) - (d) a t o t a l of N source elements are d i s t r i b u t e d over the t e s t a i r f o i l , i t s f l a p s , the t r a n s v e r s e w a l l s l a t s , and the s o l i d w a l l s e c t i o n s . In a d d i t i o n t h e r e i s an unknown vortex s t r e n g t h d e n s i t y on the t e s t a i r f o i l and on each of i t s f l a p s . For example, f o r a s i n g l e t e s t a i r f o i l (no f l a p s ) , i n f r e e a i r , equations (3.17,3.18) reduce t o , r e s p e c t i v e l y : N N •I A..a. - y I B. . = Usin8., i = l , 2 , . . . N , (3.19) j = l 3 1 3 k=l K l 1 and J <V +V AJ + Y J 1 ( A k U + A k L ) = " U(cose u +cose L) . (3.20) j = l k=l For a s i n g l e t e s t a i r f o i l i n (b) , (c) , or (d) above, the equations (3. 17,3. 13) reduce to, r e s p e c t i v e l y : N NA I A .a. - y I B, . = UsinG , i=l,2,...N (3.21) j = l 3 3 k=l K 1 1 27 a n d N NA * ( B D U + B j L ) a J + \ ^ ( A k U + A k L } = " U ( c o s 0 u + c o s 0 L ) . (3.22) j = l J J k = l Here t h e r e a r e NA s o u r c e (and v o r t e x ) elements on t h e t e s t a i r f o i l and (N-NA) s o u r c e elements on the a p p r o p r i a t e s o l i d w a l l s e c t i o n s . The s u b s c r i p t s 0 and L i n d i c a t e t h e c o n t r o l p o i n t s a d j a c e n t to the t r a i l i n g edge on the upper and l o w e r s u r f a c e s r e s p e c t i v e l y of the s i n g l e t e s t a i r f o i l . The r e p r e s e n t a t i o n of an a i r f o i l i n ground e f f e c t i s d i f f e r e n t here than i n the method used by. o t h e r s , f o r example Jacob and S t e i n b a c h [ 2 3 ] and M a v r i p l i s [ 2 4 ] . T h e i r a p p r o a c h i s t o use a second a i r f o i l i n t h e "image" p o s i t i o n so t h a t t h e l o w e r s t r a i g h t s o l i d boundary i s a " r e f l e c t i o n p l a n e " . Hence t h e boundary c o n d i t i o n of z e r o f l o w normal t o t h i s s t r a i g h t s o l i d boundary i s r e p r e s e n t e d e x a c t l y . I f t h e r e a r e N s o u r c e and N v o r t e x elements d i s t r i b u t e d o v e r the t e s t a i r f o i l and i t s f l a p s , t h e number of e q u a t i o n s t o be s o l v e d i s e x a c t l y t w i c e t h e number t o be s o l v e d f o r the same a i r f o i l / f l a p c o n f i g u r a t i o n i n an unbounded stream ( f r e e a i r ) . Hence t h e r e i s a s a v i n g by t h e p r e s e n t method where the s o l i d l o w e r boundary can be r e p r e s e n t e d by l e s s t h a n N s o u r c e e l e m e n t s . The shape of the s o l i d l o w e r boundary i s a r b i t r a r y i n t h e p r e s e n t method; t h e r e i s no i n h e r e n t r e q u i r e m e n t t h a t the boundary be s t r a i g h t as i s t h e case f o r a " r e f l e c t i o n p l a n e " . 28 i i i H§thods o f N u m e r i c a l S o l u t i o n A s s e m b l i n g t h e J g u a t i o n s . . A c o m p u t e r p r o g r a m i s u s e d t o c o n s t r u c t t h e m a t r i c e s A and B. T h i s p r o g r a m i s w r i t t e n i n FORTRAN f o r t h e OBC IBM 370/168 s y s t e m . The p r o g r a m i n p u t s a r e t h e c o o r d i n a t e s , l e n g t h s and o r i e n t a t i o n s o f t h e s o u r c e and v o r t e x e l e m e n t s on t h e a i r f o i l , i t s f l a p s , t h e a i r f o i l - s h a p e d w a l l s l a t s , a n d t h e s o l i d w a l l s e c t i o n s . T h e s e m a t r i c e s A and B a r e t h e n u s e d t o a s s e m b l e t h e c o e f f i c i e n t s c ^ o f unknowns i n t h e N+M e q u a t i o n s ( 3 . 1 7 , 3 . 1 8 ) . T y p i c a l l y N+M i s a b o u t 4 0 0 , h e n c e t h e m a t r i c e s A a n d B e a c h c o n t a i n more t h a n 150,000 n o n - z e r o n o n s y m m e t r i c e n t r i e s . I n f a c t t h e m a t r i c e s A and B a r e u s e d t o a s s e m b l e a t h i r d m a t r i x C s u c h t h a t t h e s y s t e m o f e q u a t i o n s t o b e s o l v e d i s w r i t t e n C ( a , y ) = d ; t h u s a l l t h r e e m a t r i c e s ( o r p a r t s t h e r e o f ) must r e s i d e i n memory s i m u l t a n e o u s l y . The a c t u a l c o m p u t a t i o n a l c a p a c i t y ( u s e a b l e memory) i s a b o u t 250,000 e n t r i e s , h e n c e s u c h l a r g e m a t r i c e s must be p a r t i t i o n e d f o r c o m p u t a t i o n i n b l o c k s a n d t e m p o r a r y s t o r a g e on p e r i p h e r a l d e v i c e s s u c h a s m a g n e t i c d i s c s . The m a t r i c e s A a n d B d e s c r i b e t h e r e l a t i v e g e o m e t r y o f t h e s o u r c e and v o r t e x e l e m e n t s , t h a t i s , t h e i r r e l a t i v e p o s i t i o n a n d o r i e n t a t i o n . T h e s e m a t r i c e s must be r e c a l c u l a t e d c o m p l e t e l y f o r e a c h c h a n g e i n r e l a t i v e g e o m e t r y , f o r e x a m p l e , a c h a n g e i n a i r f o i l i n c i d e n c e , s i z e o r w a l l OAR. I n t h e o r i g i n a l method o f H e s s and S m i t h [ 1 8 ] a n d i n s i m i l a r m e t h o d s i n p r e s e n t u s e ( J a c o b a n d S t e i n b a c h [ 2 3 ] , M a v r . i p l i s [ 2 4 ] , L a b r u j e r e [ 2 5 ] , Henshaw [ 2 0 , 2 1 ] ) , t h e s o l u t i o n o f (3.17,3.18) u s e s n u m e r i c a l s u p e r p o s i t i o n o f t h r e e " b a s i c 29 f l o w " s o l u t i o n s . T h i s i s p o s s i b l e f o r i s o l a t e d a i r f o i l s , a s t h e r e i s no c h a n g e i n r e l a t i v e g e o m e t r y when t h e a i r f o i l i n c i d e n c e i s c h a n g e d . T h i s i s a l s o p o s s i b l e f o r an a i r f o i l w i t h f l a p s a s l o n g a s t h e a i r f o i l - f l a p s c o m b i n a t i o n c h a n g e s i t s a t t i t u d e a s a w h o l e , t h a t i s , w i t h o u t a c h a n g e i n r e l a t i v e g e o m e t r y . The t h r e e " b a s i c f l o w " s o l u t i o n s a r e a s f o l l o w s . The f i r s t i s due t o a u n i f o r m s t r e a m o n s e t f l o w p a r a l l e l -to t h e a i r f o i l c h o r d . The s e c o n d i s due t o a u n i f o r m s t r e a m o n s e t f l o w p e r p e n d i c u l a r t o t h e a i r f o i l c h o r d . The t h i r d i s due t o a p u r e c i r c u l a t o r y o n s e t f l o w t h a t c o r r e s p o n d s t o p u r e c i r c u l a t i o n a b o u t t h e a i r f o i l . I f more t h a n one a i r f o i l i s p r e s e n t ( i . e . a f l a p ) , t h e r e i s more t h a n one i n d e p e n d e n t c i r c u l a t o r y o n s e t f l o w . The t h r e e " b a s i c f l o w s " f o r e a c h a i r f o i l a r e t h e n l i n e a r l y c o m b i n e d t o s a t i s f y t h e K u t t a c o n d i t i o n a t t h e t r a i l i n g e d g e a n d t o g i v e a p r e s c r i b e d i n c i d e n c e o r l i f t c o e f f i c i e n t . Henshaw [ 2 0 , 2 1 ] , f o r e x a m p l e , l i n e a r l y c o m b i n e s t h e s e c o n d and t h i r d s o l u t i o n s t o y i e l d a f o u r t h s o l u t i o n w h e r e b y a K u t t a c o n d i t i o n i s s a t i s f i 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 . L i n e a r c o m b i n a t i o n s o f t h e f i r s t a n d f o u r t h s o l u t i o n s y i e l d t h e f l o w a t any i n c i d e n c e w i t h a K u t t a c o n d i t i o n a u t o m a t i c a l l y s a t i s f i e d . T h i s p r o c e d u r e i s n o t p o s s i b l e w i t h an a i r f o i l b e t w e e n s o l i d w a l l s . S i n c e t h e s o l i d w a l l s a r e n o t c l o s e d l i f t i n g b o d i e s , t h e y do n o t h a v e v o r t e x e l e m e n t s d i s t r i b u t e d o v e r t h e i r s u r f a c e s . Hence t h e r e i s no c i r c u l a t o r y f l o w a s s o c i a t e d w i t h t h e s o l i d w a l l s . F o r a c h a n g e i n a i r f o i l i n c i d e n c e , s i z e o r w a l l . OAR, t h e m a t r i c e s A and B must be c a l c u l a t e d a f r e s h . 30 R e f e r r i n g t o e q u a t i o n s ( 3 . 1 7 , 3 . 1 8 ) , t h e s y s t e m o f e g u a t i o n s t o be s o l v e d i s w r i t t e n b e l o w . The s u b s c r i p t s • j , i ' o f t h e m a t r i x e l e m e n t s o f A, B, a n d C a r e n o t o f s t a n d a r d n o t a t i o n , b u t a r e r e v e r s e d . T h i s i s d e l i b e r a t e , a s i n c r e a s e d e f f i c i e n c y o f s t o r a g e a n d s u m m a t i o n o f m a t r i x e l e m e n t s i s t h e n p o s s i b l e u n d e r t h e FORTRAN c o m p i l e r s . W i t h FORTRAN c o m p i l e r s , i n c o n v e n t i o n a l f o r m , t h e e l e m e n t s o f a m a t r i x a r e s t o r e d c o l u m n by c o l u m n . T h a t i s , t h e e l e m e n t s o f a m a t r i x a r e a s s i g n e d s e q u e n t i a l l y t o s t o r a g e b e g i n n i n g w i t h e l e m e n t (1,1) and p r o c e e d i n g t h r o u g h a l l v a l u e s o f t h e ( l e f t - most) f i r s t s u b s c r i p t , t h e n i n c r e a s i n g t h e s e c o n d s u b s c r i p t by 1, a n d r e p e a t i n g . I f A, B a n d C a r e s m a l l m a t r i c e s , t h e y may a l l r e s i d e s i m u l t a n e o u s l y i n r e a l memory. I n t h i s s i t u a t i o n , r o w - w i s e a c c e s s o f t h e e l e m e n t s o f a m a t r i x e n t a i l s a c c e s s i n g e l e m e n t s w h i c h a r e a l r e a d y i n memory. H o w e v e r , i f t h e m a t r i c e s a r e l a r g e , o n l y a p o r t i o n o f e a c h m a t r i x w i l l r e s i d e i n r e a l memory a t a n y one i n s t a n t . T h i s i s due t o t h e IBM v i r t u a l memory o p e r a t i n g s y s t e m w h i c h e m p l o y s some k i n d o f p a g i n g s y s t e m f o r d y n a m i c s t o r a g e a l l o c a t i o n . F o r more d e t a i l s , , s e e M o l e r [ 2 6 ] , T h u s a c c e s s i n g o f e l e m e n t s i n a row r e q u i r e s a t r a n s f e r i n a n d o u t o f r e a l memory o f b l o c k s o f e a c h m a t r i x , b l o c k by b l o c k , u n t i l t h e a c c e s s i n g i s c o m p l e t e d . T h i s i s n o t e f f i c i e n t . H ence a m a t r i x must be s t o r e d row by r o w . T h i s i s a c c o m p l i s h e d b y r e v e r s i n g t h e s u b s c r i p t s o f t h e e l e m e n t s o f a m a t r i x s o f o r e x a m p l e , t h e e l e m e n t s o f t h e f i r s t row a r e t h e n ( 1 , 1 ) , ( 2 , 1 ) , ( 3 , 1 ) , . . . ( N , 1 ) . T h i s s u b s c r i p t i n g i s e v i d e n t i n e g u a t i o n s (4. 1) . 31 The P.L/1 c o m p i l e r s s t o r e m a t r i c e s row by r o w , s o t h i s p r o b l e m d o e s n o t a r i s e when a c c e s s i n g e l e m e n t s i n t h e same row o f a m a t r i x , b u t w o u l d i f i t were n e c e s s a r y t o a c c e s s e l e m e n t s i n t h e same c o l u m n . The s y s t e m o f e q u a t i o n s t o be s o l v e d i s w r i t t e n : C l , i a i + C 2 , 1 Q 2 + - ' - + C N , l a N + C N + l , l Y l + ' - - + CN+M,1 YM = d l C l , 2 a i + C 2 , 2 a 2 + - - ' + CN,2 aN + C N + l , 2 Y l + " - + C N + M , 2 Y M = d z C l , N a i + C 2 , N a 2 + - - + CN,N aN + C N + l , N Y l + " - + CN+M,NYM = dN^- 1> C 1 , N + I a i + C 2 , N + I a 2 + - * • + C N , N + i a N + C N + l , N + l Y l + - ' • + C N + M , N + l Y M = d N + l C1,N+M 0 1 + C2,N+M a 2 + ' '* + CN,N+M aN + CN+l,N+M Y l + * *-+CN+M,N+MYM_dN+M' where t h e m a t r i x C a n d t h e c o l u m n v e c t o r d i n t h e s y s t e m C ( 0 , Y ) - d a r e a s s e m b l e d f r o m t h e m a t r i c e s A and B by means o f e q u a t i o n s ( 3 . 1 7 , 3 - 1 3 ) , t h a t i s , C j i A j : L j = 1, 2 , . . . N; i=l,2,...N- R(k) I " I B . k=l,2,...M; j=N+k; i=l,2,'...N , m = l 32 B . j = l,2 t ' • • N;r=l,2,...M; i=N+r (4.2) c. R(k) I' (A_ +A ) k=l,2,...M;j=N+k;r=l,2,...M;i=N+r m= 1 r r and Usin6. i = l , 2 t ' ' ' N [ -ucose -ucose r=l,2 , . . . M; i=N+r (4.3) r Thus the computation of the matrix C r e q u i r e s a c c e s s t o both matrices A and B. When these m a t r i c e s a re l a r g e , C must be assembled i n b l o c k s . For d e t a i l s see the FORTRAN program i n Appendix 2. The a l t e r n a t i v e i s : c a l c u l a t e A and. B, w r i t e B i n t o p e r i p h e r a l s t o r a g e , d e - a l l o c a t e the memory as s i g n e d t o B, a l l o c a t e memory f o r C, c a l c u l a t e a l l p a r t s o f C t h a t i n v o l v e A, d e - a l l o c a t e the memory assign e d t o A, a l l o c a t e memory f o r B, read B i n t o memory from p e r i p h e r a l s t o r a g e and c a l c u l a t e a l l p a r t s of C that i n v o l v e B. T h i s a l t e r n a t i v e i s s i m p l e r t o program and appears i n Appendix 10. The l a r g e number of summations of matrix elements i n the same row i s e v i d e n t i n equations (4.2); t h i s i s the main so u r c e of i n e f f i c i e n c i e s under the present paging s y s t e o . 33 4_-_2 S o l v i n g the Eguations-_ U s u a l l y the s o l u t i o n of the complete system of N + M equations i s o b t a i n e d d i r e c t l y by G a u s s - e l i m i n a t i o n methods. For a FORTRAN G a u s s - e l i m i n a t i o n s u b r o u t i n e t h a t t a k e s account of the above mentioned paging system, see Moler [2 6 ] or Appendix 3. Another d i r e c t method, used by Hess and Smith [ 1 8 ] i s the s u c c e s s i v e row v e c t o r o r t h o g o h a l i z a t i o n process of P u r c e l l [ 2 7 ] . In t h i s method an augmented matrix i s t r e a t e d row by row such t h a t a s e r i e s of v e c t o r s o r t h o g o n a l to each row v e c t o r of the augmented matrix i s c o n s t r u c t e d . The r i g h t - h a n d s i d e v e c t o r 'd* i s used to c o n s t r u c t a s e t of N+M v e c t o r s i n (N+M +1)-dimensional space, ( G l i ' C 2 i ' C 3 i C N + M , i ' - d i } . i = l , 2 , . . . N + M . The s o l u t i o n v e c t o r (a,y) °f equations (4. 1) i s such t h a t the v e c t o r Cai, a 2 , a 3 f . . . , o^, y i , Y2,..-, y M , 1) (**-5) i s o r t h o g o n a l t o a l l the v e c t o r s of (4.4). The process of s o l v i n g e q u a t i o n s (4.1) i s e q u i v a l e n t t o d e t e r m i n i n g an (N + M+1)- d i m e n s i o n a l v e c t o r o r t h o g o n a l to the N+M v e c t o r s of (4.4) with u n i t y as i t s (N+M+1)-th component. Each row of the c o e f f i c i e n t matrix C i s used at o n l y one stage of the process, and i s not needed be f o r e or a f t e r t h a t 34 stage. Thus C i s t r a n s f e r r e d from v i r t u a l to r e a l memory a row at a time, with each row occupying the same s t o r a g e l o c a t i o n as t h a t of the p r e v i o u s row. Thus an i n s i g n i f i c a n t amount of st o r a g e i s r e q u i r e d f o r storage of a s i n g l e row. However, the components of a l l the or t h o g o n a l v e c t o r s w i l l tend to be i n r e a l memory, s i n c e they are used r e p e t i t i v e l y . The maximum t o t a l number of components occurs when the process i s about h a l f - f i n i s h e d ; the t o t a l number of memory l o c a t i o n s r e q u i r e d i s approximately (N + M) 2/4. Thus the number of equations which can be s o l v e d by t h i s process ( f o r a gi v e n c o m p u t a t i o n a l c a p a c i t y ) i s about twice t h a t f o r a G a u s s - e l i m i n a t i o n p r o c e s s . For a FORTRAN s u b r o u t i n e based on t h i s procedure, see Appendix 3. I n d i r e c t i t e r a t i v e methods such as s u c c e s s i v e - o v e r - r e l a x a t i o n (SOB) are a l s o p o s s i b l e . For a d i s c u s s i o n of such methods, see Hess and Smith [ 1 8 ] . U n s u c c e s s f u l attempts we're made t o use SOR; the method was abandoned. In g e n e r a l , the matrix C i s d i a g o n a l l y dominant, t h a t i s , the d i a g o n a l elements, 2TT, are the l a r g e s t i n the m a t r i x . However, by examining the r e l a t i o n s (4.2), i t i s seen t h a t the summations over B .. , A „ and A _ pro v i d e l a r g e elements i n t h e mi mu mL ( l a s t ) (N+M)-th column of C. In g e n e r a l , t h e , sum of a l l the d i a g o n a l e n t r i e s i s approximately equal t o the sum of a l l t h e o f f - d i a g o n a l e n t r i e s . In e s s e n t i a l l y a l l cases,, the matrix i s n o n - s i n g u l a r except f o r very t h i n bodies, such as cusped t r a i l i n g edges. Here the source and vortex elements on the two s u r f a c e s are almost c o i n c i d e n t so matrix s i n g u l a r i t i e s (rows l i n e a r l y dependent)- can creep i n . For a d i s c u s s i o n of such 35 problems, see Hess and Smith [18]. The summations N M R(k) v n. I A. . a . - I y. Y B k=l m=l m i UsinS. l (4.6) and N y B . . a j=l 1 1 M R(k) k=l m=l , i i / " • + Ucos6 D I D i . i i - , k_J:-, m i 1 (4.7) p r o v i d e the net normal and t a n g e n t i a l v e l o c i t i e s a t c o n t r o l p o i n t s »i' due to a l l source and vortex elements • j ' and ' m' r e s p e c t i v e l y , and the uniform onset flow U. The s o l u t i o n t o the s e t of eguations (4.1) i s checked by computing the v e l o c i t i e s V n , V t at each p o i n t of a p p l i c a t i o n of the z e r o n o r m a l - v e l o c i t y i i or Kutta boundary c o n d i t i o n s . At a l l c o n t r o l p o i n t s on s o l i d s u r f a c e s , V i s z e r o , and the l o c a l p r e s s u r e c o e f f i c i e n t C i s 1 c a l c u l a t e d from : = 1 - V. U (4.8) The r e s u l t i n g v a l u e s of C p are i n t e g r a t e d n u m e r i c a l l y (by t r a p e z o i d a l r u l e or the f i t t i n g of c u b i c s p l i n e s ) around the t e s t a i r f o i l and f l a p contours to determine the l i f t , drag and nose-up midchord and g 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 , from the e x p r e s s i o n s 1 c c|>C dx. P J 3 1 c 4>C dy. 36 C M . = 772 <|>C (x.dx.+y.dy.) , (4.9) 'MoT C where dx =ds.cos9. and dy = ds s i n 6 (4.10) 3 3 3 3 3 3 and i n t e g r a t i o n s are performed c l o c k w i s e around the p o l y g o n a l c o n t o u r s . From a c a l c u l a t i o n of the net c i r c u l a t i o n about a l i f t i n g body, re p r e s e n t e d by NA source and vortex elements. f . y ^ t N A r = Av-<U = ov. ds. = I V, As., (4.11) J J t . i . ̂ , t . I X 1 = 1 X I and s i n c e the l i f t c o e f f i c i e n t C i s r e l a t e d t o the t o t a l c i r c u l a t i o n r by C i s giv e n by ~ N A °L = Uc" ^ V t / S i * ^- 1 3> 1=1 l By s u b s t i t u t i o n of (4.7), i t can be shown t h a t (4.13) reduces t o C = ^JH-(perimeter of the body) , (4.14) Ii U C - where y i s the vortex s t r e n g t h d e n s i t y f o r the body under c o n s i d e r a t i o n . The C v a l u e s c a l c u l a t e d from (4.9) and (4.13) are e q u i v a l e n t only f o r an i s o l a t e d a i r f o i l . I f a second body or a 37 boundary i s present the two v a l u e s are not e q u a l . The c a l c u l a t i o n of the c i r c u l a t i o n about a p a r t i c u l a r body then depends on the s i z e of the contour of i n t e g r a t i o n . The i n t e g r a l (4.11) reduces to the c o r r e c t value only as the contour s i z e s h r i n k s to z e r o . In the f o l l o w i n g pages, a l l C v a l u e s quoted a r e c a l c u l a t e d from e x p r e s s i o n ( 4 . 9 ) . I t must be emphasized t h a t on the body s u r f a c e elements, the flow f i e l d s o l u t i o n i s v a l i d only at the c o n t r o l p o i n t s ; f o r " o f f - c e n t e r " p o i n t s any c a l c u l a t e d s u r f a c e v e l o c i t i e s are meaningless. F i g u r e 3 . 4 shows how v e l o c i t i e s vary with p o s i t i o n , on a s u r f a c e element. On a g i v e n s u r f a c e element, i f t h e normal v e l o c i t y i s p r e s c r i b e d as z e r o , i t i s i n g e n e r a l non-zero a t a l l p o i n t s of the element except the c o n t r o l p o i n t . At the edges of the s u r f a c e elements the t a n g e n t i a l v e l o c i t y approaches i n f i n i t y because of the s i n g u l a r i t y i n the e x p r e s s i o n ( 3 . 8 ) and/or the d i s c o n t i n u i t y i n s u r f a c e s l o p e . However, the flow f i e l d s o l u t i o n i s u n i f o r m l y v a l i d a t a l l o f f - s u r f a c e p o i n t s . Hence at a f i e l d p o i n t ' i ' the v e l o c i t y components . p a r a l l e l and p e r p e n d i c u l a r to the streamwise d i r e c t i o n can be computed. The l o c a l flow . d i r e c t i o n can be c a l c u l a t e d and, by s t e p p i n g from p o i n t t o p o i n t , a p a r t i c u l a r s t r e a m l i n e can be t r a c k e d . The a l g o r i t h m i s given i n Appendix 4 . A l t e r n a t i v e l y the e x p r e s s i o n f o r the stream f u n c t i o n g i v e n i n Appendix 1 can be s o l v e d i t e r a t i v e l y f o r say the y - c o o r d i n a t e , a t a g i v e n x - c o o r d i n a t e , t o l o c a t e p o i n t s on a p a r t i c u l a r s t r e a m l i n e , t h a t i s , a l o c u s of p o i n t s (x,y) can be found along 33 w h i c h t h e c o m p u t e d v a l u e o f t h e s t r e a m f u n c t i o n i s a c o n s t a n t . 39 5.. R e s u l t s of the New Theory. The use of t h i s type of two-dimensional s u r f a c e s i n g u l a r i t y d i s t r i b u t i o n method, to c a l c u l a t e p r e s s u r e d i s t r i b u t i o n s on i s o l a t e d bodies, i s well e s t a b l i s h e d . For a comparison o f pressure d i s t r i b u t i o n s o b tained from t h i s method, with p r e s s u r e d i s t r i b u t i o n s d e r i v e d from experiments and other two-dimensional p o t e n t i a l flow t h e o r i e s , see Hess and Smith [ 1 8 ] . For purposes of comparison here, the s l o p e o f the c u r v e o f l i f t c o e f f i c i e n t C L as a f u n c t i o n o f a i r f o i l i n c i d e n c e w i l l be used. As an example of the present t h e o r e t i c a l method, f r e e - a i r l o a d i n g s were c a l c u l a t e d f o r the NACA-0015 a i r f o i l , u s i n g 50 c o n t r o l p o i n t s to re p r e s e n t the p r o f i l e o f the a i r f o i l . The c o o r d i n a t e s of the 50 c o n t r o l p o i n t s used a re given i n T a b l e 1. The v a l u e s of the l i f t c o e f f i c i e n t s so o b t a i n e d a re l i s t e d i n Tabl e 3, as. a f u n c t i o n of the i n c i d e n c e a . By exact c u r v e - f i t t i n g a polynomial of order 5 through the 6 p o i n t s a t 0,2,3,5,8, and 10 degrees i n c i d e n c e , the l i f t - c u r v e s l o p e a t zero degrees was found t o be 0.1193 The co r r e s p o n d i n g v a l u e a t +3 degrees i s 0.1229. These v a l u e s would be s l i g h t l y h i g h e r i f a l a r g e r number of c o n t r o l p o i n t s was used. From t h i n a i r f o i l theory (see Pope [ 2 8 ] ) t h i s l i f t - c u r v e s l o p e m f o r symmetrical a i r f o i l s i s g i v e n by (per ra d i a n ) m = 2TT 1 + . 7 7 3 - c U + (»773|) 2 J (5.1) where t/c i s the maximum-thickness t o chord r a t i o . For the NACA- 0015 a i r f o i l , the value o f is i s 6.919 per r a d i a n or 0.1208 per 40 degree. The agreement between the two t h e o r i e s i s good. S t r i c t l y the e x p r e s s i o n (5.1) g i v e s the t h e o r e t i c a l l i f t - c n r v e s l o p e o n l y at z ero degrees, s i n c e an u n d e r l y i n g assumption i s t h a t m i s independent of i n c i d e n c e . The present theory r e p r e s e n t s a uniform flow of i n f i n i t e e x t e n t , past a s e t of m u l t i p l e a i r f o i l s , and f l a t s u r f a c e s a l i g n e d with the d i r e c t i o n of the u n d i s t u r b e d f l o w . S i t h t h i s c o n f i g u r a t i o n ( Figure 3 . 1 ) , l o a d i n g s were c a l c u l a t e d f o r d i f f e r e n t t e s t a i r f o i l s d e v e l o p i n g high l i f t c o e f f i c i e n t s i n the presence of the above mentioned w a l l c o n f i g u r a t i o n . I t c o n s i s t s of a s o l i d lower w a l l i n c o n j u n c t i o n with a t r a n s v e r s e l y - s l o t t e d upper w a l l , with v a r i o u s upper w a l l OAEs. T h i s c o n f i g u r a t i o n w i l l be r e f e r r e d t o by the a b b r e v i a t i o n TSOSL., meaning t r a n s v e r s e l y - s l o t t e d upper, s o l i d lower. By comparing the l i f t c o e f f i c i e n t i n the windtunnel, C T , t o the f r e e - a i r v a l u e , C T , T F the r e s u l t s i n d i c a t e d t h a t a t r a n s v e r s e l y - s l o t t e d w a l l of about 70%OAR gave very s m a l l l i f t c o r r e c t i o n s (C -C ) f o r a l l the L T L F a i r f o i l s c o n s i d e r e d , at a l l l i f t c o e f f i c i e n t s , and up t o a i r f o i l s i z e s , c/H, as l a r g e as u n i t y . F i g u r e 5.1 shows the c a l c u l a t e d r a t i o of l i f t c o e f f i c i e n t s as a f u n c t i o n of a i r f o i l s i z e , c/H, f o r t h r e e a i r f o i l s . The f i r s t a i r f o i l i s a 14% t h i c k , 4.6% caaber Clark-Y ( r e p r e s e n t e d by 50 c o n t r o l points) a t zero and 20 degrees i n c i d e n c e (the f a c t t h a t the a c t u a l a i r f o i l would be s t a l l e d at t h i s i n c i d e n c e i s of no consequence f o r the present purpose). The second a i r f o i l i s an NACA-23012 at 8 degrees i n c i d e n c e , with a 2 5 . 6 % ( o v e r a l l chord length) s l o t t e d f l a p d e f l e c t e d 20 degrees (represented by 41 46 c o n t r o l p o i n t s o n t h e main a i r f o i l and 35 on t h e f l a p ) . The t h i r d a i r f o i l i s an NACA-0015 a t 3 d e g r e e s i n c i d e n c e - T h i s r a t i o o f l i f t c o e f f i c i e n t s i s shown - f o r two. w a l l c o n f i g u r a t i o n s . One i s w i t h two s o l i d w a l l s , a n d t h e s e c o n d i s a 70%OAH TSUSL w a l l c o n f i g u r a t i o n . I t i s s e e n t h a t w h e r e a s t h e l i f t c o r r e c t i o n f o r an a i r f o i l t e s t e d b e t w e e n s o l i d w a l l s , c o u l d be mora t h a u 5 0 % o f t h e t r u e , f r e e - a i r v a l u e , t h e p r e s e n t t h e o r y f o r one s o l i d w a l l a n d one s l o t t e d w a l l o f OAS n e a r .701 p r e d i c t s l i f t c o r r e c t i o n s o f l e s s t h a n 1%, f o r a i r f o i l s i z e c/H l e s s t h a n 1.0. I n d e e d , i t a p p e a r s t h a t a s l i g h t l y l o w e r v a l u e o f OAS w o u l d s h i f t t h e c u r v e s up s l i g h t l y , and g i v e l i f t c o r r e c t i o n s o f l e s s t h a n a b o u t <A% f o r c / f l l e s s t h a n 0.8. F o r t h e d e t a i l s o f t h e number o f w a l l s l a t s , number o f c o n t r o l p o i n t s u s e d , s l a t s i z e and s p a c i n g s , f o r a n y a i r f o i l - w a l l c o n f i g u r a t i o n t e s t e d t h e o r e t i c a l l y , a n d r e f e r r e d t o h e r e i n , s e e T a b l e 3. A p r i n c i p a l r e a s o n f o r 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 . i s t o o b t a i n p r e s s u r e d i s t r i b u t i o n s f o r use i n subsequent a e r o d y n a m i c a n a l y s i s . F i g u r e 5.2 shows a c o m p a r i s o n o f t h e p r e s s u r e d i s t r i b u t i o n s c a l c u l a t e d by . t h e p r e s e n t t h e o r e t i c a l method on t h e C l a r k - Y a i r f o i l a t 20 d e g r e e s i n c i d e n c e i n t h e p r e s e n c e o f a 70%OA.R TSUSL w a l l c o n f i g u r a t i o n , a n d i n f r e e a i r . The l i f t c o e f f i c i e n t s a r e t h e same, t h a t i s , t h e r e i s z e r o l i f t - c o r r e c t i o n . The f i g u r e i s p l o t t e d i n t e r m s o f t h e u s u a l n o n - d i m e n s i o n a l p r e s s u r e c o e f f i c i e n t C p . The f i g u r e shows t h a t t h e d i s t o r t i o n o f t h e p r e s s u r e . d i s t r i b u t i o n on t h e . a i r f o i l i n t h e p r e s e n c e o f t h i s TSUSL w a l l c o n f i g u r a t i o n i s s m a l l , e v e n a t t h e 42 high C of 3.09. The pressure d i s t r i b u t i o n f o r f r e e a i r shows L F l a r g e r negative pressures over the forward s e c t i o n of the a i r f o i l and a lower negative pressure over the a f t e r s e c t i o n , than i n the pressure d i s t r i b u t i o n f o r the TSUS.L w a l l c o n f i g u r a t i o n . The r e s u l t i n g midchord nose-up p i t c h i n g moment c o e f f i c i e n t f o r f r e e a i r i s 0.603, which i s t h e r e f o r e l a r g e r than the corresponding value of 0.573 f o r the TSOSL w a l l c o n f i g u r a t i o n . The r a t i o of midchord p i t c h i n g moment c o e f f i c i e n t s i n the t u n n e l , C , to the f r e e - a i r value C , i s MoT Mo p 0.95 here, which suggests t h a t t h i s z e r o - l i f t - c o r r e c t i o n TSU5L w a l l c o n f i g u r a t i o n might a l s o f i n d use as a low moment- c o r r e c t i o n t e s t c o n f i g u r a t i o n . A flow f i e l d comparison of a TSUSL w a l l c o n f i g u r a t i o n and f r e e a i r was made as f o l l o w s . The flow speed and d i r e c t i o n were computed at p o i n t s on f l a t s u r f a c e s p a r a l l e l to the plane of the t e s t s e c t i o n w a i l s , and s l i g h t l y i n s i d e the w a l l s . A s i m i l a r computation was performed f o r the f r e e - a i r case, f o r two f l a t s u r f a c e s with the same c o r r e s p o n d i n g p o s i t i o n s r e l a t i v e to the t e s t a i r f o i l . The r e s u l t s are shown i n F i g u r e 5.3, f o r the same zero l i f t - c o r r e c t i o n case d e s c r i b e d above, i n terms of the v a r i a t i o n of C along these s u r f a c e s . Q u a l i t a t i v e l y the f l o w f i e l d s are s i m i l a r i n t h a t a d e t a i l e d c a l c u l a t i o n u s i n g pressures and momentum f l u x e s f o r a r e c t a n g u l a r c o n t r o l volume about the. t e s t a i r f o i l c o n f i r m s the value of the a i r f o i l l i f t c o e f f i c i e n t w i t h i n 5%. 43 Experiments to Ver ify_ the New Theory. J2il T e s t s e c t i o n DesJ.g_ru The success of ,the proposed t r a n s v e r s e l y - s l o t t e d upper and s o l i d lower w a l l (TSOSL) t e s t c o n f i g u r a t i o n depends on the exp e r i m e n t a l v e r i f i c a t i o n of the present t h e o r y . I n i t i a l experiments were performed i n the o c t a g o n a l t e s t s e c t i o n of an e x i s t i n g low-speed c l o s e d - c i r c u i t windtunnel (Figure 6.1, P l a t e 1). Th i s t u n n e l has a t e s t s e c t i o n 915mm wide by 686mm deep, over a l e n g t h of 2.59m, and produces a very uniform flow, with a tu r b u l e n c e l e v e l l e s s than 0.1%, over a windspeed range of zero to 50m/s. The t e s t s e c t i o n has 152 by 152mm c o r n e r f i l l e t s ( a c t u a l l y tapered downstream t o compensate f o r boundary l a y e r growth), so t h a t the o c t a g o n a l c r o s s s e c t i o n area i s 0.582m2. As these i n i t i a l experiments were encouraging, the e x i s t i n g t e s t s e c t i o n was m o d i f i e d to ac c e p t a two-dimensional t e s t s e c t i o n i n s e r t . T h i s i n s e r t i s 915mm wide by 388mm deep i n c r o s s s e c t i o n , and 2.59m l o n g . T e s t a i r f o i l s are mounted v e r t i c a l l y on the yaw- t u r n t a b l e of a six-component windtunnel balance, at the midpoint of the t e s t s e c t i o n , and spanned the 388mm depth. One s i d e - w a l l was surrounded by a 0.39 by 0.30 by 2.44m plenum, and c o u l d be f i t t e d with a i r f o i l - s h a p e d s l a t s of NACA-0015 s e c t i o n ( P l a t e s 2,3) and chords of 46 or 92mra, at zero i n c i d e n c e . A f u l l range of w a l l open area r a t i o s (OAR) c o u l d be t e s t e d , 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 t u r n were s e p a r a t e d by wooden spa c e r s i n an aluminum channel r e c e s s e d i n the s i d e - w a l l frame. . M o d i f i c a t i o n s to the e x i s t i n g windtunnel c o n s i s t e d o f an i n s e r t e d nozzle and d i f f u s e r s e c t i o n (2.6m length) i n a d d i t i o n to the 388 by 915mm t e s t s e c t i o n . The design of the t h e o r e t i c a l shape of converging s e c t i o n s of c i r c u l a r c r o s s s e c t i o n to produce s p a t i a l l y uniform flow c o n d i t i o n s at e x i t i s w e l l e s t a b l i s h e d ; f o r example,.see Smith and Wang [ 2 9 ] . The approach here was to use the "same c r o s s s e c t i o n a l area v a r i a t i o n (in the streamwise d i r e c t i o n ) f o r an e q u i v a l e n t r e c t a n g u l a r c r o s s s e c t i o n as f o r the t h e o r e t i c a l c i r c u l a r c r o s s s e c t i o n . For the d e t a i l s of the d e s i g n , see Appendix 5. The t e s t s e c t i o n f l o o r and c e i l i n g are p a r a l l e l and s o l i d ; no s t r u c t u r a l or mechanical compensation f o r boundary l a y e r growth i s attempted. P i t o t s t a t i c tube t r a v e r s e s i n the empty t e s t s e c t i o n (with two s o l i d walls) where a t e s t a i r f o i l would be mounted, i n d i c a t e t h a t the t e s t s e c t i o n windspeed i s s p a t i a l l y uniform to w i t h i n 0.3% i n the c e n t r a l " c o r e " f l o w , o u t s i d e the w a l l boundary l a y e r s . F i g u r e 6.2 shows a t y p i c a l " c o r e " flow p i t o t s t a t i c tube windspeed t r a v e r s e . Boundary l a y e r p i t o t s t a t i c tube measurements i n the empty t e s t s e c t i o n , again where the t e s t a i r f o i l would be mounted, i n d i c a t e a displacement t h i c k n e s s of the o r d e r of 12mm, over a range of windspeeds c o v e r i n g the range of Reynolds numbers r e q u i r e d f o r v a r i o u s t e s t a i r f o i l s . F i g u r e 6.3 shows a t y p i c a l boundary l a y e r p i t o t s t a t i c tube windspeed t r a v e r s e . I n i t i a l experiments without the plenum gave i n c o n s i s t e n t trends i n the data taken as f u n c t i o n s of the w a l l OAR, with a s l o t t e d - w a l l of OAR 10% or g r e a t e r . The flow e x i t i n g from the U5 t e s t s e c t i o n t h r o u g h t h e s l o t t e d w a l l , u p s t r e a m o f t h e t e s t m o d e l , a t l a r g e OAR was n o t c o n s t r a i n e d t o r e - e n t e r t h e t e s t s e c t i o n t h r o u g h t h e s l o t t e d w a l l d o w n s t r e a m o f t h e t e s t a i r f o i l . C o n s e r v a t i o n o f mass f l o w was p r e s e r v e d by a l a r g e i n f l u x o f a i r i n t o t h e d i f f u s e r s e c t i o n t h r o u g h a b r e a t h e r s l o t b e t w e e n t h e t e s t s e c t i o n e x i t and d i f f u s e r s e c t i o n e n t r a n c e . T h i s p r o b l e m was c u r e d by u s i n g a p l e n u m s u r r o u n d i n g t h e t r a n s v e r s e l y - s l o t t e d w a l l . C o n s i s t e n t t r e n d s i n d a t a t a k e n a s f u n c t i o n s o f w a l l OAR a r e now a c h i e v a b l e w i t h a n y OAR w h a t s o e v e r . 4 6 6.i2 A i r f o i l Models Te s t e d . A l t o g e t h e r nine d i f f e r e n t a i r f o i l s were t e s t e d , with l i f t , drag and p i t c h i n g moment data taken f o r each a i r f o i l . I n a d d i t i o n , s u r f a c e pressure measurements were made on two of the a i r f o i l s . Four a i r f o i l s of NACA-0015 s e c t i o n , 383mm span and 153, 307, 462, and 616mm chord (model s i z e c/H of 0.17, 0.34, 0.51 and 0.67 r e s p e c t i v e l y ) , were machined from .. s o l i d aluminum b i l l e t s to c l o s e t o l e r a n c e s , on a n u m e r i c a l l y - c o n t r o l l e d m i l l i n g machine. 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 h o l e i n the t e s t s e c t i o n f l o o r with 3mm c l e a r a n c e a l l around between the c i r c u l a r h o l e and the mounting spar ( P l a t e 4)- The gaps between the t i p of the t e s t a i r f o i l and the f l o o r or c e i l i n g were l e s s than 2.5mm on a l l t e s t s . Four laminated wood a i r f o i l s of 14% t h i c k n e s s , 4.6% camber Clark-Y s e c t i o n , 683mm span and 227, 354, 481 . and 608mm chord ( a i r f o i l s i z e s c/H o f 0-25, 0.39, 0.53 and 0.66 r e s p e c t i v e l y ) were a l s o t e s t e d . S i n c e these a i r f o i l s extended o u t s i d e t h e t e s t s e c t i o n , they were f i t t e d with l a r g e (716mm diameter) c i r c u l a r aluminum e n d p l a t e s 3mm t h i c k ( P l a t e 5)- The e n d p l a t e s f i t f l u s h with the t e s t s e c t i o n f l o o r and c e i l i n g i n t o c i r c u l a r stepped r e c e s s e s , of 724mm diameter and 6ram depth..Thus the t e s t a i r f o i l i n c i d e n c e c o u l d be v a r i e d by r o t a t i n g the t e s t a i r f o i l - e n d plate combination about the v e r t i c a l a x i s , without a l l o w i n g the endplates t o touch the t e s t s e c t i o n f l o o r or c e i l i n g . A s i n g l e laminated wood a i r f o i l of 11% t h i c k n e s s , 2-3% 47 camber Joukowsky s e c t i o n , 683mm span and 307mm chord ( a i r f o i l s i z e c/H of 0.34) was a l s o t e s t e d . I t was s i m i l a r l y f i t t e d with c i r c u l a r e n d p l a t e s . The t r a i l i n g edge r e g i o n was t h i c k e n e d t o 3.8mm to allow pressure taps to be l o c a t e d t h e r e . T a b l e 1 c o n t a i n s the t h e o r e t i c a l and mo d i f i e d p r o f i l e c o o r d i n a t e s f o r t h i s a i r f o i l . For the f i v e a i r f o i l s f i t t e d with e n d p l a t e s , i t was necessary t o measure the l o a d i n g s on the endplates themselves. While under t e s t , depending on the t e s t a i r f o i l i n c i d e n c e , flow c o u l d be d e t e c t e d out of the t e s t s e c t i o n through t h e gap between the c i r c u l a r e n d p l a t e s and the c i r c u l a r r e c e s s e s i n the t e s t s e c t i o n f l o o r and c e i l i n g - . T h i s flow would c e r t a i n l y a f f e c t the l o a d i n g s on the t e s t a i r f o i l - e n d p l a t e c o m b i n a t i o n , p a r t i c u l a r l y the l o a d i n g on an end p l a t e . Thus t h e r e would be a l o a d i n g on an en d p l a t e due to the outflow through the gap i n a d d i t i o n to the expected s k i n - f r i c t i o n drag f o r c e i n t h e streamwise d i r e c t i o n . The l o a d i n g on a s i n g l e endplate was t h e r e f o r e determined by mounting the endpl a t e on the windtunnel balance with t h e upper s u r f a c e f l u s h with the t e s t s e c t i o n f l o o r . Each.of t h e f i v e a i r f o i l s was then suspended v e r t i c a l l y above the en d p l a t e t o e s t a b l i s h the c o r r e c t flow f i e l d over the endpl a t e a t a l l a i r f o i l i n c i d e n c e s . The gap between the t i p of the t e s t a i r f o i l and the endplate was a d j u s t e d by r a i s i n g or l o w e r i n g the a i r f o i l . Measurements of the l o a d i n g on the endplate were made f o r each a i r f o i l over a complete range of t e s t a i r f o i l i n c i d e n c e and f o r v a r y i n g t i p - e n d p l a t e gaps. The r e s u l t s were e x t r a p o l a t e d 48 to zero gap t o deduce that p o r t i o n o f the l o a d i n g s on the t e s t a i r f o i l / e n d p l a t e combination which was due to the two e n d p l a t e s . The net e f f e c t on t e s t a i r f o i l l i f t and midchord p i t c h i n g moment was t y p i c a l l y l e s s than 2%; the e f f e c t on t e s t a i r f o i l drag was l a r g e , t y p i c a l l y 30%. Table 4 and F i g u r e 6.4 show t y p i c a l endplate l o a d i n g s and the co r r e s p o n d i n g u n c o r r e c t e d and c o r r e c t e d a i r f o i l l o a d i n g s f o r the Joukowsky a i r f o i l . The f a c t t h a t the l o a d i n g on the endpl a t e d i d not vary markedly f o r a gap l e s s than t h a t c o r r e s p o n d i n g t o a gap t o chord r a t i o of about 0.005, supports the c l a i m t h a t t he l o a d i n g on a two-dimensional t e s t a i r f o i l s h o u l d not be too s e n s i t i v e t o the t i p - w a l l gap, p r o v i d e d the gap i s l e s s than a c e r t a i n minimum v a l u e . The 0.67-NACA-0015 a i r f o i l was f i t t e d with 65 c e n t e r - s p a n pressure taps ( P l a t e 6 ) . The Joukowsky a i r f o i l has a 75mm wide aluminum c e n t e r - s p a n s e c t i o n housing 37 p r e s s u r e t a p s . A l l pressure taps are s u r f a c e f l u s h and have 0.5mm diameter o r i f i c e s i n metal. P l a s t i c tubes of 1.6am i n s i d e diameter and approxim- a t e l y 1.0m l e n g t h t r a n s m i t the s u r f a c e p r e s s u r e s v i a the mounting spar t o a l o c a t i o n e x t e r n a l t o the t e s t s e c t i o n . P r e s s u r e s were measured using a 48 port " S c a n i v a l v e " manual-scan pressure t r a n s d u c e r . The tubes were d i s c o n n e c t e d from t he pressure t r a n s d u c e r and t i e d to the windtunnel balance " t u r n t a b l e d u r i n g balance measurements to e l i m i n a t e any e f f e c t of t e n s i o n i n the p l a s t i c tubes. 4 9 6 ._3 Test Procedures. The t e s t s e c t i o n windspeed was deduced from a p i t o t s t a t i c tube mounted on the flow c e n t e r l i n e i n the t u n n e l n o z z l e midway between the s e t t l i n g chamber e x i t annd the t e s t s e c t i o n e n t r a n c e . Located thus, the p i t o t s t a t i c tube would be f a r enough upstream to be r e l a t i v e l y u n a f f e c t e d by t e s t model "blockage" e f f e c t s ^ A l s o 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 f a r downstream i n the n o z z l e t h a t the flow speed would produce n u m e r i c a l l y l a r g e p i t o t s t a t i c tube r e a d i n g s (mm of water). Thus a s u f f i c i e n t l y a c c u r a t e r e a d i n g , e i t h e r on a "Betz" micromanometer.. s c a l e or as a " B a r o c e l " output v o l t a g e , c o u l d be o b t a i n e d . • The n o z z l e p i t o t s t a t i c tube was c a l i b r a t e d a g a i n s t a-second p i t o t s t a t i c tube mounted i n the empty s o l i d w a l l e d t e s t s e c t i o n , on the flow c e n t e r l i n e , where the t e s t a i r f o i l s would be l o c a t e d . . The p i t o t s t a t i c tubes were connected to " B a r o c e l " pressure t r a n s d u c e r s . During t e s t s , the n o z z l e p i t o t r e a d i n g i s s i m u l t a n e o u s l y monitored on a "Betz" micromanometer. During pressure t e s t s , the t o t a l head i n the n o z z l e i s measured (with the same p i t o t s t a t i c t u be), and used as a c a l i b r a t i o n p r e s s u r e f o r the " S c a n i v a l v e " pressure t r a n s d u c e r . The. r e f e r e n c e windspeed and s t a t i c p r e s s u r e used t o reduce balance and p r e s s u r e measurements are determined as f o l l o w s . S i n c e the t o t a l head at the n o z z l e p i t o t s t a t i c tube (when th e r e i s a t e s t a i r f o i l i n p l a c e , a t i n c i d e n c e ) 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 t e s t s e c t i o n i n the v i c i n i t y o f the t e s t a i r f o i l , the r e f e r e n c e windspeed and s t a t i c p r e ssure can be deduced from the n o z z l e p i t o t s t a t i c tube measurements of t o t a l 50 head and dynamic pressure (q). T h i s e q u a l i t y of t o t a l heads can only be v e r i f i e d i n the empty t e s t s e c t i o n ( s o l i d walls) with no t e s t a i r f o i l i n p l a c e ; such measurements i n d i c a t e d t h a t the t e s t s e c t i o n t o t a l head was lower than the n o z z l e t o t a l head by .1 p a r t i n 200. Let H, p and q be the t o t a l head, s t a t i c pressure';, and dynamic pr e s s u r e r e s p e c t i v e l y , and l e t the s u b s c r i p t s 1 and 2 r e f e r to the n o z z l e and t e s t s e c t i o n r e s p e c t i v e l y . In the empty t e s t s e c t i o n ( s o l i d w a l l s ) , a c c o r d i n g t o the above, Hi = P l + q i = H 2 = p 2 + qz - ( 6 . 1 ) I f k i s the empty t e s t s e c t i o n ( s o l i d walls) c a l i b r a t e d r a t i o of q! t o q 2 , then s i n c e only Hi and gj are measured while under t e s t , the e q u i v a l e n t empty t e s t s e c t i o n ( s o l i d walls) dynamic pressure and s t a t i c p r essure are k q x and (H.j-kqi) r e s p e c t i v e l y . . Thus the r e f e r e n c e windspeed and s t a t i c p r e s s u r e used to reduce data taken f o r any t e s t a i r f o i l , a t any i n c i d e n c e , i n the presence of any s l o t t e d - w a l l OAB, are the e q u i v a l e n t v a l u e s i n the u n d i s t u r b e d uniform stream c o n d i t i o n s t h a t would a c t u a l l y occur i n the empty ( s o l i d walls) t e s t s e c t i o n c o r r e s p o n d i n g to t h a t measured H x and q x. i n the n o z z l e . Hence the r e f e r e n c e windspeed "so "deduced i s not the a c t u a l flow speed past the t e s t a i r f o i l nor i s the r e f e r e n c e s t a t i c p r e s s u r e the a c t u a l s t a t i c pressure i n the t e s t s e c t i o n , while under t e s t . F i g u r e 6.5 shows a t y p i c a l n o z z l e - t e s t s e c t i o n windspeed 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 51 i s determined from a s t r a i g h t l i n e l e a s t - s q u a r e s c u r v e . f i t t e d through the o r i g i n . The six-component windtunnel balance has s i x independent four-arm s t r a i n gauge lo a d c e l l s so t h a t i n p r i n c i p l e , t h r e e f o r c e s and three moments can be measured and read i n d e p e n d e n t l y and s i m u l t a n e o u s l y . In p r a c t i c e , o n l y two s t r a i n gauge a m p l i f i e r s , n u l l i n g and readout u n i t s are a v a i l a b l e s i m u l t a n e o u s l y . As these a m p l i f i e r s are of the vacuum tube type, they a r e l e f t switched 'ON' at a l l times to minimize t h e d r i f t i n the zero r e a d i n g s on the readout u n i t s . The s t a n d a r d procedure f o r windtunnel balance r e a d i n g s i s to shut o f f the windspeed every 20 t o 30 minutes and r e c o r d the d r i f t s i n the z e r o r e a d i n g s . These zero d r i f t s are then a p p l i e d to each r e a d i n g i n a p r o p o r t i o n , assuming t h a t these zero d r i f t s are l i n e a r i n time and t h a t the time between r e a d i n g s i s the same. A l l pressure t r a n s d u c e r and s t r a i n gauge v o l t a g e s . a r e a m p l i f i e d and time-ayeraged to produce e s s e n t i a l l y steady 3 or 4 s i g n i f i c a n t , d i g i t r e a d i n g s on d i g i t a l v o l t m e t e r d i s p l a y s . The r e l a t i v e accuracy on a s i n g l e windtunnel balance measurement i s i n d i c a t e d as f o l l o w s . T y p i c a l v a l u e s and maximum p o s s i b l e e r r o r s are, f o r example, f o r the Joukowsky a i r f o i l a t +3 degrees i n c i d e n c e ( i n c l u d i n g endplate e f f e c t s ) , Reynolds number (Re) 0 . 5 ( 1 0 6 ) : C =0.713±0.004 (0.6%) ; C =-0.0625±0.0005 (0 . 8 % ) ; L Mc_ 4 and C=0.0286±0.0004 (1.4%) (6.2) At t h i s i n c i d e n c e and Re the net l i f t , q u a r t e r c h o r d p i t c h i n g moment, drag and windspeed can ba measured, to w i t h i n 0.2,0.4,1.0 and 0.2%, r e s p e c t i v e l y . S i m i l a r l y a value of l i f t c o e f f i c i e n t determined by i n t e g r a t i o n of a measured pressure d i s t r i b u t i o n has C =0. 709±0. 006 (0. 9%) . (6.3) In comparing c o e f f i c i e n t values taken at d i f f e r e n t a n g l e s of i n c i d e n c e t h e r e i s a maximum p o s s i b l e e r r o r of ±0.005 degrees i n the measured i n c i d e n c e due t o s l a c k i n the t u r n t a b l e worm gear mechanism. Care was taken t o always r o t a t e the t u r n t a b l e i n the same d i r e c t i o n t o minimize the e f f e c t of back l a s h i n the worm gear mechanism. T h i s p o s s i b l e e r r o r i n a i s the l a r g e s t of the p o s s i b l e e r r o r s which a r i s e i n computing the s l o p e of the C (a) curves. F i g u r e 6.6 shows a t y p i c a l C (a) curve, with e r r o r bars. L The v e r t i c a l e r r o r bars i n d i c a t e the maximum p o s s i b l e c u m u l a t i v e e r r o r o b t a i n a b l e on a s i n g l e measurement as d e s c r i b e d above. The h o r i z o n t a l e r r o r bars i n d i c a t e the maximum p o s s i b l e e r r o r i n the measured i n c i d e n c e . . The accuracy o b t a i n a b l e on a s i n g l e measurement i s good f o r the type of measurements made here. F i g u r e 6.7 shows th r e e C (ct) curves and i n d i c a t e s the L • r e p e a t a b i l i t y o b t a i n a b l e i n measurements taken i n three c o n s e c u t i v e runs under i d e n t i c a l t e s t c o n d i t i o n s and p r o c e d u r e s . The degree of r e p e a t a b i l i t y o b t a i n a b l e g i v e s c o n f i d e n c e i n measurements taken on a s i n g l e run (which i s not repeated) , f o r the type of measurements made here. The l e v e l of t u r b u l e n c e i n the two-dimensional t e s t s e c t i o n i n s e r t was n o t . m e a s u r e d . I t i s a s s u m e d t h a t . t h i s l e v e l m i g h t be a s h i g h a s i n t h e u n m o d i f i e d , o c t a g o n a l c r o s s s e c t i o n , t h a t i s 0 . 1 % . I t i s p r o b a b l y l e s s d u e t o t h e e x p e c t e d r e d u c t i o n i n a n y s t r e a m w i s e f l u c t u a t i o n s d u e t o t h e i n c r e a s e d c o n t r a c t i o n r a t i o , now 11.8 v e r s u s 7 f o r t h e o c t a g o n a l c r o s s s e c t i o n . 54 Zi. ExjBsrimental R e s u l t s . The r e s u l t s of the experiments using the nine a i r f o i l s are l i s t e d i n Tables 5, 6 and 7. The r e s u l t s a re t a b u l a t e d as c o e f f i c i e n t s of l i f t , drag, midchord and q u a r t e r c h o r d p i t c h i n g moments, a c c o r d i n g to t e s t a i r f o i l s i z e (c/H) and t e s t Reynolds number (Re), as f u n c t i o n s of t e s t a i r f o i l i n c i d e n c e (a) . The l i f t - c u r v e s l o p e dc /da i s c a l c u l a t e d from a l e a s t - s q u a r e f i t s t r a i g h t l i n e on an i n t e r v a l of 10 degrees i n c i d e n c e , f o r measurements i n 1 degree increments from 2 degrees below the angle of zero l i f t t o 8 degrees above the z e r o l i f t a n g l e . For the NACA-0015 a i r f o i l s with a z e r o - l i f t a n g l e of 0<>, m i s c a l c u l a t e d on (-2°,8°). For the Clark-Y a i r f o i l s with a z e r o - l i f t angle of about -6.3 degrees, m i s c a l c u l a t e d on (-8°,2°). For the Joukowsky a i r f o i l with a z e r o - l i f t angle of about -3.8 degrees, m i s c a l c u l a t e d on '•(-6°,4°). From the f i t t e d s t r a i g h t l i n e the z e r o - l i f t angle can be determined as the a - i n t e r c e p t when C L i s z e r o . In a s i m i l a r manner the s l o p e o f the midchord p i t c h i n g moment cur v e , dc /da, can be determined. The p o s i t i o n Mo of the aerodynamic c e n t e r x a c / c with r e s p e c t t o the a i r f o i l l e a d i n g edge i s x a C _ Xo _ _T /7 1\ c c m where x 0 , m, c, and x are the d i s t a n c e from the l e a d i n g edge to the a x i s of measurement of the moment c o e f f i c i e n t C , the l i f t -Mo cur v e s l o p e , the chord and the midchord p i t c h i n g moment-curve s l o p e , r e s p e c t i v e l y . Table 8 c o n t a i n s v a l u e s of the l i f t - c u r v e ' s l o p e , the z e r o - l i f t angle and the p o s i t i o n of the aerodynamic 55 c e n t e r f o r the n i n e a i r f o i l s and the v a r i o u s w a l l c o n f i g u r a t i o n s t e s t e d . F i g u r e 7.1 shows a t y p i c a l v a r i a t i o n of C (a) with upper L w a l l open-area r a t i o (OAR). The s t r a i g h t l i n e p l o t t e d c orresponds t o e s t a b l i s h e d experimental f r e e - a i r l i f t - c u r v e s l o p e s (Jacobs and Sherman [ 3 1 ] , R i e g e l s [ 3 2 ] ) . The z e r o - c o r r e c t i o n OAR appears to l i e between zero and 40%; t h i s f a c t i s a l s o apparent i n F i g u r e 7.5 and w i l l d i s c u s s e d t h e r e . The r e s u l t i n g v a l u e s of the l i f t - c u r v e s l o p e s , m, f o r any of the a i r f o i l s a r e , to the degree of accuracy r e q u i r e d here, q u i t e s e n s i t i v e t o the c h o i c e of range of i n c i d e n c e over which the l e a s t - s q u a r e s s t r a i g h t l i n e curve f i t t i n g i s done. For example, f o r the 0. 67-N AC.A-00 15 a i r f o i l between two s o l i d w a l l s , the value of m computed on (-2°,8°) i s 0.1114, while t h e v a l u e on ( 0 - o , 1 0 O ) i s 0.1156 and the value on ( 0 0 , 1 2 ° ) i s 0.1138. T h i s i s p a r t i c u l a r l y n o t i c e a b l e f o r t h i s a i r f o i l as t h e r e i s a pronounced jog i n the Ci,(a) curve i n t h i s range o f Reynolds numbers. T h i s jog i s a t t r i b u t e d to (see T a n i [30]) the f o r m a t i o n of a laminar s e p a r a t i o n bubble by the s e p a r a t i o n of the l a m i n a r boundary l a y e r near the l e a d i n g edge and subsequent reattachment downstream. As the Re i s i n c r e a s e d , the jog becomes p r o g r e s s i v e l y l e s s pronounced. F i g u r e 7.2 shows a comparison of two s e t s o f measured va l u e s of l i f t - c u r v e s l o p e m, with the c o r r e s p o n d i n g s l o p e s from the present theory, f o r a 14% 0.53-Clark-Y a i r f o i l , as a f u n c t i o n of the upper w a l l OAR. The l a r g e w a l l s l a t s (92mm) were used f o r one s e t of measured v a l u e s , and the s m a l l s l a t s (46am) 56 f o r the other s e t . The o r d i n a t e s are normalized by the v a l u e of the l i f t - c u r v e s l o p e , m , i n the presence of two s o l i d w a l l s (zero OAB). The t e s t Re i n both s e t s of measurements was 0.5(10*), based on the t e s t a i r f o i l chord. The t h e o r e t i c a l v a l u e s of l i f t - c u r v e s l o p e m are determined from a l e a s t - s q u a r e s s t r a i g h t l i n e f i t through t h r e e v a l u e s of l i f t c o e f f i c i e n t C > computed a t -8, -3 and +2 degrees i n c i d e n c e . F i g u r e 7.3 shows s i m i l a r r e s u l t s f o r the 0.67-NACA-00 15 a i r f o i l t e s t e d at a Re of 1.0 ( 1 0 6 ) . The measured l i f t - c u r v e s l o p e s a r e shown f o r t e s t s with both t h e l a r g e and s m a l l w a l l s l a t s . The t h e o r e t i c a l v a l u e s o f l i f t - c u r v e s l o p e m a r e determined from a l e a s t - s q u a r e s s t r a i g h t l i n e f i t through t h r e e v a l u e s of l i f t c o e f f i c i e n t C , computed a t -2, +3 and +8 degrees i n c i d e n c e . F i g u r e s 7.2 and 7.3 show t h a t t h e t h e o r e t i c a l v a l u e s of m/m are h i g h e r than the e x p e r i m e n t a l v a l u e s , f o r both s a i r f o i l s , and f o r a l l OABs. T h i s d i f f e r e n c e i s about 2.8%, a t 70%OAR, and w i l l be accounted f o r by two e x t e n s i o n s to the p r e s e n t theory i n §§8.1 and 8.2. F i g u r e 7.4 shows e x p e r i m e n t a l v a l u e s o f l i f t - c u r v e s l o p e , m, (per degree) f o r f o u r s i z e s of NACA-0015 a i r f o i l i n the presence of w a l l s of d i f f e r e n t OAR, 0, 60, 70, and 80S, u s i n g the l a r g e s l a t s . A l l t e s t s f o r the t h r e e l a r g e r a i r f o i l s (c/H of 0.67, 0.51, and 0.34) were run at a Re of .0.5(10*). T h i s Re c o u l d not be reached f o r the s m a l l e s t a i r f o i l (c/H of 0.17), which was t e s t e d at a Re of 0.3(10*). The data f o r t h i s a i r f o i l were then a d j u s t e d t o correspond to the 0.5(10*) Re, u s i n g p u b l i s h e d m (Re) data f o r the NACA-0015 a i r f o i l (see Jacobs and 57 Sherman [ 3 1 ] ) . The a d j u s t e d data are the f l a g g e d p o i n t s i n f i g u r e 7.4. The r e s u l t s show a convergence toward a f r e e a i r (zero c/H) l i f t - c u r v e s l o p e value of 0.093, i n good agreement with [ 3 1 ] . The r e s u l t s i n d i c a t e zero l i f t - c o r r e c t i o n s f o r an upper w a l l OAB between 60 and 70%, i n agreement with the , p r e d i c t i o n s of the p r e s e n t theory. F i g u r e 7.5 shows the c o r r e s p o n d i n g e x p e r i m e n t a l v a l u e s of m f o r f o u r s i z e s of Clark-Y a i r f o i l i n the presence of w a l l s of d i f f e r e n t OAR, using the l a r g e s l a t s . : A l l t e s t s f o r the t h r e e l a r g e r a i r f o i l s (c/H of 0.66, 0.53, and 0.39) were run a t a Re of 0.5 (10*). T h i s He c o u l d not be reached f o r t h e s m a l l e s t (c/H of 0.25) a i r f o i l , which was t e s t e d a t a Re of 0.45(10*) - The data f o r t h i s a i r f o i l were not a d j u s t e d t o the 0.5 (10*) Re as a (Sa)- i n f o r m a t i o n f o r the 14% t h i c k C l a r k - Y s e c t i o n i s s c a r c e . : The r e s u l t s show a convergence toward a f r e e a i r (zero c / f i ) l i f t - c u r v e s l o p e v a l u e of 0.096, which agrees f a v o u r a b l y w i t h the i n f o r m a t i o n t h a t i s a v a i l a b l e ( S i l v e r s t e i n [ 3 3 ] ) . An e x t r a p o l a t i o n of the curve of m (He) of F i g u r e 11 of [ 33 ], f o r an 11.7% t h i c k C l a r k - Y s e c t i o n of aspect r a t i o 6 g i v e s a value o f ra 0.071 a t a Re of 0.5(10*). By u s i n g t h e t h e o r e t i c a l r e l a t i o n (5.1) to e s t i m a t e an e q u i v a l e n t value f o r 14% t h i c k n e s s , then c o r r e c t i n g t o i n f i n i t e a spect r a t i o , the e s t i m a t e d value of m f o r the 14% t h i c k n e s s a t a Re of 0.5(10*) i s 0.096. Here z e r o l i f t - c o r r e c t i o n s are i n d i c a t e d f o r an upper w a l l OAR l e s s than 60%. E x p e r i m e n t a l l y , from both F i g u r e 7.1 and 7.5, the z e r o - c o r r e c t i o n OAR appears to ±>e somewhat l e s s than 60%, based on a 58 f r e e a i r l i f t - c u r v e s l o p e of 0.096- I f a lower value such as 0.092 were chosen, the z e r o - c o r r e c t i o n OAR would be about 60%. The m-values f o r the s m a l l e s t a i r f o i l s h o u l d be lowered due to the d i f f e r e n c e i n the t e s t Re. P r e v i o u s t e s t s £ 3 6 ] with the s m a l l e s t a i r f o i l have been u n r e l i a b l e . I t appears t h a t t h i s a i r f o i l has a r e l a t i v e l y s m a l l e r nose r a d i u s than the l a r g e r a i r f o i l s , which might account f o r the h i g h e r m-values. F i g u r e 7.6 g i v e s a comparison of e x p e r i m e n t a l p r e s s u r e d i s t r i b u t i o n s on the Joukowsky a i r f o i l , a t +3 degrees i n c i d e n c e i n the presence of two s o l i d w a l l s , and the 70%OAR, TSOSL w a l l c o n f i g u r a t i o n , u s i n g the l a r g e w a l l s l a t s . The comparison s u p p o r t s the t h e o r e t i c a l p r e d i c t i o n t h a t t h e c h i e f e f f e c t of the s l o t t e d w a l l i s t o lower the n e g a t i v e p r e s s u r e s over the upper s u r f a c e of the a i r f o i l , without a p p r e c i a b l y modifying the p o s i t i v e p r e s s u r e s on the un d e r s u r f a c e , or g e n e r a l l y d i s t o r t i n g the d i s t r i b u t i o n . The c l o s e agreement of C va l u e s from balance measurements with those o b t a i n e d by i n t e g r a t i o n o f the p r e s s u r e c o e f f i c i e n t , C , i n d i c a t e s s a t i s f a c t o r y two-dimensional f l o w P c o n d i t i o n s . C u b i c - s p l i n e p o l y n o m i a l s a re used t o c u r v e - f i t the d i s t r i b u t i o n of C f o r i n t e g r a t i o n . P In F i g u r e 7.7,. the pre s s u r e data from F i g u r e 7.6 f o r two s o l i d w a l l s are c o r r e c t e d t o e q u i v a l e n t f r e e - a i r c o n d i t i o n s by c o n v e n t i o n a l w a l l - c o r r e c t i o n theory {Pankhurst and Holder '£34]). The a c t u a l c o r r e c t i o n formulae used are reco r d e d i n Appendix 7. These c o r r e c t e d data are compared with the 70% OAR pressure data as taken. The 70%OAR data are seen t o agree q u i t e c l o s e l y with the c o r r e c t e d s o l i d w a l l data. The c o r r e c t e d s o l i d w a l l value of 59 C i s 0.675, while the 70%OAR value of C T i s 0.651; t h i s again i n d i c a t e s t h a t zero l i f t - c o r r e c t i o n t e s t c o n d i t i o n s w i l l occur at an upper wall OAR between 60 and 70%. The hump i n the pressure d i s t r i b u t i o n s toward the r e a r upper s u r f a c e i s caused by the t h i c k e r t r a i l i n g edge of the e x p e r i m e n t a l a i r f o i l r e p l a c i n g the t h e o r e t i c a l t r a i l i n g edge cusp. Por completeness, to show how the use: of such a TSUSL w a l l c o n f i g u r a t i o n a f f e c t s other aerodynamic d a t a , some t y p i c a l c urves of p i t c h i n g moment and drag c o e f f i c i e n t s are shown, as f u n c t i o n s of a i r f o i l i n c i d e n c e - F i g u r e s 7.8 and 7.9 show the midchord p i t c h i n g moment c o e f f i c i e n t c and d r a g , c o e f f i c i e n t C_ r e s p e c t i v e l y f o r the Mo D 0.53-Clark-Y a i r f o i l , as a f u n c t i o n of a i r f o i l i n c i d e n c e , and upper w a l l OAR. The v a l u e s a t 70%OAR (near f r e e - a i r c o n d i t i o n s ) f o r C and C_ agree w e l l with c o r r e s p o n d i n g v a l u e s from D . M 0 mxn F i g u r e 68 of P i n k e r t o n and Greenberg [ 3 5 ] , and Lim [ 3 6 ] . The v a l u e s of a t a given i n c i d e n c e i n i t i a l l y i n c r e a s e and then decrease, with i n c r e a s i n g OAR. T h i s same behaviour of C D was observed f o r the ot h e r a i r f o i l s t e s t e d , both with and without e n d p l a t e s , and appears t o be a pro p e r t y of t h i s p a r t i c u l a r t e s t c o n f i g u r a t i o n . 60 Si . E x t e n s i o n s t o the New Theory., What i s a v a i l a b l e at present i s a p o t e n t i a l flow theory f o r a t r a n s v e r s e l y - s l o t t e d upper and s o l i d lower w a l l c o n f i g u r a t i o n (TSOSL), t h a t , f o r an upper w a l l OAB of between 60 and 70% p r e d i c t s low l i f t - c o r r e c t i o n s f o r a v a r i e t y of a i r f o i l s , s i z e s and i n c i d e n c e s . However, e x p e r i m e n t a l l y , the OAB at which z e r o l i f t - c o r r e c t i o n o ccurs i s l e s s than t h e OAR p r e d i c t e d t h e o r e t i c a l l y . Thus the s l o t t e d w a l l behaves e x p e r i m e n t a l l y as i f i t were o p e r a t i n g at a l a r g e r OAB. The result,, at a given OAR, i s t h a t the r a t i o s , of t h e t h e o r e t i c a l values of l i f t - c u r v e s l o p e m or l i f t c o e f f i c i e n t C L , normalized by t h e i r c o r r e s p o n d i n g t h e o r e t i c a l f r e e - a i r v a l u e s , are too high. T h i s i s because the TSOSL w a l l c o n f i g u r a t i o n theory i s a p o t e n t i a l flow r e p r e s e n t a t i o n of a v i s c o u s flow f i e l d . For example, because of v i s c o s i t y , the w a l l s l a t s a r e n o t d e v e l o p i n g t h e i r f u l l c i r c u l a t i o n s p r e d i c t e d by the p o t e n t i a l flow theory s i n c e they are o p e r a t i n g i n a Re range of 37,500 t o 180,000 (with r e s p e c t to t h e i r chord l e n g t h s ) . , Thus the net c i r c u l a t i o n on the s l o t t e d w a l l i s l e s s than the c i r c u l a t i o n p r e d i c t e d t h e o r e t i c a l l y . Hence t h a t p o r t i o n of the measured, l i f t a t a g i v e n OAR which i s due t o the c i r c u l a t i o n on the w a l l s l a t s , w i l l be much l e s s than t h a t p o r t i o n of the l i f t p r e d i c t e d t h e o r e t i c a l l y . In other words, i n order t o p r e d i c t t h e o r e t i c a l l y t h e l i f t t h a t would a c t u a l l y be measured e x p e r i m e n t a l l y on an a i r f o i l i n 61 t h i s TSUSL w a l l c o n f i g u r a t i o n , t h e r e are two v i s c o u s flow f i e l d s to be accounted f o r . The f i r s t i s the usual f r e e a i r v i s c o u s flow f i e l d about the i s o l a t e d a i r f o i l . Now a second v i s c o u s flow f i e l d has been i n t r o d u c e d , t h a t of the flow through the TSUSL w a l l c o n f i g u r a t i o n . Thus a complete s o l u t i o n would r e q u i r e the a p p l i c a t i o n of v i s c o u s theory to the t e s t a i r f o i l , the w a l l s l a t s and any shear l a y e r s which develop i n the plenum. Let C ( ;E) and C { ;T) denote l i f t c o e f f i c i e n t s o b t a i n e d from experiment and theory r e s p e c t i v e l y , and l e t C (OAR; ) and C (F; ) denote l i f t c o e f f i c i e n t s from OAR and f r e e a i r t e s t c o n f i g u r a t i o n s r e s p e c t i v e l y . For a p a r t i c u l a r OAS, i t i s r e a s o n a b l e to assume t h a t the r a t i o C (OAB;E) :C (OAB;T) w i l l be equal to the r a t i o C (F; E) : C (F;T) , p a r t i c u l a r l y i n a low- Li Li c o r r e c t i o n t e s t environment. In t h i s case the v i s c o u s e f f e c t s on each r a t i o w i l l be the same. Thus the r a t i o of C (E) t o C (T) w i l l be the same f o r f r e e - a i r t e s t s or f o r t e s t s at whatever OAR produces near f r e e - a i r t e s t c o n d i t i o n s . T h e r e f o r e i t i s s t i l l u s e f u l t o compare the r a t i o of C (OAS;E):C (F;E) , with the r a t i o of C (OAR; T) ;C (F; T) . L» Li . What i s then d e s i r e d i s a p o t e n t i a l flow c a l c u l a t i o n t h a t w i l l account f o r the v i s c o u s e f f e c t s present e x p e r i m e n t a l l y which are due o n l y t o the TSUSL w a l l c o n f i g u r a t i o n . That i s , the u s u a l v i s c o u s flow a n a l y s i s f o r the t e s t a i r f o i l w i l l s t i l l be r e q u i r e d . 62 8.1 P o t e n t i a l Flow C o n s i d e r a t i o n s of V i s c o u s E f f e c t s . There are two ways to extend the pr e s e n t theory toward such a p o t e n t i a l flow c a l c u l a t i o n . One way i s to account f o r the e f f e c t of v i s c o s i t y on the t e s t a i r f o i l and w a l l s l a t s , through the f o r m a t i o n of v i s c o u s boundary l a y e r s on these s t r e a m l i n e d shapes. The second way i s to make the geometry of the fl o w r e p r e s e n t a t i o n more l i k e t h a t which 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 the t e s t s e c t i o n , w i t h the plenum s u r r o u n d i n g the s l o t t e d w a l l . The f o l l o w i n g d i s c u s s i o n a p p l i e s t o both the t e s t a i r f o i l and the a i r f o i l - s h a p e d w a l l s l a t s . F or boundary l a y e r e f f e c t s , only completely a t t a c h e d f l o w s are c o n s i d e r e d , t h a t i s , boundary l a y e r flows which s e p a r a t e from an a i r f o i l s u r f a c e are not c o n s i d e r e d . Thus, s i n c e the flow l e a v i n g the a i r f o i l t r a i l i n g edge i s a t t a c h e d , the v e l o c i t i e s and pr e s s u r e s a t the upper and lower s u r f a c e s a d j a c e n t to the t r a i l i n g edge, must be e q u a l . T h i s statement i s t r u e on a time-average, s i n c e the f o r m a t i o n o f a vortex s t r e e t i n the a i r f o i l wake r e q u i r e s the shedding o f c o n s e c u t i v e t r a i l i n g v o r t i c e s of a l t e r n a t e s i g n . For a g i v e n a i r f o i l shape and i n c i d e n c e , t h e r e i s a unique c i r c u l a t i o n t h a t w i l l s e t equal the t r a i l i n g edge v e l o c i t i e s on the upper and lower s u r f a c e s . T h i s i s the p h y s i c a l r e a s o n i n g behind t h e t h e o r e t i c a l r e l a t i o n known as the K u t t a c o n d i t i o n , t h a t i s , t h a t the v e l o c i t i e s at the upper and lower 'surfaces a d j a c e n t t o the t r a i l i n g edge (of any body which possesses a sharp t r a i l i n g edge) must be e q u a l . E x p e r i m e n t a l l y the measured value o f l i f t (which i s 63 p r o p o r t i o n a l t o the c i r c u l a t i o n ) i s always lower than the l i f t p r e d i c t e d by the u s u a l p o t e n t i a l flow t h e o r i e s ; the r a t i o of these l i f t s might be, f o r example, k. What then i s r e q u i r e d i s a procedure to reduce the usual t h e o r e t i c a l value of t h e c i r c u l a t i o n r t o a f r a c t i o n k of r . • From the uniqueness of the value of the c i r c u l a t i o n f o r a gi v e n a i r f o i l shape and i n c i d e n c e , the only p o s s i b i l i t y of red u c i n g the c i r c u l a t i o n , i f the Kutta c o n d i t i o n i s r e t a i n e d , i s to reduce the i n c i d e n c e , and/or a l t e r the shape of the p r o f i l e - . The r e d u c t i o n of i n c i d e n c e approach i s o f t e n used i n t h e comparison of the t h e o r e t i c a l and e x p e r i m e n t a l p r e s s u r e d i s t r i b u t i o n s . The t h e o r e t i c a l p r e s s u r e s are c a l c u l a t e d a t a reduced i n c i d e n c e , such t h a t the t h e o r e t i c a l l i f t (determined by i n t e g r a t i o n of the pre s s u r e d i s t r i b u t i o n s ) a t the reduced i n c i d e n c e i s the same as the measured l i f t at the measured incidence.•, T h i s procedure i s not completely s a t i s f a c t o r y as the magnitude of the n e g a t i v e p r e s s u r e . peaks i n the t h e o r e t i c a l p ressure d i s t r i b u t i o n s (near the a i r f o i l l e a d i n g edge on the upper surface) may not equal the magnitude of the peaks a c t u a l l y measured.: P h y s i c a l l y the t h i n laminar boundary l a y e r , which forms beginning a t t h e forward s t a g n a t i o n p o i n t on the u n d e r s i d e of the a i r f o i l , grows i n t h i c k n e s s as i t rounds the l e a d i n g edge and passes through the negative p r e s s u r e peaks. The ' e f f e c t of t h i s t h i n boundary l a y e r i s to i n c r e a s e the r a d i u s of c u r v a t u r e of the a i r f o i l l e a d i n g edge so t h a t the flow v e l o c i t i e s t h e r e are much lower than those p r e d i c t e d by t h e o r y . Thus the e f f e c t of the t h i n boundary l a y e r over the l e a d i n g edge i s t o reduce 64 the magnitude of the negative p r e s s u r e peaks below the magnitudes p r e d i c t e d t h e o r e t i c a l l y . Reducing the a i r f o i l i n c i d e n c e f o r purposes of the t h e o r e t i c a l c a l c u l a t i o n w i l l reduce the magnitude of these negative p r e s s u r e peaks, but not to the same degree as a c t u a l l y occurs due t o the presence of the boundary l a y e r . However, i f the c i r c u l a t i o n i s not f i x e d by t h e Kutta c o n d i t i o n , i t i s p o s s i b l e t o s p e c i f y the c i r c u l a t i o n o t h e r w i s e . The c i r c u l a t i o n may be determined f o r the t h e o r e t i c a l c a l c u l a t i o n from the measured l i f t . The t h e o r e t i c a l p r e s s u r e s are then c a l c u l a t e d at the measured i n c i d e n c e . The v e l o c i t i e s a t the t r a i l i n g edge w i l l no l o n g e r be e q u a l , but the t h e o r e t i c a l l i f t o b t a ined from the i n t e g r a t i o n of the p r e s s u r e d i s t r i b u t i o n can be made to e q u a l the e x p e r i m e n t a l l y measured l i f t . The procedure f o r t h i s c a l c u l a t i o n i s presented i n Appendix 8. F i g u r e 8.1 compares the r e s u l t i n g t h e o r e t i c a l and measured pressure d i s t r i b u t i o n s f o r the 0.67-NACA-0015 a i r f o i l a t 10 degrees i n c i d e n c e i n the presence of two s o l i d w a l l s . The t h e o r e t i c a l r e s u l t s by t h i s method are p h y s i c a l l y u n s a t i s f a c t o r y i n the v i c i n i t y of the a i r f o i l t r a i l i n g edge as such l a r g e n e g a tive p r e s s u r e s a r e i n p r a c t i c e not found t h e r e . I t i s p o s s i b l e t o lower t h e c i r c u l a t i o n developed by a l t e r i n g the shape of the a i r f o i l p r o f i l e . T h i s procedure i s j u s t i f i a b l e p h y s i c a l l y i f the a l t e r a t i o n s a r e i n keeping w i t h those which occur when the boundary l a y e r e f f e c t i v e l y m o d i f i e s the p r o f i l e shape. R e c a l l t h a t f o r an a i r f o i l , t he e f f e c t s o f v i s c o s i t y a r e 65 c o n f i n e d to the t h i n boundary l a y e r adjacent to the a i r f o i l s u r f a c e , and the flow o u t s i d e the boundary l a y e r can be c o n s i d e r e d as i r r o t a t i o n a l . Thus the boundary l a y e r m o d i f i e s the p r o f i l e shape and the p o t e n t i a l flow i s c a l c u l a t e d about t h i s modified shape, using the u s u a l e q u a l - v e l o c i t y Kutta c o n d i t i o n - The f o l l o w i n g procedure f o l l o w s and extends t h a t of P i n k e r t o n [37] whose a n a l y s i s i s f o r two-dimensional a i r f o i l s e c t i o n s t o be mapped conf o r r a a l l y onto a m o d i f i e d c i r c l e by means of the c l a s s i c a l Theodorsen method. F i g u r e 8.2 shows t h e o r i g i n a l p r o f i l e of the NACA-0015 a i r f o i l and the r e s u l t i n g m o d i f i e d p r o f i l e which r e s u l t s from the pr e s e n t procedure when t h i s a i r f o i l i s at 10 degrees i n c i d e n c e between two s o l i d w a l l s . T h i s i s the modified p r o f i l e used i n the c a l c u l a t i o n of the 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 i n F i g u r e 8.3. with r e s p e c t t o the d i r e c t i o n of the approach fl o w , the modified p r o f i l e i s e f f e c t i v e l y at a s l i g h t l y lower i n c i d e n c e and has s l i g h t l y l e s s camber than the o r i g i n a l p r o f i l e . The a c t u a l p r o f i l e ( o r i g i n a l p r o f i l e s w o l l e n by the a d d i t i o n o f the boundary l a y e r displacement t h i c k n e s s ) about which a p o t e n t i a l flow c a l c u l a t i o n might be c o n s i d e r e d , would be b l u n t at t h e t r a i l i n g edge and would have the t h i c k n e s s of the wake at t h a t p o i n t . The t h i c k n e s s of the boundary l a y e r on the upper s u r f a c e i s g r e a t e r than the t h i c k n e s s on the lower s u r f a c e , except f o r a symmetrical a i r f o i l i n f r e e a i r or between two s o l i d w a l l s , a t zero l i f t . T h e r e f o r e i f the t r a i l i n g edge was imagined to be taken ( v e r t i c a l l y ) as the midpoint of the wake, and the a f t e r p o r t i o n of the p r o f i l e were f a i r e d t o t h a t p o i n t , the r e s u l t i n g shape would be s i m i l a r to the e f f e c t i v e p r o f i l e of F i g u r e 8.2- 66 The a i r f o i l i s supposed to be at i n c i d e n c e a - P o i n t s on the p r o f i l e are to be r o t a t e d about the p r o f i l e l e a d i n g edge i n p r o p o r t i o n to t h e i r d i s t a n c e from the p r o f i l e l e a d i n g edge- The d i r e c t i o n of r o t a t i o n i s such to reduce the e f f e c t i v e i n c i d e n c e , of the p r o f i l e - As not a l l p o i n t s are r o t a t e d by the same amount, t h i s i s not a r i g i d body r o t a t i o n about the p r o f i l e l e a d i n g edge. Thus the e f f e c t i v e camber of the p r o f i l e i s reduced as the t r a i l i n g e dge.is " r a i s e d " more than o t h e r p o i n t s on the p r o f i l e . The d e t a i l s of t h i s " r a i s e d " t r a i l i n g edge procedure a r e presented i n Appendix 9- F i g u r e 8.3 shows a comparison of the r e s u l t i n g t h e o r e t i c a l and measured p r e s s u r e d i s t r i b u t i o n s f o r the 0.67-NACA-0015 a i r f o i l at 10 degrees i n c i d e n c e i n t h e presence of two s o l i d w a l l s . The t h e o r e t i c a l r e s u l t s by t h i s procedure are q u i t e s a t i s f a c t o r y except f o r the n e g a t i v e pressure peak near the a i r f o i l l e a d i n g edge. The magnitude of the t h e o r e t i c a l peaks i s s t i l l l a r g e r than the magnitude a c t u a l l y measured, and w i l l always be so f o r any procedure t h a t does not i n c r e a s e the r a d i u s o f c u r v a t u r e of the a i r f o i l l e a d i n g edge to the same degree t h a t a c t u a l l y o c c u r s due t o the boundary l a y e r t h e r e . The above procedure was a p p l i e d t o the a i r f o i l - s h a p e d w a l l s l a t s to see what the e f f e c t would be of r e d u c i n g the c i r c u l a t i o n s t h e r e while m a i n t a i n i n g the f u l l K utta c i r c u l a t i o n on the t e s t a i r f o i l . For the case of the 0.66-Clark-Y a i r f o i l a t 20 degrees i n c i d e n c e i n the presence of a 70%OAH TSOSL w a l l c o n f i g u r a t i o n , t h e t e s t a i r f o i l l i f t c o e f f i c i e n t was reduced 67 from 3.010 to 2.935, t h a t i s , by about 2.5%. T h i s c a l c u l a t i o n used a value of k of 0.80 on a l l the w a l l s l a t s . Thus the e f f e c t on the t e s t a i r f o i l of the n e g l e c t of v i s c o u s e f f e c t s on the w a l l s l a t s alone i s s m a l l i n comparison with the n e g l e c t o f v i s c o u s e f f e c t s on the t e s t a i r f o i l i t s e l f . A complete a c c o u n t i n g f o r the boundary l a y e r e f f e c t s even f o r completely a t t a c h e d flows must i n v o l v e the c a l c u l a t i o n s of the l a m i n a r boundary l a y e r growth, t r a n s i t i o n t o a t u r b u l e n t boundary l a y e r , and the growth o f the wake. Seebohm [ 3 8 ] has performed such c a l c u l a t i o n s and found good agreement w i t h experiments. His procedure i s as f o l l o w s . The u s u a l p o t e n t i a l flow c a l c u l a t i o n s a r e performed f o r the a i r f o i l p r o f i l e , t h e growth of the boundary l a y e r and wake are c a l c u l a t e d . The a i r f o i l p r o f i l e i s s w o l l e n by the boundary l a y e r d i s p l a c e m e n t t h i c k n e s s and i s extended downstream t o r e p r e s e n t the wake. Seebohm develops a c o n d i t i o n to f i x the c i r c u l a t i o n (which reduces t o the e q u a l - v e l o c i t y K u t t a c o n d i t i o n f o r no boundary l a y e r and no wake) based on the p r e s s u r e d i f f e r e n c e a c r o s s the wake, c a l c u l a t e d at two p o i n t s ( v e r t i c a l l y ) above and below the a i r f o i l t r a i l i n g edge, a t the outer edges of the boundary l a y e r s . The l a s t two s t e p s are i t e r a t e d on u n t i l t h e p r e s s u r e d i f f e r e n c e a c r o s s the wake no l o n g e r changes on i t e r a t i o n . Such c a l c u l a t i o n s c o u l d be i n c l u d e d i n the present t h e o r y , on the w a l l s l a t s . A method of h a n d l i n g the s e p a r a t e d flow r e g i o n over t h e upper s u r f a c e of an a i r f o i l at high i n c i d e n c e , approaching maximum l i f t , has been developed by Jacob and S t e i n b a c h [ 2 3 ] . 68 The s e p a r a t i n g s t r e a m l i n e d e p a r t s t a n g e n t i a l l y f r o m t h e u p p e r s u r f a c e s i n c e t h e o b s e r v e d p r e s s u r e s c o r r e s p o n d t o a s m o o t h n o n - z e r o v e l o c i t y d i s t r i b u t i o n a r o u n d t h e , s e p a r a t i o n p o i n t . The s e p a r a t i n g s t r e a m l i n e h a s ah a p p r o x i m a t e l y c o n s t a n t p r e s s u r e v a r i a t i o n a l o n g i t s l e n g t h , n e a r t h e a i r f o i l , t o r e p r e s e n t t h e a p p r o x i m a t e l y c o n s t a n t p r e s s u r e i n t h e r e a l d e a d a i r r e g i o n i n t h e s e p a r a t e d wake. T h i s i s a c c o m p l i s h e d by r e q u i r i n g e q u a l p r e s s u r e s a t a p o i n t a t t h e t r a i l i n g e d g e , and a t a s e c o n d p o i n t a b o v e t h e t r a i l i n g e d g e , on t h e s e p a r a t i n g s t r e a m l i n e . The c i r c u l a t i o n i s t h u s f i x e d , a n d t h e method i s s e e n t o b e . s i m i l a r t o t h e p r o c e d u r e o f Seebohm. A g a i n t h e b o u n d a r y l a y e r c a l c u l a t i o n s must be p e r f o r m e d t o c a l c u l a t e t h e p o s i t i o n o f t h e s e p a r a t i o n p o i n t , and t h e p r e s s u r e on t h e s e p a r a t i n g s t r e a m l i n e . A g a i n s u c h a p r o c e d u r e c o u l d be i n c o r p o r a t e d i n t o t h e p r e s e n t m e t h o d , f o r t h e w a l l s l a t s , i f d e s i r e d . 69 8.2 The Flow i n the Plenum: The Bounding Shear Layer. The second e x t e n s i o n of the present theory i s t o make the geometry of the flow r e p r e s e n t a t i o n more l i k e t h a t which a c t u a l l y occurs e x p e r i m e n t a l l y i n the t e s t s e c t i o n , with t h e plenum surrounding the s l o t t e d w a l l - F i g u r e 8-4 compares the flow r e p r e s e n t a t i o n of the present theory with the p h y s i c a l f l o w which a c t u a l l y occurs i n the t e s t s e c t i o n . The p r e s e n t t h e o r y r e p r e s e n t s a uniform flow of i n f i n i t e e x t e n t , past a s e t of m u l t i p l e a i r f o i l s , and f l a t s u r f a c e s a l i g n e d with the d i r e c t i o n of the undisturbed f l o w . Hence the energy l e v e l s of the. f l o w s i n s i d e and o u t s i d e t h e t e s t s e c t i o n are the same. F i g u r e 8.5 shows a shear l a y e r formed i n s i d e the plenum chamber surrounding the upper s l o t t e d w a l l . T h i s shear l a y e r i s formed as the o u t f l o w i n g a i r from the t e s t s e c t i o n , upstream of the t e s t a i r f o i l , mixes with and e n t r a i n s the o t h e r w i s e s t a g n a n t a i r i n the plenum. T h i s shear l a y e r t h e r e f o r e d i v i d e s the two f l o w s , the high-energy flow which e x i s t s i n the t e s t s e c t i o n , and the zero-energy stagnant flow of the plenum. I n a p o t e n t i a l f l o w t h e o r y , t h i s s h e a r l a y e r c o u l d be i d e a l i z e d as a c o n s t a n t - pressure f r e e s t r e a m l i n e which leaves the t e s t s e c t i o n at the upstream end of t h e s l o t t e d w a l l , e n t e r s the plenum and r e a t t a c h e s to the s o l i d w a l l s e c t i o n at the downstream end of the s l o t t e d w a l l . The p o s i t i o n o f , and the p r e s s u r e v a r i a t i o n a long such a s t r e a m l i n e are i n i t i a l l y unknown. The i n i t i a l and t e r m i n a l p o s i t i o n s , i n c l i n a t i o n s and p r e s s u r e s c o u l d be estimated from a flow f i e l d c a l c u l a t i o n which omits r e p r e s e n t a t i o n of t h i s s t r e a m l i n e e n t i r e l y . 70 An i t e r a t i v e procedure c o u l d be developed t o c a l c u l a t e the p o s i t i o n of t h i s s t r e a m l i n e i n a segmented s t e p - b y - s t e p approach. , The p o s i t i o n of a segment would be assumed- The r e s u l t i n g flow i n c l i n a t i o n s would be c a l c u l a t e d in. the v i c i n i t y of the supposed s t r e a m l i n e and compared w i t h the d i r e c t i o n the s t r e a m l i n e i s t a x i n g t h e r e . These s t e p s c o u l d be i t e r a t e d on u n t i l a, s a t i s f a c t o r y s t r e a m l i n e p o s i t i o n was found. Such procedures are used f o r l o c a t i n g s t r e a m l i n e s i n v o r t e x wakes, but are c o m p u t a t i o n a l l y time-consuming. As the approach to a z e r o l i f t - 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 i s made, the o v e r a l l flow f i e l d w i l l approach c l o s e l y t h a t of the t e s t a i r f o i l i n a f r e e - a i r t e s t environment. An obvious p o s i t i o n t o take f o r such a s t r e a m l i n e i s t h e r e f o r e the p o s i t i o n t h a t the p a r t i c u l a r s t r e a m l i n e i n the f r e e - a i r case o c c u p i e s which passes through the two p o i n t s d e f i n e d by the ends of the s o l i d w a l l s e c t i o n s at the b e g i n n i n g and end of t h e s l o t t e d w a l l . In order to i n v e s t i g a t e the e f f e c t of i n c l u d i n g a c o n s t a n t - p r e s s u r e f r e e s t r e a m l i n e i n the present t h e o r y , an a n a l y t i c two- d i m e n s i o n a l p o t e n t i a l flow r e p r e s e n t a t i o n was developed- The t e s t a i r f o i l was r e p r e s e n t e d by a s i n g l e v o r t e x , and the w a l l s l a t s by a s e t of v o r t i c e s near the c o n s t a n t p r e s s u r e boundary- F i g u r e A6.2 shows the m u l t i p l y - i n f i n i t e s e t of v o r t e x images r e q u i r e d to s a t i s f y the boundary c o n d i t i o n s . On the s o l i d lower w a l l , the boundary c o n d i t i o n i s zero d i s t u r b a n c e v e l o c i t y normal to the w a l l . On the c o n s t a n t pressure boundary, the l i n e a r i z e d c o n d i t i o n of c o n s t a n t pressure can be expressed v i a B e r n o u i l l i ' s 71 e q u a t i o n a s r e q u i r i n g z e r o d i s t u r b a n c e v e l o c i t y i n t h e s t r e a m w i s e ' d i r e c t i o n . Thus i n s o l i d b o u n d a r i e s , t h e a p p r o p r i a t e i m a ge o f a v o r t e x i s a v o r t e x o f e q u a l but o p p o s i t e c i r c u l a t i o n ; t h e i m a g e i n a c o n s t a n t p r e s s u r e b o u n d a r y h a s i d e n t i c a l c i r c u l a t i o n . The d e t a i l s o f t h e image s y s t e m a n d t h e e g u a t i o n s i n v o l v e d a r e i n A p p e n d i x 6. The r e s u l t s shown i n F i g u r e 8.6 c o m p a r e -the p r e s e n t a n a l y t i c i m a g e r e p r e s e n t a t i o n w i t h t h e a n a l y s i s o f H a v e l o c k [ 5 ] , f o r a f l a t p l a t e (a) between two s o l i d w a l l s , (b) i n g r o u n d e f f e c t , (c) b e t w e e n a s o l i d l o w e r b o u n d a r y and a c o n s t a n t p r e s s u r e u p p e r b o u n d a r y , and (d) b e t w e e n two c o n s t a n t p r e s s u r e b o u n d a r i e s (open j e t ) . The i m a g e r e p r e s e n t a t i o n i s u n s a t i s f a c t o r y a s i t p r e d i c t s a l o w e r l i f t t h a n f o r e i t h e r t h e g r o u n d e f f e c t o r t h e open j e t c a s e ; t h e e x p e r i m e n t s i n d i c a t e l i f t s a b o v e t h e g r o u n d e f f e c t v a l u e s . T h a t t h e l i f t i s s o l o w i s a r e s u l t , o f t h e c o n s t a n t p r e s s u r e b o u n d a r y c o n d i t i o n w h i c h r e q u i r e s t h e t a n g e n t i a l d i s t u r b a n c e v e l o c i t y t o be z e r o t h e r e . The c o r r e s p o n d i n g v a l u e o f t h e p r e s s u r e c o e f f i c i e n t , C p , i s z e r o . By t r a c k i n g a s t r e a m l i n e ( i n t h e TSOSL w a l l c o n f i g u r a t i o n t h e o r y ) w h i c h l e a v e s t h e t e s t s e c t i o n t h r o u g h t h e s l o t t e d w a l l 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 r e - e n t e r s t h e t e s t s e c t i o n d o w n s t r e a m o f t h e t e s t a i r f o i l , t h e t h e o r e t i c a l v a r i a t i o n o f p r e s s u r e a l o n g - s u c h a s t r e a m l i n e i s known. F i g u r e 8.7 shows s u c h a p r e s s u r e v a r i a t i o n , a n d , e x c l u d i n g t h e l a r g e n e g a t i v e p r e s s u r e e x c u r s i o n s a s t h e f l o w . a c c e l e r a t e s when i n the- v i c i n i t y o f t h e w a l l s l a t s , t h e a v e r a g e . v a l u e - o f . t h e p r e s s u r e c o e f f i c i e n t on t h i s s t r e a m l i n e i s 72 about -0.25. Hence any r e p r e s e n t a t i o n which uses a zero C value P i s i n c o r r e c t . \ The TSUSL wall c o n f i g u r a t i o n theory i n d i c a t e s t h a t the w a l l s l a t s c o l l e c t i v e l y have a net c o u n t e r - c l o c k w i s e c i r c u l a t i o n , s i n c e there are a l a r g e r number of s l a t s immersed i n the r e - e n t e r i n g flow than i n the e x i t i n g f l o w . However the. image r e p r e s e n t a t i o n c o r r e c t l y p r e d i c t s the e f f e c t on the a i r f o i l l i f t of i n c l u d i n g the c i r c u l a t i o n on the w a l l s l a t s v i a the s e t of v o r t i c e s adjacent to the c o n s t a n t p r e s s u r e boundary. For example, f o r a f l a t p l a t e at 26.3° i n c i d e n c e (see Appendix 6 and F i g u r e 8.6), the image r e p r e s e n t a t i o n p r e d i c t s t h a t the l i f t i s depressed 25% below the f r e e a i r value; the c o n t r i b u t i o n from the w a l l s l a t v o r t i c e s i s 2% and t h a t from the c o n s t a n t p r e s s u r e boundary i s .23%. The comparison would be b e t t e r f o r the f l a t p l a t e at 20° i n c i d e n c e , but the c a l c u l a t i o n s use w a l l s l a t c i r c u l a t i o n values from a Clark-Y a i r f o i l a t 20° i n c i d e n c e and 7 0%OAR. However, any s i n g l e p o t e n t i a l flow 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 cannot model the d i v i s i o n of the two flows of very d i f f e r e n t energy l e v e l , t h a t i s , t o t a l head, c o r r e c t l y . Suppose that the p o s i t i o n of the s t r e a m l i n e r e p r e s e n t i n g t h i s shear l a y e r i s known. The flow along t h i s s t r e a m l i n e s a t i s f i e s the t a n g e n t - v e l o c i t y boundary c o n d i t i o n . In a d d i t i o n , the pressure v a r i a t i o n along t h i s s t r e a m l i n e must r e f l e c t the f a c t t h a t the flow energy l e v e l on one s i d e of t h i s s t r e a m l i n e ( i n the plenum) i s z e r o . On the other s i d e (in the t e s t s e c t i o n ) the flow energy l e v e l i s t h a t of the Uniform u n d i s t u r b e d approach flow. T h i s shear l a y e r c o u l d be modelled a n a l y t i c a l l y by a vortex sheet, but t h i s • modelling was not attempted here. 7 3 P h y s i c a l l y t h e p r e s s u r e v a r i a t i o n i n t h e s h e a r l a y e r o r on a r e p r e s e n t a t i v e s t r e a m l i n e i s a r e s u l t o f t h e manner i n w h i c h t h e h i g h e n e r g y t e s t s e c t i o n f l o w l e a v e s t h e s o l i d w a l l s e c t i o n , a n d o f t h e low p r e s s u r e r e g i o n o f r e c i r c u l a t i n g f l o w s o f o r m e d i n t h e p l e n u m . To r e p r e s e n t t h e s h e a r l a y e r by t h e p r e s e n t t h e o r e t i c a l m e t h o d , c o i n c i d e n t s o u r c e a n d v o r t e x e l e m e n t s a r e d i s t r i b u t e d a l o n g a r e p r e s e n t a t i v e s t r e a m l i n e . S o u r c e e l e m e n t s a r e u s e d i n t h e u s u a l manner t o e n s u r e t h a t t h e f l o w i s t a n g e n t i a l t o t h e s u r f a c e . V o r t e x e l e m e n t s , a l l o f d i f f e r e n t s t r e n g t h d e n s i t i e s , a r e u s e d t o s e t p r e s c r i b e d v a l u e s o f t h e t a n g e n t i a l v e l o c i t y a t e a c h c o n t r o l p o i n t on t h e s u r f a c e . T h u s i f t h e s h e a r l a y e r i s r e p r e s e n t e d by, s a y , S c o n t r o l p o i n t s , t h e r e a r e a n a d d i t i o n a l unknown S s o u r c e and S v o r t e x s t r e n g t h d e n s i t i e s . T h us . t h e r e a r e S a d d i t i o n a l z e r o n o r m a l - v e l o c i t y e q u a t i o n s , and S a d d i t i o n a l p r e s c r i b a d - t a n g e n t i a l - v e l o c i t y e g u a t i o n s t o be s o l v e d . S i n c e i t i s u s u a l t o p r e s c r i b e t h e p r e s s u r e v a r i a t i o n o n t h e s u r f a c e r a t h e r t h a n t h e t a n g e n t i a l v e l o c i t y v a r i a t i o n , t h e t a n g e n t i a l v e l o c i t y 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 s h a v e t h e f o r m , f r o m (3. 1 4 , 3 . 1 8 ) : N M R(k) S 7 ( B . U . + A . V . ) % v r i l r i i-* r = l (8.1) = - U c o s e . ± / ( l - C ) , i = l , 2 1 p. , . . . s. Here y and v a r e t h e s o u r c e and v o r t e x s t r e n g t h d e n s i t i e s r e s p e c t i v e l y on t h e s t r e a m l i n e r e p r e s e n t i n g t h e s h e a r l a y e r . F o r 74 c a l c u l a t i o n s i n the same sense as the flow d i r e c t i o n , the squareroot term i s p o s i t i v e i f the source and vortex elements are d i s t r i b u t e d s e q u e n t i a l l y . The i n i t i a l and t e r m i n a l p o s i t i o n s o f t h i s s t r e a m l i n e a r e known, and the co r r e s p o n d i n g i n c l i n a t i o n s and p r e s s u r e s c o u l d be estimated from the flow c o n d i t i o n s on the s o l i d w a l l s e c t i o n s . I t remains o n l y t o s p e c i f y the v a r i a t i o n of p r e s s u r e (between two known end values) along the s t r e a m l i n e . Values of s p r e a d i n g c o e f f i c i e n t s f o r unbounded shear l a y e r s are to be found i n the l i t e r a t u r e , ( f o r example, [40]) and i f estim a t e s c o u l d be made of the e f f e c t of confinement which occurs here i n the plenum, such e s t i m a t e s c o u l d then be used t o s p e c i f y the p o s i t i o n of a s t r e a m l i n e r e p r e s e n t i n g the shear l a y e r . I t i s proposed that the p r e s s u r e v a r i a t i o n along the s t r e a m l i n e r e p r e s e n t i n g the shear l a y e r i s a f r e e parameter i n the a n a l y s i s here. That i s , s e v e r a l p r e s s u r e v a r i a t i o n s were assumed and s p e c i f i e d , and the computations performed. The s t r e a m l i n e t r a c k i n g procedure of Appendix 4 was used t o compute the v a r i a t i o n of pressure along the s t r e a m l i n e s of s i m i l a r p o s i t i o n and shape as are r e q u i r e d f o r s p e c i f i c a t i o n o f the p o s i t i o n of the shear l a y e r . T h i s pressure v a r i a t i o n i s shown on F i g u r e 8.7. T h i s s t r e a m l i n e l e a v e s the upstream upper s o l i d w a l l s e c t i o n and r e - e n t e r s the t e s t s e c t i o n downstream of the t e s t a i r f o i l , t h u s a p p a r a n t l y " e n t r a i n i n g " some of the e x t e r i o r flow. I t would be 75 expected, from c o n t i n u i t y , t h at t h i s s t r e a m l i n e should end on the downstream upper s o l i d w a l l s e c t i o n . Hess [ 2 2 ] e x p e r i e n c e d s i m i l a r "leakage" f l o w s i n the c a l c u l a t i o n of the i n t e r i o r flow i n a r i g h t angle bend. At f i r s t i t appeared that t h e r e were too few s o u r c e elements on the s o l i d w a l l s e c t i o n s , and t h a t the s o l i d w a l l s e c t i o n s were too s h o r t . These s e c t i o n s were lengthened from two t e s t a i r f o i l chord l e n g t h s to t e n , and the number of s o u r c e elements on a s e c t i o n was i n c r e a s e d from 20 t o 50. The "entrainment" e f f e c t was s t i l l p r e s e n t . T h i s a p p a r a n t l y i s a common problem f o r i n t e r i o r flow c a l c u l a t i o n s u sing f l a t d i s t r i b u t e d source elements with c o n s t a n t uniform s o u r c e s t r e n g t h d e n s i t i e s ; the problem i s e l i m i n a t e d by u s i n g c u r v e d source elements with l i n e a r or p a r a b o l i c v a r i a t i o n s of s o u r c e s t r e n g t h d e n s i t i e s over the elements. The "entrainment'•• f l o w r a t e here i s about 5% of the net flow r a t e ; i n Hsss's example the "leakage" flow r a t e was 12%. With the end v a l u e s of pressure s p e c i f i e d , s e v e r a l p r e s s u r e v a r i a t i o n s were t r i e d such t h a t the average p r e s s u r e along the s t r e a m l i n e r e p r e s e n t i n g the shear l a y e r was s i m i l a r t o t h e average pressure on the t r a c k e d s t r e a m l i n e s . The r e s u l t s i n F i g u r e 8.8 show how the a i r f o i l l i f t c o e f f i c i e n t v a r i e s with t h e assumed value of C . T h i s r e p r e s e n t a t i o n of the shear l a y e r i s P used i n the f o l l o w i n g s e c t i o n to compare the t h e o r e t i c a l and measured values of l i f t - c u r v e s l o p e f o r two a i r f o i l s . 75 8^2 Summary... A comparison of the curves of F i g u r e s 7.2 and 7.3 i n d i c a t e s t h a t at an upper w a l l OAS of 70%, the e x p e r i m e n t a l value of the r a t i o of the l i f t - c u r v e s l o p e s ra/m i s about 2.8% lower than the s t h e o r e t i c a l v a l u e , f o r both the Clark-Y and the NACA-0015 a i r f o i l s . Thus t h e r e i s a s m a l l r e s i d u a l d i f f e r e n c e of 2.8% to be accounted f o r by the e x t e n s i o n s to the the o r y o u t l i n e d i n §8.1 and 8.2. Assuming t h a t t h e r e are curves s i m i l a r t o the curves of F i g u r e 8.8 of the r a t i o of l i f t c o e f f i c i e n t s C /C f o r the L T L F present case, the r e s i d u a l 2.8% c o u l d be accounted f o r by an i n c r e a s e i n C of about +6%, f o r example from -0.33 to -0.31. A P value of C of -0.31 corresponds to the p r e s s u r e l e v e l P e s t a b l i s h e d by the r e c i r c u l a t i n g flow i n the plenum. F.ecall t h a t an adjustment of the same order c o u l d be accounted f o r with the procedure of §8.1, t h a t i s , by r e d u c i n g the c i r c u l a t i o n s on the w a l l s l a t s . Thus the d i f f e r e n c e between the t h e o r e t i c a l and experimental c u r v e s of F i g u r e s 7.2 and 7.3 at any OAR can be accounted f o r i n the theory by a combination o f (i) the r e d u c t i o n of the c i r c u l a t i o n s on the w a l l s l a t s by modifying t h e i r e f f e c t i v e p r o f i l e s , and ( i i ) by the r e p r e s e n t a t i o n . o f the shear l a y e r i n the plenum by a s i n g l e s t r e a m l i n e of assumed p o s i t i o n and streamwise pressure v a r i a t i o n . These e x t e n s i o n s are p r e l i m i n a r y ; f u r t h e r work i s r e q u i r e d t o e s t a b l i s h a s a t i s f a c t o r y theory. 77 9j. C o n c l u s i o n s . A two-dimensional theory which 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 windtunnel t e s t c o n f i g u r a t i o n has been developed. The t h e o r y i s an e x t e n s i o n of the two-dimensional p o t e n t i a l flow theory based on the method of 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 extended theory takes i n t o c o n s i d e r a t i o n not only a wide range of a i r f o i l s i z e s and shapes, but a l s o the e f f e c t on the a i r f o i l l o a d i n g s of d i f f e r e n t windtunnel w a l l c o n f i g u r a t i o n s . The r e s u l t s of the t h e o r e t i c a l study i n d i c a t e t h a t f o r two- 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 , a windtunnel c o n s i s t i n g of a s o l i d w a l l o p p o s i t e the pressure s i d e of the a i r f o i l , and a t r a n s v e r s e l y s l o t t e d w a l l , the s o l i d p o r t i o n s of which are symmetrical a i r f o i l - s h a p e d s l a t s at zero i n c i d e n c e , with open area r a t i o between 60 and 70 p e r c e n t , o p p o s i t e the s u c t i o n s i d e of the a i r f o i l , w i l l y i e l d u n c o r r e c t e d pressure d i s t r i b u t i o n s and l i f t c o e f f i c i e n t s which are w i t h i n a few percent of 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 remain 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 wide range of a i r f o i l s i z e s and a n g l e s of i n c i d e n c e . Experiments c a r r i e d out on a number of a i r f o i l s f o r a Reynolds number range of 300,000 to 1 m i l l i o n (based, on. the a i r f o i l c h o r d ) , i n a two-dimensional t e s t c o n f i g u r a t i o n support the p r e d i c t i o n s of the theory. E xperimental work showed t h a t t h e c o r r e c t i o n - f r e e t e s t c o n f i g u r a t i o n c o u l d be a c hieved with a s l o t t e d w a l l c o n s i s t i n g of symmetric a i r f o i l shaped s l a t s a t zero i n c i d e n c e , when the s l o t t e d s e c t i o n was surrounded by a 78 plenum chamber. The above theory was then extended t o account f o r v i s c o u s e f f e c t s on the w a l l s l a t s , and the e f f e c t of the shear l a y e r which forms i n the plenum chamber. Measurements taken with the c o r r e c t i o n - f r e e w a l l c o n f i g u r a t i o n of the l i f t , drag and p i t c h i n g moments f o r nine d i f f e r e n t a i r f o i l s which ranged i n s i z e (chord to h e i g h t r a t i o ) from-0.17 to 0.67, showed good agreement with e s t a b l i s h e d f r e e a i r v a l u e s . Furthermore, measurements of the p r e s s u r e d i s t r i b u t i o n on two a i r f o i l s , with the c o r r e c t i o n - f r e e w a l l c o n f i g u r a t i o n , showed good agreement with pressure d i s t r i b u t i o n s measured i n s o l i d w a l l c o n f i g u r a t i o n s and c o r r e c t e d by st a n d a r d methods. The l o w - c o r r e c t i o n t e s t c o n f i g u r a t i o n theory which has been t e s t e d and v e r i f i e d i n the work r e p o r t e d i n t h i s t h e s i s can be developed t o p r o v i d e 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 i n e x i s t i n g windtunnels which can be mo d i f i e d t o ac h i e v e the 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 . Such t e s t s would otherwise r e q u i r e e l a b o r a t e t e s t f a c i l i t i e s or complex c o r r e c t i o n procedures. Appendix 1. The I n t e g r a t i o n o f a Three Dimensional P o i n t Source t o a Two Dimensional F l a t D i s t r i b u t e d Source Element The p o t e n t i a l cj> a t a p o i n t P due t o a source s t r e n g t h d e n s i t y d i s t r i b u t i o n a(Q) over a s u r f a c e S i s , from (3.2), <MP) q ( Q ) ,r (PQ) dS , ( A l . l ) where r(PQ) i s the d i s t a n c e from the p o i n t P t o the p o i n t Q. From F i g u r e A l . l , the p o t e n t i a l a t P(x,y,0), f o r u n i t s t r e n g t h d e n s i t y a, i s As <}>(x,y,0) = As 2 d? _„ /(x-a2+y2+c2 (A1.2) The i n n e r i n t e g r a t i o n sums over a l l i n f i n i t e s i m a l source . elements d£d£; t o produce t h a t p a r t o f the p o t e n t i a l a t P due to a l i n e source element o f width d£. The o u t e r i n t e g r a t i o n sums over a l l such l i n e source elements t o produce the p o t e n t i a l a t P due to a f l a t d i s t r i b u t e d source element o f wid t h As, and o f c o n s t a n t u n i f o r m s t r e n g t h d e n s i t y . The v e l o c i t y components i n the x and y d i r e c t i o n s r e s p e c - t i v e l y , induced a t P by a source element o f width As, a r e : 80 V = x V 34) 3 x 3(j) 3y + + As 2 (x - C)I(x,?/y) d£, As 2 As 2 y l ( x , 5 , y ) d£ , As 2 ( A 1 . 3 ) (A1.4) where I(x,£,y) = !«»• ((x-?) 2+y 2+? 2) 3/2 (A1..5) Now _3 ( ( x - o 2 + y 2 ) / ( ( x - ? ) 2 + y 2 + c 2 ) j . ( ( x - £ ) 2 + y 2 + s 2 ) ^ (A1.6) t h e r e f o r e I ( x , £ , y ) = ( ( x - 0 2 + y 2 ) / ( ( x - £ ) 2 + y 2 + C 2 ) - ~ ( x - ? ) 2 + y 2 (A1.7) A l s o ^ ( - l o g ( ( x - ? ) 2 + y 2 ) ) = + 2 ( X " S ) , 9 5 ( x - ? ) 2 + y 2 (A1.8) ( t a n " 1 ^ 1 ) = 3 5 C-x i y J ( 5 - x ) 2 + y 2 (A1.9) Th e r e f o r e V As •log( (x - 5 ) 2+y 2) As ' 2 +log [ ( x - ^ ) 2+y 2J (Al.10) 81 V = 2tan Y 5-x t y As r , A s X + - — As ' 2 2 (tan -1 l y J - tan -1 x — As y J ) . ( A l . l l ) In a two di m e n s i o n a l i n c o m p r e s s i b l e , i r r o t a t i o n a l flow, the stream f u n c t i o n ty and the p o t e n t i a l ty are conjugate harmonic f u n c t i o n s . Thus the Cauchy-Riemann equations r e q u i r e dty _ 3<j> 3y 3x' H i 3x 9(j) 3y (A1.12) Hence ty may be expressed as ijj(x,y) - o (|i)dy + f (x) - j ( | | ) d x + g(y) , (A1.13) where f and g are a r b i t r a r y f u n c t i o n s and ty0 i s an a r b i t r a r y c o n s t a n t . Some u s e f u l i d e n t i t i e s a r e : l o g ( y 2 + a 2 ) = - ^ ( y l o g (y 2+a 2) - 2y + 2atan 1 ) , (A1.14) tan -1 (xtan 1 — - ^-log ( x 2 + a 2 ) ) , 3x' ( A 1 . 1 5 ) tan ''"A ± tan XB = tan x ( j ^ g ) • 1 rA±B (A1.16) T h e r e f o r e , from Al.10, H = s f f " Y A + x B + c ) < ( A 1 . - 1 7 > where A = l o g ( x + % . 2 + y 2 ̂ As { ( x ~ ) 2 + y 2J C = Astan B = 2tan 2xy -1 yAs [ x 2 + y 2 - ( ^ ) 2 J (A1.18) l x 2 " Y 2 - ( f ) 2 J A c c o r d i n g t o equations (Al.13) and (A1.17), the stream f u n c t i o n f o r a f l a t d i s t r i b u t e d source element o f wi d t h As and u n i t s t r e n g t h d e n s i t y i s iMx fy) = tyo + A - B + C. (A1.19) As i n e q u a t i o n ( A l . 1 3 ) , the p o t e n t i a l <j> may be expressed as (J)(x,y) - <J)o = - ( | i ) d y + f ( x ) = (||)dx.+ g ( y ) , (A1.20) where f and g are a r b i t r a r y f u n c t i o n s , and (J>0 i s an a r b i t r a r y c o n s t a n t . The a p p l i c a t i o n o f the t h r e e r e l a t i o n s ( A l . 1 4 ) , (A1.15), and (A1.16) to (Al.19) r e s u l t s i n 83 ( A1.21) D = f l o g ( ( x ^ ) V ) [ ( x - ^ ) V ! ^ T h e r e f o r e t h e p o t e n t i a l f u n c t i o n f o r a f l a t d i s t r i b u t e d s o u r c e e l e m e n t o f w i d t h As a n d u n i t s t r e n g t h d e n s i t y i s (J)(x,y) = c f > o + x A + yB + D - 2As . ( A 1 . 2 2 ) The c o r r e s p o n d i n g r e s u l t s f o r a u n i t s t r e n g t h d e n s i t y f l a t v o r t e x e l e m e n t c a n be w r i t t e n i m m e d i a t e l y s i n c e i n a t w o - d i m e n s i o n a l i n c o m p r e s s i b l e i r r o t a t i o n a l f l o w 4> ( v o r t e x ) = - ^ ( s o u r c e ) , i> ( v o r t e x ) = +cj) ( s o u r c e ) . (A1.23) 84 Appendix 2._ h Procedure f o r Block Commutation of M a t r i c e s Aj_ 8 and C__ When the m a t r i c e s A., B and C are l a r g e , C might be assembled from A and B by p a r t i t i o n i n g C i n t o b l o c k s as f o l l o w s . Memory i s a l l o c a t e d f o r A, B and C a c c o r d i n g t o the s i z e "of the l a r g e s t b l o c k . * j = l , 2, . . .N* j=N+l,N+2,-. .N+M* i=l,2,..NL4 i=NSUl,NSU2 i=NKA,M r A. . D i R ( k ) r - y B . * L , m i . m = l { i i -j •; A . . ] l R ( k ) c - y B . * L -, m i m = l ' 'B . +B.T r r f 1 R ( k ) >• y (A T T +A _ L , mil mL m = l r r ::B. +B . nU jL • r r R ( k ) ' I (A +A )^ ^ , mU mL . m = l r r t e s t a i r f o i l , f l a p s and s o l i d w a l l s s l a t s r - t h s l a t r - t h a i r f o i l o r f l a p The s u b s c r i p t ' i ' r e f e r s t o the row number i n the matrix C. The meanings f o r the v a r i a b l e s NL4, NSD1, NSB2, NK1, NK2 and NKA are given i n the programs which f o l l o w . Subroutine SUB1 c a l c u l a t e s A and B i n b l o c k s as needed t o assemble the b l o c k s of C. SUB 1 i s c a l l e d twice t o s e t up the zero n o r m a l - v e l o c i t y boundary c o n d i t i o n e g u a t i o n s on a l l s o l i d s u r f a c e s . I t i s c a l l e d f i r s t f o r the t e s t a i r f o i l and f l a p s , and a l l s o l i d w a l l s e c t i o n s and a second time f o r the w a l l s l a t s . S ubroutine SUB2 uses A and B to s e t up the Kutta c o n d i t i o n 85 e q u a t i o n s on t h e w a l l s l a t s . S u b r o u t i n e SUB3 d o e s t h e . saiae f o r t h e t e s t a i r f o i l and f l a p s . The FORTRAN c o d e d v e r s i o n s o f SUB1, SUB2 a n d SUB3 f o l l o w . The s u b r o u t i n e s READER (and WRITER) r e a d ( w r i t e ) a m a t r i x f r o m ( o n t o ) a p e r i p h e r a l s t o r a g e d e v i c e , s u c h a s a m a g n e t i c d i s c . 1 C SUB1 CALCULATES A £ B IN BLOCKS 6 SETS UP -EON'S ZERO NORM VEL ON ALL SOLID 2 C SURFACES 3 SUBROUTINE S U B 1 ( A , B , C , X X , Y Y , D S , C S , S I , N , L , M f N A , N A F 2 , L 1 , L 2 , N S P S , 4 1 NSLATtNS - U l t N K l ) 5 C A,B - MATRICES OF INFLUENCE CCEFFS FOR SOURCE £ VORTEX ELEMENTS 6 c C- MATRIX FOR SYSTEM OF EQN 1S C*SIG=D 7 c SIG(M) - UNKNOWNS IN SYSTEM G*SIG=D - PART IS GAM 8 c GAM(NAFt-NSLAT) - VORTEX STRENGTHS ON KUTTA BODY. 9 c M - TOTAL U UNKNCWS= TOTAL M EQUATIONS 10 c NAF = #TE.ST A I R F O I L S I FLAPS WITH KUTTA CONDITIONS APPLIED 11 c NSU1,NSU2 - SET OF EQN'S FOR ZERO NORMAL VELOCITY ON SLATS. 12 c NK1,NK2 - SET OF EQN'S FOR KUTTA CONDITIONS ON WALL SLATS. 13 c XX,YY - CONTROL POINT COORDINATES; DS - ELEMENT LENGTH 14 c NSLAT - •# OF SLATS WITH KUTTA CONDITIONS; NSPS - U ELEM'S PER SLAT 15 c CS S I - SIN,CCS CF ELEMENT INCLINATION 16 c NA(KJ - RANGE OF CONTROL POINT #'S FOR K-TH TEST AIRFOIL OR FLAP I . E . 1,50 17 c N - TOTAL # CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES 18 c L - BLOCK S I Z E 19 c L 1 , L 2 - RANGE OF CONTROL PT H'S FOR BLOCK - SET OF EQN #S ALSO 2 0 REAL A ( N , L ) , B ( N , L ) , C ( M , L ) , X X ( N ) , Y Y ( N ) , D S ( N ) , C S { N ) , S I ( N ) 21 INTEGER NA(NAF2\ 22 NSU2=NSU1+NSPS*NSLAT-1 23 NK2=NSU2=NSLAT 24 NAF=NAF2/2 25 c CALCULATE A AND B. 26 DO .1 I=L1,L2 27 K=I-L1+1 28 DO 1 J=1,N 29 I F { J . N E . I ) GO TO 2 30 A I J t K ) = 6 . 2 8 3 1 9 2 7 31 eu,K j = o . 32 GO TO 1 33 2 DXJ = XX(I }-XX{J) 34 DYJ= YY{I ) - Y Y ( J ) 35 X J = D X J * C S ( J ) + D Y J * S I ( J ) 3 6 Y J = D Y J * C S t J J - D X J * S I ( J ) 3 7 DSJ2=0S( J ) / 2 . 38 Y J S = Y J * Y J 39 S=XJ-rDSJ2 40 T=XJ-DSJ2 I Q ft CM I joo I 3 |>- 4- C O >-+ •H- oo —> + •»• 00 o o _ J II X — - ) - 5 - J OO 00 —• oo ' 00 l_) Q I + CM •— — Z 00 oo < u o < — — CM >-« 00 II OO <_} >- I) II —<—>—> zz -> a CM co - J - >i- <j- <r 00 L U OO * >- »—* t—t <J X 00 a. _t il X LU CO C£ 3 Z Z < CO cC CC LU LU ai oi 3: ?. ro L J O lT\ O I s - -4- -4- CO QT- O -J- vf lA OO LU o < U - r> 00 a oo o oo L U > cc o z a 0£ LU M cc a 00 2 - 3 C J II LU - 3 a . <J- ZZ> a LU OO on 00 < 00 a CD CC a H s: o a e> 2: r- — CC < O O - I • LL OO Of 2 LU 00 •• • - —I f-.Z II < Of 00 —I LU ^ lA o Q LU 00 CM COivf LA 43 i r\ L o l i n i n i r \ I 00 id •«• 00 .-t a . I 00 00 Z Q. + O0 Z ZD > OO z 11 11 ^ _J •XL M co 0* LO lT\ i i w i i —- II CO I » co O ^ 00 II II co a .co 00 o 00 r - i CM s0 O 00 a . < 1 u. o j OO o cc 00 —I LU > 2: ex a z a a: LU M or a a . a . < z 00 •• CO - 1-1 cfl Z l| I! O Z — LU »—4 » Q . CO - 3 3 — a 0 1 — 0 LU OO m u r- ro <r on -0 O z i i I <r> co i i i i < < z z II II —1 CM i i i i r - c o i i + U - <t z t CM i i o O II CO o 00 a o 00 LU II ZD — Z - 3 Z w a o o cr« c o c o CM 87 L L . < a 2: * LJ — a s: r - O CX >»• <t Qi O- L U r- OO >—1 >- ai X ^ z r - Qi _ l ZZ) HI J h y— < L U >-i o CX oi 3 r— r«- r > f>- f— CO 79 80 93 94 95 96 97 98 99 100 101 102 103 104 105 106 10 7_ 108 109 110 111 112 113 114 115 116 SUB2 SETS UP KUTTA EQN'S DN WALL SLATS.' SUBROUTINE SUB2(A,B,C,NA,N,L,M tNAF2,NSLAT,NSPStNSU1,NK1) X f ~~~ REAL A ( N , L ) , B ( f l 7 T T 7 C ( K T N S n m 82 INTEGER P » Qt NA(NAF2 I 83 C A» B - MATRICES -OF INFLUENCE COEFFS FOR SOURCE £ VORTEX ELEMENTS 84 C C- MATRIX FOR SYSTEM OF EQN'S C*SIG=D 85 C M - TOTAL ti UNKN0WS= TOTAL # EQUATIONS 86 C NAF=#TEST A I R F O I L S £ FLAPS WITH KUTTA CONDITIONS APPLIED ~87 C NSU1,NSU2 - SET OF EQN'S FOR ZERO NORMAL VELOCITY ON SLATS. 88 C NK1,NK2 — SET OF EQN'S FOR KUTTA CONDITIONS ON WALL SLATS. 89 C NSLAT - # OF SLATS M Jj-_ J_UJ_T A_ CONDITIONS; NSPS - # ELEM'S PER SLAT ~9Q~~- C _ l ^ T i n — : n R A N G E ~ O F CONTROL POINT #»S FOR K—TH TEST AIRFOIL OR FLAP I . E . 1,50 91 C . N - TOTAL # CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES 92 C L=NSPS*NSLAT - BLOCK S I Z E ; : : NAF=NAF2/2 NSU2=NSU1+NSPS*NSLAT-1 NK1=NSU2+1 NK2=NSU2-fNSLAT C READ IN THAT PART OF A £B WHICH CONTAINS THE INFLUENCE COEFFS FOR V E L O C I T I E S C INDUCED AT THE CONTROL POINTS ON THE WALL SLATS. CALL READER(A,N,L ) CALL READER I B , N , L ) SET UP EQNS F_Q_R KUTTA CN WALL SLA_TS_ ' "_TQ i i=i, N SLA T P=1+NSPS*(1-1) Q=P+NSPS-1 C P , Q - T.E. CONTROL PT #S FOR SLATS C 2 LOOP - TANG'L VELS DUE TO ALL SOURCE ELEM'S. ' DO 2 J=L ,N • 2 C C ( J , I ) = B ( J , P ) + B t J , Q ) 4 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON SLATS DO 4 KS=1,NSLAT . . J=NK1+KS-1 KK=NSU1+NSPS*(KS-l) KL=KK+NSPS-i SA = 0. DO 3 K=K'K , KL SA=SA+A(K,P)+A(K,Q) 117 4 C ( J , I ) = S A 118 C 6 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON AIRFOILS £ F L A P S . -co- co 119_ _10_ DO 6 KN- 1, NAF . 1 2 0 " K3=2*(NAF+1-KN)-1 121 K1=NAIK33 122 K2=NMK3 + 1) : ; __ 123 J-=N-NAF + KN 124 SA=0. 125 DO 5 K=K1, K2 ; - . . 126 5 SA = SA + A(K,P)+A{K,Q ) 12 7 6 C U » I ) = SA 128 1 CONTINUE __ 129 C WRITE THIS PORTION OF C INTO A F I L E . 130 CALL WRITER(C,M,NSLAT) 131 . RE TURN - ; 132 ' END 133 C SUB3 SETS UP KUTTA EQN'S ON TEST A I R F O I L S & F L A P S . 134 SUBROUTINE SUB3(A,B,C,NA,NTE,N,M,NL4,NAF,NAF2,NSPStNSLAT,NSU1,NK1 ) 135 REAL A ( N , N L 4 ) , B ( N ,NL4) ,C(M,NAF) 136 INTEGER NA{NAF2>,NTECNAF» 1.3 7 C A t B - MATRICES OF IN FLU EN CE COEF'FS FOR SOURCE & VORTEX ELEMENTS ____ 138 C C- MATRIX FOR SYSTEM OF EQN'S C*S.IG = D 139 C SIG(M) - UNKNOWNS IN SYSTEM C*SIG=D - PART IS GAM 140 C GAM ( NA F + NS LAT ) - VORTEX STRENGTHS ON KUTTA BODY. _ 141 C NL4 - M OF CONTROL POINTS ON TEST A I R F O I L , FLAPS L SOLID WALL SECTIONS 142 C M - TOTAL # UNKNOWS= TOTAL EQUATIONS 143 C NA F-ffTEST A I R F O I L S £ FLAPS WITH KUTTA CONDITIONS APPLIED 144 C NSLAT - # OF SLATS WITH KUTTA CONDITIONS; NSPS - # ELEM'S PER SLAT 145 G NA{K) - RANGE OF CONTROL POINT #'S FOR K-TH TEST AIRFOIL OR FLAP I . E . 1,50 146 C N - TOTAL H CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES 147 - C NT E{K I - CONTROL PT # FOR UPPER T.E. ON K-TH TEST A I R F O I L OR FLAP 148 hSU2=NSUl+NSPS*NSLAT-l !_ 9_ M<l = NSU2+._ . ' ; . 150'" C READ IN THAT PART" OF A &B WHICH CONTAINS THE INFLUENCE COEFFS FOR THE 151 C CONTROL POINTS ON THE TEST A I R F O I L S I F L A P S . 152 CALL READER ( A ,N ,NL4) : 153 CALL READER{B,N,NL4J 154 C SET UP EGN'S FOR KUTTA GN A I R F O I L S AND FLAPS 15_5 '_„__.___ JJOJL J<|_f i_NAF ; ' _ _ _ '156"" " ' I=KN" " . ' " • 157 MT=NTE(NAF+l-KN) 15 8 MTE = MT+1 . • o CTi 1 5 g c MT t MT E ARE #S FDR T.E. CONTROL PTS. ON TEST A I R F O I L S t FLAPS. 160 C 2 LCGP - TANG'L VELS DUE TO ALL SOURCE ELEM'S. 16 i DO 2 J=1,N „ . 162 2 C ( J f n = B(J,Mf)+B(J»MTE) 163 164 c IF{NSLAT.EG.0) GO TO 5 4 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON WALL SLATS. i u • 165 DG 4 KS=1,NSLAT 166 167 J=NK1+KS-1 KK=NSU1-NSPS*IKS-1) Jt, W I 168 KL=KK+NSPS-1 169 L70 SA = /. DO 3 K=KK,KL i 1 w 171 3 SA=SA+A{K,MT)+A(K,MTE) 172 173 4 c C ( J , U = S A 9 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON TEST AI R F O I L S & FLAPS. 174 5 DO 9 KM=1 »NAF 175 K 3 = i N A F - r l - K M ) - l 176 K1=NA(K3) 177 K2=NA{K3+1) 178 J=M-NAF-rKM 179 SA^O. ,—, . 180 181 ; 8 DO 8 K=K1 tK2 SA=SA^A(K,MT)-AIK,MTE) 18 2 9 C(J» I ) = SA 183 184 1 C CONTINUE WRITE THIS LAST PART OF C INTO A F I L E . 185 CALL WRIT ER{C,M» NAF) _ 186 Hfcll 18 7 END. O 9i A£_endix 3_ Two M e t h o d s o f S o l v i n _ t h e S y s t e m s o f S i m u l t a n e o u s L i l ^ e a E l i a e b j c a i c E j _ u a t i o n j _ _ ___ G a u s s i a n E l i m i n a t i o n . T h i s PORTRAN c o d e d s u b r o u t i n e p r o v i d e s f o r s i g n i f i c a n t s a v i n g s when s o l v i n g a l a r g e number o f e q u a t i o n s u n d e r a v i r t u a l memory o p e r a t i n g s y s t e m w h i c h e m p l o y s some k i n d o f p a g i n g s y s t e m f o r d y n a m i c s t o r a g e a l l o c a t i o n . To i l l u s t r a t e t h e r e v e r s e d s u b s c r i p t n o t a t i o n , t h e s y s t e m o f e q u a t i o n s i s w r i t t e n : c i io-. + c 2 \ 0 2 +c 3 i.a 3=d 1. C i 2 0 i + C 2 2 0 " 2 +C32a3 = d 3 ( A 3 . 1) C i 3 a i +c 2 3 a 2 + c 3 3 a 3 = d 3 The p r i n t o u t o f t h e s u b r o u t i n e ATXB f o l l o w s . 92 3__2 S u c c e s s i v e Row V e c t o r Q r t h o _ o n a l i _ P r o c e s s . The m a t r i x C f o r t h e s y s t e m C ( a , y ) = d i s augmen t e d by t h e r i g h t h and s i d e v e c t o r »d' t o f o r m t h e e g u a t i o n s C i \ 0 i + C 2 lO" 2+C 3 iO 3~d l t = 0 c i 2 0 " i + C 2 2 0" 2+c 3 2cr 3 - d 2 t=0 (A3.2) C 1 3 C T 1 + C 2 3 a 2 + C 3 3 ^ 3 - d 3 t = 0 A s e t o f N v e c t o r s i n ( N + 1 ) - d i m e n s i o n a l s p a c e i s f o r m e d : ( C l i ' C 2 i ' C 3 i ' * * * C N i ' ~ d i ) i=1,2,...N (A3.3) The s o l u t i o n v e c t o r o f . e q u a t i o n s (A3.3) i s s u c h t h a t t h e v e c t o r { o l f a 2 , a 3 , . . , o ^ t 1) (A3. 4) i s o r t h o g o n a l t o a l l t h e v e c t o r s o f ( A 3 . 3 ) . The p r o c e s s o f s o l v i n g e q u a t i o n s (A3.2) i s e q u i v a l e n t t o d e t e r m i n i n g a n (N+1)- d i m e n s i o n a l v e c t o r o r t h o g o n a l t o t h e N v e c t o r s o f (A3.3) a n d h a v i n g u n i t y a s i t s (N+1)st c o m p o n e n t . L e t U. d e n o t e t h e j - t h row o f t h e a u g m e n t e d m a t r i x . A t e a c h 3 s t a g e j , ( j = 1 # 2, . . . N+ 1) , a s e t o f v e c t o r s f o r ±= 1, 2, . . . N + 2- j i s c o n s t r u c t e d . A l l (N+2-j) v e c t o r s i n t h i s j - t h s e t a r e c o n s t r u c t e d t o be o r t h o g o n a l t o t h e ( j - 1 ) t h row o f t h e a u g m e n t e d m a t r i x , t h a t i s , a 1-. I n f a c t , a l l (N + 2 - j ) v e c t o r s i n ' t h i s j - t h s e t a r e o r t h o g o n a l t o a l l o f t h e f i r s t ( j - 1) r o w s o f t h e au g m e n t e d m a t r i x , t h a t i s " , t h e v e c t o r s U^,.. . U Thus v i l . ( t h e l a s t member o f t h e ( p r e s e n t ) j - t h s e t o f N+2-j v e c t o r s V"? ) , i s a v e c t o r w h i c h i s o r t h o g o n a l (by c o n s t r u c t i o n ) t o __A o r f h e f i r s t (j-1) v e c t o r s 0^, 0 ... U ^, a n d t h u s i s 93 o r t h o g o n a l t o a l l o f t h e f i r s t r o w s o f t h e a u g m e n t e d m a t r i x . N+1 U l t i m a t e l y t h e r e i s a s i n g l e v e c t o r w h i c h i s o r t h o g o n a l t o a l l M r o w s o f t h e a u g m e n t e d m a t r i x . T h i s i s t h e s o l u t i o n t o t h e s y s t e m o f e q u a t i o n s , a s t h e ( N + 1 ) t h ( l a s t ) N+1 c o m p o n e n t o f w i l l be u n i t y . I n f a c t , a t e a c h s t a g e ' j ' t h e l a s t c o m p o n e n t o f V ^ + 2 _ j i s u n i t y . T o a c t u a l l y c o n s t r u c t t h e s e t o f v e c t o r s a t t h e j - t h s t a g e , s c a l a r m u l t i p l i e r s c ^ 1 m u s t b e c a l c u l a t e d s u c h t h a t t h e s e t o f v e c t o r s v i = c i " l v r 1 + v i + i (A3.5) i s o r t h o g o n a l t o t h e ( j - 1 ) t h row o f t h e a u g m e n t e d m a t r i x , t h a t i s O J _ - L " Thus c ^ i s d e f i n e d by t h e s c a l a r p r o d u c t V i * U j - l = 0 " ( A 3 . 6 ) Hence c~! 1 i s c a l c u l a t e d f r o m i c i ~ 1 = - ( V i i i * u j - i ) / ( V i ~ 1 * D j - i ) ( a 3 - 7 ) To i n i t i a t e t h e p r o c e s s , t h e s e t o f v e c t o r s V 1 i s c h o s e n t o be t h e s e t o f " u n i t " v e c t o r s ( 0 , 0 ,. . . 1. . . 0, 0) , where 1 i s t h e i - t h c o m p o n e n t . T h i s p r o c e d u r e i s b e s t i l l u s t r a t e d by an e x a m p l e . F o r t h e s y s t e m o f e q u a t i o n s x+y-z=2 2x+y+z=1 (A3.8) x + y-t- 2z=- 1 T h e a u g m e n t e d m a t r i x i s 94 1 1 - 1 - 2 2 1 1 - 1 1 1 2 1 (A 3 . 9) and 0 = ( 1 , 1 , - 1 . - 2 ) r U =(2,1,1,-1) a n d 0.= ( 1 , 1 , 2 , 1 ) . (A3-. 10) F o r j = 1 , N + 2 - j i s 4, s o t h e r e ' a r e 4 " u n i t " v e c t o r s v-., s o vj-=(1, 0 , 0 , 0 ) , v j=(0,1,0,0) , ^ = ( 0 , 0 , 1 , 0 ) , V ^ ( 0 , 0 , 0 , 1 ) . (A3.11) 1 z j 4 F o r j = 2 , N+2-j i s 3, s o t h e r e a r e 3 v e c t o r s V 2 a n d 3 m u l t i p l i e r s c . U s i n g t h e r e v e r s e d s u b s c r i p t n o t a t i o n , i •V-+*U =a 1 1 1 1 a n d r ± *U =a i+1 1 1+1,1 t h e r e f o r e i I + I , 1 11 and ^ = - 1 , c^=1, c 1 = 2 . 1 ' 2 3 T h e r e f o r e V 2 = c 1 V 1 + V 1 i i 1 i+1 s o V 2 - ( - 1 , 1 , 0 , 0 ) , ¥^=(1,0,1,0) and ^ = ( 2 , 0 , 0 , 1 ) . 2 A l l t h r e e V a r e s e e n t o be o r t h o g o n a l t o 0 . i 1 (A3. 12) (A3. 13) (A3. 14) (A3. 15) (A3. 16) F o r j = 3 , N + 2 - j i s 2, s o t h e r e a r e 2 v e c t o r s a n d m u l t i p l i e r s c . . H e r e • I V l * ° 2 = - 1 a n d C i ' V i + l * D 2 2 2 t h e r e f o r e c_=3 and c =3. T h e r e f o r e ^ 2 2 2 Vf = c f v f + VT. . 1 X 1 x+1 s o V 3= (-2,3,1,0) a n d V 3 = (-1, 3, 0 ,1) . B o t h V 3 a r e s e e n t o be o r t h o g o n a l t o U a n d II , i 1 2 (A3.17) (A3. 18) (A 3 . 1 9) (A3. 20) 3 F o r j = 4 , N+2-j i s 1, s o t h e r e i s a s i n g l e m u l t x p l x e r c ^ a n d v e c t o r V . H e r e V 3 * 0 =3, V 3*U =3, s o c 3 = 1 . 1 3 2 3 1 (A3. 2 1) 9 5 T h e r e f o r e V^- (T , 0,-1,1) i s the s o l u t i o n v e c t o r f o r the system (A3.8) as .x=1, y=0, and z=-1. The FORTRAN programs f o r r e a d i n g and w r i t i n g m a t r i c e s o n t o p e r i p h e r a l s t o r a g e d e v i c e s a r e h i g h l y system-dependent, so t h e program f o r t h i s v e c t o r method i s not g i v e n h e r e . 96 AE_.§____ __ A __________ ________ _________ T h i s a l g o r i t h m i s used to t r a c k s t r e a m l i n e s from a s p e c i f i e d s t a r t i n g p o i n t , given an increment i n the x - d i r e c t i o n . The method uses the s u b r o u t i n e THETA to c a l c u l a t e the f l o w d i r e c t i o n , v e l o c i t y components, v e l o c i t y magnitude, and p r e s s u r e c o e f f i c i e n t at each p o i n t . From a s t a r t i n g p o i n t ( x _ , y _ ) , the flow d i r e c t i o n 81 i s c a l c u l a t e d , and used to l o c a t e the next p o i n t (xi+Ax,yi+Axtan8!). The flow d i r e c t i o n 9 2 i s c a l c u l a t e d t h e r e . The two flow d i r e c t i o n s are averaged to g i v e 8, and t h e y- c o o r d i n a t e i s changed so that the next p o i n t i s now (xi+Ax,yi+Axtan0). The flow d i r e c t i o n 9 2 i s c a l c u l a t e d t h e r e , and so on. An o u t l i n e of the FORTRAN coded s u b r o u t i n e THETA f o l l o w s . •s- fr*: z ' _ w cc >< - z nj •-CD 0 . •«-*• Z > 0 0 CO z ir-X- X CS 0 to 0 *—V z _1 z zz *—1 •«* IO —* «* 0 _ >v r-l. _ •• * —* >•*_ • _;••<-« &' il •. ""̂  t i l _> If f-f ZD < 0 fe: * _ • + !-« cc _ _ _ _ w CO ^ _5 -rr CD -*r 11 1 H .<£ »—> _: i II ZD _! io. _ . 0 II t_ II _ O l» CO £_ _ z to->-+ a: • • BR _3 _•' < il II z Ml _S Z . II CDl Z . t_ H r - l H j j - T H NV <r«i i r - CO 'TJ''"-* — — CD 03| _ II -3 -l> ; £_ I! 2 l H II ft':. 11: 11 _ . «"» at _ o _|! -—'.: or. <: _ il w V T X .XI • .XI ^ jr—t *" to X • 'ir* •»! Z X —•_-*] «-*• O _ l <C| * - 1^. s >-* a: o £_ o o 3 : :t_: z Ir-t .;—• CO 00 ^ M - < •"; ;: X • y-i —1 i rr —* w fc— cc zz <; O H 2: i — -t> Z5 •r -^^tal E — Q O ' 1_ Z O X QL. Ui . " •04-* 188 SUBROUTINE THETA {X,Y,THE,CP,XX,YY,DS,CS,S1,N,VNT,VTT,VM,A,B,SIG, 189 1 GAM,NA,GAMM,M) 190 REAL X X ( N ) , Y Y I N ) , D S I N J * C S ( N ) , S M N ) , A < N) , B (N) , S IG (N i , GAM { N) 19 1 INTEGER P, Q 19 2 COMMONOAO NA,NL4,NSLAT,NSPS 193 C I X , Y ) - CALCULATING FLOW PARAM * S HERE. 19 4 C XX,YY - CONTROL POINT COORDS FOR ALL SOURCE £ VORTEX ELEMS: DS IS ELEM LENGT 195 c SI,CS - SIN,COS ELEM INCLINATION 196 c N - TOTAL 3 CONTROL POINTS 197 c A,B - INFLUENCE COEFFS FOR VELOCITIES INDUCED 2 <X,Y) DUE TO ALL SOURCE £ 198 c VORTEX ELEM'S. 199 DO 1 J= 1 i N 200 DXJ = X - X X J J ) 201 DYJ = Y - Y Y T J ) 202 XJ = D X J * C S I J ) + D Y J 4 S U J ) 203 Y J = D Y J * C S I J ) - D X J * S I I J ) 204 DS J2 = D S ( J ) /2 . 20 5 Y J S = Y J * Y J 206 S=XJ+DSJ2 20 7 T = XJ-DSJ2 208 PHIX-=ALOG( (S*S + Y J S ) / ( T * T + Y J S J ) 209 PHI Y = 2 . * A T A N 2 ( ( D S 1 J ) * Y J ) , ( X J * X J + Y J S - D S J 2 * D S J 2 ) ) 210 A ( J J =. P F I Y * C S < J ) + P H I X * S I ( J ) 211 1 BUI = P H I X * C S ( J ) - P H I Y * S I U ) 212 c SIG - SOURCE £ VORTEX STRENGTHS 213 c GAM - VORTEX STRENGTHS FOR TEST AI R F O I L £ SLATS 214 c GA MM - TEST A I R F O I L VORTEX STRENGTH 215 c VNSTiVTST - TOTAL NORMAL £ TANG'L VELS 3 t X , Y ) DUE TO SOURCE ELEMS. 216 VNST = 0 217 VTST = 0 218 DO 2 J= I ,N 219 VNST = VNST + M J J * S I G 1 J ) 220 2 VTST = VTST t e U)*S I G ( J ) 221 AS=0. 222 BS = 0. 223 C NSALT,NSPS - # SLATS,# SOURCE ELEMS PER SLAT 224 IF(NSLAT.EQ.O) GO TO 3 225 CO 4 J = l ,NSLAT 226 C NL4 - # OF CONTROL POINTS ON TEST A I R F O I L , FLAPS 6 SOLID WALL SECTIONS 227 P=NL4+NSPS*J CO 228 C=P-NSPS+1 229 C P,Q - 1ST £ LAST CONTROL PTS ON A SLAT. 230 C AP,8P - NORM £ TANG VELS 3 (X,Y) DUE TO VORTEX ELEMS ON SLATS 231 AP = 0. 232 BP = 0. 233 • DO 5 M=P,Q 234 A P = A P + A { M } 235 5 BP = BP+B-(M1 23 6 AP=AP*GAM-l J ) 23 7 BP = BP*GAV.{ J) 238 AS=ASfAP 239 4 BS=BS+BP 240 C . A T i B T - NORM £ TANG VELS 3 (X,Y.) DUE TO VORTEX ELEMS ON TEST A I R F O I L . 241 AT=0. 242 B7-0. 243 C NA - # CONTROL PTS ON TEST A I R F O I L . 244 ' DO 6. J - l i N A 245 A T = A T +A f J ) 246 6 BT=BT+B1J) 247 AT=AT*GAMM 248 8T=BT*GAMM 249 C VNVTrVTVT - TOTAL NORM £ TANG VEL 2. (X,Y) DUE TO VORTEX ELEMS. 250 VNVT=-BS-BT 251 VTVT=AS+AT 252 C VNOT.VTVT - NORM £ TANG UNIFORM ONSET FLOW V E L S . 253 VNOT=0. 254 VTOT=U 255 C VNTiVTT - TOTAL NORM £ TANG VEL 3 ( X , Y ) . 256 VNT=VNOT+VNVT+VNST 257 VTT=VTOT+VTVT+VTST 258 vS=vTT*VTT+vN.T*VNT 259 C THE,CP,VM - FLOW DI RECTI ON,PRESSURE, MAG(VELOC) 260 CP = .1.-VS _ 261 VM= SGRT(VS ) ~ " ~ 262 THE = ATAN2(VNT , VTT) 263 RETURN 264 E N C a 99 D e _ i _ _ o f t h e T w o - D i m e n s i o n a l N o z z l e I n s e r t . The d e s i g n - o f t h e r e c t a n g u l a r c o n t r a c t i n g s e c t i o n i n s e r t was b a s e d on t h e method o f S m i t h a n d Wang [ 29 J. The. s o l u t i o n g i v e n makes u s e o f t h e e x a c t a n a l o g y b e t w e e n t h e m a g n e t i c f i e l d t h a t i s c r e a t e d by two c i r c u l a r c o a x i a l a n d p a r a l l e l c o i l s c a r r y i n g an e l e c t r i c c u r r e n t , and t h e v e l o c i t y f i e l d t h a t i s c r e a t e d by two a n a l a g o u s r i n g v o r x i c e s . E x p e r i m e n t a l l y , t h e h i g h u n i f o r m i t y o f t h e m a g n e t i c f i e l d o v e r a c o r e a r e a (when t h e c o i l s a r e s u i t a b l y a r r a n g e d ) i s w e l l known. E s s e n t i a l l y t h e same u n i f o r m i t y i n t h r o a t s p e e d w i l l o c c u r i n t h e c a s e o f r e a l a i r f l o w s , p r o v i d e d t h e n o r m a l l y f a v o u r a b l e p r e s s u r e g r a d i e n t a l o n g t h e c o n t r a c t i n g s u r f a c e i s n o t d i s t u r b e d . A p r e c i s e a p p l i c a t i o n w o u l d r e q u i r e t h a t t h e c o n t r a c t i n g s u r f a c e s be s w o l l e n by t h e b o u n d a r y l a y e r d i s p l a c e m e n t t h i c k n e s s . I n p r a c t i c e , , s u c h b o u n d a r y l a y e r s , a r e q u i t e / t h i n a n d t h e t h i c k n e s s i n c r e a s e s s l o w l y . N o r m a l l y , no a p p r e c i a b l e e r r o r i s made i n n e g l e c t i n g t h e b o u n d a r y l a y e r d i s p l a c e m e n t t h i c k n e s s . L e t T be t h e s t r e n g t h o f a r i n g v o r t e x o f r i n g r a d i u s ' a 1 , w i t h t h e p l a n e o f t h e v o r t e x r i n g n o r m a l t o t h e z - a x i s ( F i g u r e A5.1), The v o r t e x r i n g i s c e n t e r e d on t h e z - a x i s , a d i s t a n c e ' b' f r o m t h e o r i g i n . The ( a x i s y m m e t r i c ) s t r e a m f u n c t i o n i s t h e n - , . . F r a i M r , z ) • = -2pr 2TT COS8 _, - , - r- ... d0 , (A5- 1) 0 PQ where 100 PQ 2 = ( z - b ) 2 + a 2 + r 2 - 2ar cos9 (A5.2) = ( ( z - b ) 2 + ( a + r ) 2 ) ( l - k 2 c o s 2 ( f ) } , 4ar ( ( z - b ) 2 + ( a + r ) 2 ) ' (AS: 3) Then ( r , z) ar 4 V ( (z-b) 2+(a+r) 2) -4 • (1,-2) k 2 j 0 2TT / ( l - k 2 c o s 2 a ) da da 0 / ( l - k 2 c o s 2 a ) (A5.4) aF/r r k SE/_(< I-_>*-"> where a = - (A5.5) N o n - d i m e n s i o n a l i z i n g , l e t Z = f , R = f , B = K i> c l c l - c l _n_ ar ' (A5.6) and l e t F(k) = (1-| 2)K - E) , (A5.7) w i t h 1 0 1 ;2 = iR . ( A 5 _ 8 ) ( Z - B ) 2 + ( 1 + R ) 2 Tnen $(R,Z) = /R F ( k ) , (A5.9) where K and E are the complete e l l i p t i c i n t e g r a l s of the f i r s t and second k i n d , r e s p e c t i v e l y . To generate the uniform flow a t the e x i t of the c o n t r a c t i n g s e c t i o n , the s t r e a m l i n e s must be p a r a l l e l t h e r e . Hence an i d e n t i c a l r i n g v o r tex must be cen t e r e d at (0,0,-b). The r e s u l t i n g stream f u n c t i o n i s t h e r e f o r e u>(R,Z) = /R (F (k J ) + F ( k 2 ) ) , (AS. 1 0 ) w h e r e i 2 4R , 2 4 R , , c „ k i =• , k 2 = . ( A 5 . 1 1 ) ( Z - B ) 2 + ( 1 + R ) 2 ( Z + B ) 2 + ( 1 + R ) 2 Reference [ 2 9 ] s t a t e s t h a t when b i s 0.46936a the v e l o c i t y d i s t r i b u t i o n over a flow core area of r a d i u s 0.42241a i n the median plane between the vortex r i n g s w i l l be uniform to w i t h i n 1 p a r t i n 500. The e l l i p t i c i n t e g r a l s can be e v a l u a t e d simply from polynomial approximations 17.3.34 and 17.3.36 of [ 4 1 ] . A program was w r i t t e n to search f o r the value of R which, f o r a gi v e n 1 0 2 value of Z gives the same value of ty as through a s t a r t i n g point ( E 0 , Z o ) - In t h i s way the coordinates (R,Z) of the stream surface were generated, using a value of ty of 0.10510. R 0 i s chosen to be 0.30 (see Figure 6 of [40]) to o b t a i n a thr o a t f l o w u n i f o r m i t y w i t h i n 0.23. Thus ro i s 0.30a. The req u i r e d t e s t s e c t i o n entrance area i s 0. 34 8m2. Hence ur2} i s 0.348m2. Therefore r 0 i s 333mm and 'a' i s 1110mm. The nozz l e l e n g t h i s chosen t o be 1.52m, due to p h y s i c a l r e s t r i c t i o n s i n the e x i s t i n g converging s e c t i o n , hence the nozzle entrance area Tix^ i s f i x e d at 1.068m2. Hence r , i s 583mm, or 1.751r 0 or 0.525a. Therefore S i i s 0.525, z i i s 4.53ro or 1.37a. Hence Z i i s 1.37. The t a b l e of nozzle coordinates f o l l o w s . Z , z, R, r are as above, and w and h are the e x i s t i n g width of the c o n t r a c t i n g s e c t i o n and height of the new nozzle i n s e r t r e s p e c t i v e l y . A~7T r 2 z . z R r = w h w h 0.0 0 mm .3 00 333mm 0.34 84m2 9 1 4mm 381mm 0. 1 111 .300 3 33 .3484 915 381 0.2 222 .300 333 . 3484 916 381 0. 3 333 .301 334 . 3507 917 383 0.4 444 .304 337 . 3575 920 389 0.5 555 .309 343 .3706 927 400 0.6 666 .318 353 . 3924 934 421 0.7 777 .331 367 . 4248 942 452 0. 8 888 .343 386 ' . 4700 952 • 496 0.9 999 .370 411 .5298 962 554 1.0 1100 .395 440 .6065 979 626 1.1 1221 .4 26 473 .7023 1002 7 10 1.2 1332 .460 511 .8199 103 6 803 1.3 1443 .498 55 3 .9617 1082 902 1.37 1524 .525 583 1.068 1 123 951 103 AH An.aly_t.ic R e p r e s e n t a t i o n of a L i f t i n g Vortex between a S o l i d a XEiLHSversexy-Slgtted and a Constant P r e s s u r e Boundary.. 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 two-dimensional p o t e n t i a l flow "method of images" model f o r a l i f t i n g a i r f o i l between a s o l i d lower boundary and a t r a n s v e r s e l y - s l o t t e d upper boundary c o n s i s t i n g of a i r f o i l - s h a p e d s l a t s . A c o n s t a n t p r e s s u r e boundary o u t s i d e the s l o t t e d w a l l r e p r e s e n t s a f r e e s t r e a m l i n e t h a t " d i v i d e s the t e s t s e c t i o n flow from the plenum flow. The a i r f o i l ana the w a l l s l a t s are r e p r e s e n t e d by p o i n t v o r t i c e s which are "imaged" a p p r o p r i a t e l y . The image of a vortex, i n a s o l i d boundary i s a vortex of equal but o p p o s i t e c i r c u l a t i o n ; the image i n a constant pressure boundary i s a v o r t e x of i d e n t i c a l c i r c u l a t i o n . From [ 4 1 ] , the complex p o t e n t i a l f o r an i n f i n i t e v e r t i c a l row of p o i n t v o r t i c e s of the same s i g n , spaced a d i s t a n c e 'd' a p a r t , i s where the " c e n t r a l " vortex i s at zo, and the s t r e n g t h K i s r e l a t e d to the c i r c u l a t i o n r ( p o s i t i v e c l o c k w i s e ) by with r e f e r e n c e to F i g u r e s A6.1, A6.2, the image system f o r F(z) = Klog s i n h - r ( z - z 0 ) ( A 6 . 1) ( A 6 . 2 ) a s i n g l e vortex between the s o l i d and c o n s t a n t p r e s s u r e 104 b o u n d a r i e s i s t h e sura o f f o u r s e t s o f i m a g e s . U s i n g t h e n o t a t i o n o f F i g u r e A6.2, two s e t s a r e o f p o s i t i v e c i r c u l a t i o n , " c e n t e r e d " w i t h z 0 v a l u e s o f a i and ( a + 2 b ) i . The o t h e r two s e t s a r e o f n e g a t i v e c i r c u l a t i o n , " c e n t e r e d " a t - a i and - (a + 2b) i . A l l f o u r s e t s have t h e same s p a c i n g , 4 (a + b) . The c o m p l e t e s y s t e m of images f o r t h e s i n g l e v o r t e x immersed i n a u n i f o r m flow U (from l e f t t o r i g h t ) h as t h e complex p o t e n t i a l F(z) = Uz + K l o g s i n h A ( z - a i ) + K l o g sinhA(z-(a+2b)i) ( A 6 . 3 ) - K l o g sinhA(z+ai) — K l o g sinhA(z+ (a+2b) i ) ., where A = irS+bT • ( A 6* 4 ) The complex v e l o c i t y w(z) i s the d e r i v a t i v e o f F(z) with r e s p e c t t o z. Hence f o r the s i n g l e v o r t e x , w(z) = U + KA(cothA(z-ia) + cothA(z-i(a+2b)) (A6. 5) - cothA(z+ia) - cot h A ( z + i (a+2b) )) ». To c a l c u l a t e t h e f o r c e on the v o r t e x r e p r e s e n t i n g the a i r f o i l , due t o t h e e f f e c t o f t h e two b o u n d a r i e s , the B l a s i u s r e l a t i o n {[41]) i s u s e d , 105 D - i L = w2 (z) dz, (A6.6) where D and L a r e the f o r c e s i n t h e X and Y d i r e c t i o n s . , r e s p e c t i v e l y . I n t e g r a t i o n i s performed about a c o n t o u r e n c l o s i n g o n l y the a i r f o i l v o r t e x . To e v a l u a t e t h i s i n t e g r a l u s i n g r e s i d u e s , w i t h the. a i r f o i l v o r t e x a t i h , t h e c o e f f i c i e n t i n the L a u r e n t s e r i e s e x p a n s i o n o f w 2 ( z ) of the term i n 1 / ( z - i h ) i s r e q u i r e d . The L a u r e n t s e r i e s e x p a n s i o n f o r the c o t h f u n c t i o n about i h i s _„ / -„\ 1 . ( z - i h ) z - i h , ,,r n s coth(z-xh) - r—- + - . — - - - — j - = — - + ... (A6.7) z-xh 3 45 The r e q u i r e d c o e f f i c i e n t i s 2UK + 2 K 2 A ( c o t h A ( z - i ( a + 2 b ) ) - c o t h A ( z + i a ) - c o t h A ( z + i ( a + 2 b ) ) ) . . (A6.8) The r e s i d u e a t i h i s 2UK + 2 K 2 A i ( + c o t ( 2 A b ) + c o t ( 2 A a ) + c o t ( 2 A ( a + b ) ) ) , (A6.9) s i n c e c o t h ( i z ) = - i c o t (z) . (A6. 10) T h e r e f o r e \ • D - i l . = ( ^ ) 2 i r i (Residue (ih) ) . (Ab.11) S u b s t i t u t i n g f o r K i n terms of. r , D i s z e r o , and 106 L - L 0 ( l - j j ^ c s c ( k a ) c s c ( k b ) ) , (A6.12) w here k - 2TaTbT ' ( 4 6 - 1 3 ) ana L 0 = pur (A5. 14) i s the t u n n e l l i f t . T h i s i s the e x p r e s s i o n f o r the r e d u c t i o n i n l i f t e x perienced by a s i n g l e p o i n t vortex between a s o l i d lower boundary and a c o n s t a n t pressure upper boundary. I f t h e v o r t e x i s midway between the two boundaries, Lo = 1 ~ 4UH ' < A 6* 1 5> Using (A6.14) and C T = T — — , (A6.16) ° | P U 2 c where •c' i s the a i r f o i l chord, Now c o n s i d e r the v o r t i c e s Y r e p r e s e n t i n g the a i r f o i l - n 1 0 7 s h a p e d w a l l s l a t s . From e q u a t i o n s ( A 8 . 4 ) , (A6.5) u s i n g . t h e n o t a t i o n o f F i g u r e A 6 . 1 , CO w ( z ) = J k B ( c o t h B ( z - 2 i h - n r ) + c o t h B (z-2i(h+£ ) - n r ) ^ n n ^ n n n n = - o o (A6.-18) - c o t h B ( z + 2 i h - n r ) - c o t h B ( z + 2 i ( h + e ) - n r ) 1 n n n. J i s t h e a d d i t i o n a l c o m p l e x v e l o c i t y due t o t h e f o u r s e t s o f i m a g e s c o r r e s p o n d i n g t o 'n* i n f i n i t e v e r t i c a l r o w s o f p o i n t v o r t i c e s y s p a c e d a h o r i z o n t a l d i s t a n c e ' r ' a p a r t . H e r e k = , B = TTOT-I r . (A6.19) n 2TT n 4 (2h+e ) n and * r ' i s r e l a t e d t o t h e s l o t t e d w a l l o p e n - a r e a r a t i o . H ence t h e c o m p l e t e c o m p l e x v e l o c i t y f i e l d f o r t h e v o r t e x r r e p r e s e n t i n g t h e t e s t a i r f o i l , t h e v o r t i c e s y r e p r e s e n t i n g t h e w a l l s l a t s , a n d t h e u n i f o r m f l o w U, n i s . w ( z ) - U + K A ( c o t h A ( z - i h ) + c o t h A ( z - i ( 3 h + 2 6 ) ) - cothA(z+ih) - cothA(z+i(3h+26))) ( A 6 . 2 0 ) CO + Y ' k B (cothB (z-2ih-nr) + cothB (z-2i(h + e )-nr) L n n v n n n n=-°° - c o t h B ( z + 2 i h - n r ) - c o t h B (z + 2 i (h+e .) - n r ) ) n n n } where 6 and e a r e t h e d i s t a n c e o f r a n d y r e s p e c t i v e l y f r o m 108 the constant pressure boundary. ' F o l l o w i n g the p r e v i o u s procedure to c a l c u l a t e the c o e f f i c i e n t of. 1 / ( z - i h ) i n the Laurent s e r i e s expansion f o r w2 ( z ) , the r e q u i r e d c o e f f i c i e n t i s , using (A6.9), 2UK + 2K 2Ai(cot(2A(h+6)) + cot(2Ah) + cot(2A(2h+6))) + 2K T k B (cothB -xh-nr) + cothB -x(h+2e )-nr) L n n A n n n n = - c o (A6.21) - cothB (+3ih-nr) - cothB (+i(3h+2e ) - n r ) K n n n . wnere A = T(2h+6T* ( A 6 * 2 2 > Mow using coth(x+iy) = c o t h ( x ) c s c 2 ( y ) - i c s c h 2 ( x ) c o t ( y ) ( A 6 . 2 3 ) c o t h 2 ( x ) + c o t 2 ( y ) t o c a l c u l a t e the r e s i d u e of w2 (z) at i h , and u s i n g (A6.12) and (A6.10) , " " L = pur - £|^-(cot (Bh) + cot(8(h+S))) - •— y y B c s c h 2 ( B n r ) F ( B ,h,e , n , r ) , (A6.24) 2TT L 1 n n n n n n=-°° and 109 D pr Y y B coth(B nr)E(B ,h,_ , n , r ) , (A6.25) " rt y~\ Y\ n n 2TT ^ 'n n n=-°° n where TT 2(2h+S)' B n 4(2h+£ ) ' n ( A 6 . 2 6 ) E(B ,h,e ,n,r) .= G(B h) + G(B (h+2e )) n n n n n + G(3B_h) + G(B y.(3h+2e_)) , n n n F(B ,h,e ,n,r) = H(B h) + H (B (h+2 e )) n n n n n H(3B h) - H(B (3h+2 e )) n n n In the above e x p r e s s i o n s (A6.27) (A6.28) G(u) = esc (u) c o t h 2 ( B n r ) + c o t 2 ( u ) n (A6.29) and H(u) = c o t (u) c o t h 2 ( B n r ) + c o t 2 ( u ) n (A6.30) When a l l o f the y are of i d e n t i c a l s t r e n g t h , and a l l o f the e ( d i s t a n c e o f Y from the con s t a n t p r e s s u r e boundary) a r e n n 110 e q u a l , t h e .drag f o r c e D i s z e r o , s i n c e c o t h (B_nr) i s an o d d f u n c t i o n of n . N u m e r i c a l l y ' i t i s s a t i s f a c t o r y t o t a k e n g r e a t e r t h a n 10, t h a t i s , 21 or more v o r t i c e s y . F o r t h e c a s e o f a f l a t p l a t e a t 26.3° i n c i d e n c e , and average v a l u e s o f y £ t a k e n f r o m s t r e a m l i n e , c a l c u l a t i o n s f o r a 0 . 6 6 - C l a r k - i a i r f o i l a t . 20° i n c i d e n c e i n t h e p r e s e n c e o f a 70%OAR.TSOSL w a l l c o n f i g u r a t i o n , the v a l u e s of the e x p r e s s i o n s a r e . 0 6 2 5 , 1 = 0 . 1 , £ = 0.67, | = 0.67. (16.31) H ence ~ - 1 " ( 0 . 1 6 8 ) ( I ) - ( - 0 . 3 1 7 ) ) c u c s = 1 - 0. 235 - 0. 020 = 0. 745 , (A6.32) an d t h e d r a g f o r c e D i s z e r o . Hence t h e e f f e c t o f t h e w a l l s l a t c i r c u l a t i o n y on t h e t e s t a i r f o i l i s s m a l l c o m p a r e d t o t h e e f f e c t o f t h e c o n s t a n t p r e s s u r e b o u n d a r y on t h e t e s t . a i r f o i l . A n o t h e r p o s s i b l e a n a l y t i c r e p r e s e n t a t i o n o f t h e b o u n d i n g s h e a r l a y e r , i s a v o r t e x s h e e t , a c r o s s which t h e r e i s a jump i n t a n g e n t i a l v e l o c i t y and t o t a l head. T h i s was n o t a t t e m p t e d h e r e . r U, = 1.40, I l l A p p e n d i x 7. S t a n d a r d S o l i d W a l l C o r r e c t i o n s The f o l l o w i n g s e v e n e x p r e s s i o n s a r e r e p r o d u c e d f r o m p a g e 382 o f [ 3 4 ] , a n d a r e t h e e x p r e s s i o n s u s e d f o r t h e c a l c u l a t i o n s h e r e i n . T h e y a r e w r i t t e n f o r t h e i n c i d e n c e c o r r e c t i o n a p p l i e d a s a n e q u i v a l e n t c h a n g e i n l i f t . The c o r r e c t i o n s a r e t o b e a p p l i e d a t t h e m e a s u r e d i n c i d e n c e . The s u b s c r i p t s T a n d F i m p l y m e a s u r e d a n d e q u i v a l e n t f r e e - a i r v a l u e s r e s p e c t i v e l y . The f i r s t f i v e e x p r e s s i o n s a r e , r e s p e c t i v e l y , t h e c o r r e c - t i o n s t o be a d d e d ( r e g a r d l e s s o f s i g n ) t o t h e m e a s u r e d v a l u e s o f w i n d s p e e d , i n c i d e n c e a n d l i f t , q u a r t e r c h o r d p i t c h i n g moment, a n d d r a g c o e f f i c i e n t s . AU = U F - U T = e U T (A7.1) Aa = a_ - a_ = 0. (A7.2) A C L " = - 2 E V K V 2 ¥ ( V 4 C M C T > £ ? " <"- 3> A C H C = C M = "Sic = - 2 e C M c + ! V ^ ( W ' ^ f ' ( A 7 - 4 ) 4 4 F 4 T 4 T 4 T A C D = C D - CD T = - 2 £ C D T + 27(CL + 4 C M c > ( C L ~ If D> ' <A7-5>" - T T I n t h e a b o v e e q u a t i o n s , e i s c o m p o s e d o f t h e c o r r e c t i o n s 112 f o r wake and s o l i d blockage, and i s g i v e n by 1 rc^ e = f f e ) C D _ + A K ' (A7.6) where K = ^ ( | ) \ (A7.7) and c/H i s the model s i z e . The q u a n t i t y A i s o b t a i n e d from F i g . 6 . 8 of [ 3 9 ] . In p r a c t i c e , the a - d e r i v a t i v e v a l u e s are determined g r a p h i c a l l y . Otherwise the f o l l o w i n g v a l u e s may be used: = 27T, = 0 , _ M ^ = 0 , ^ = . 2 . ( A 7 . 8 ) The f o l l o w i n g c o r r e c t i o n i s a p p l i e d t o the measured p r e s s u r e d i s t r i b u t i o n s (at the measured i n c i d e n c e ct^) : 113 _E___S__ __ _ _______ _______ _ _ _ _ _ i _ _ _ 2 _ __________ ____ ___ ________ L i f t . In t h i s method, the usual Kutta c o n d i t i o n , of equal v e l o c i t i e s on the upper and lower surfaces of a l i f t i n g body, adjacent to the t r a i l i n g edge i s abandoned. The c i r c u l a t i o n i s to be determined from the measured l i f t . To s i m p l i f y the equations, consider a s i n g l e t e s t a i r f o i l between s o l i d w a l l s . The a i r f o i l i s represented by N c o n t r o l p o i n t s , and the two plane s o l i d walls by M-N a d d i t i o n a l c o n t r o l p o i n t s . Thus the t o t a l number of c o n t r o l p o i n t s i s li. There are t h e r e f o r e Pi. unknown source s t r e n g t h d e n s i t i e s o and a s i n g l e unknown vortex stre n g t h d e n s i t y y on the a i r f o i l . Thus H+1 equations are r e q u i r e d to determine the M+1 unknowns. There are M c o n t r o l points at which the norma 1 - v e l o c i t y boundary c o n d i t i o n must be s a t i s f i e d ; these y i e l d M equations. In the n o t a t i o n of §3.2, the t a n g e n t i a l v e l o c i t y a t a p o i n t • i ' i s The (M+1)st equation u s u a l l y c o n t a i n s the Kutta c o n d i t i o n at the . a i r f o i l t r a i l i n g edge. The usual Kutta c o n d i t i o n i s expressed as M N ( A 8 . 1) = - V (A8. 2) x i + 1 114 w h i c h r e s u l t s i n t h e e q u a t i o n M N | 1 ( B j i + B j i + l ^ 0 j . + . Y k I 1 ( A k i + A k i + l ) = - U ( c o s 0 i + c o s 9 i + 1 ) ( A 8 . 3 ) f o r t h e c o n t r o l p o i n t s ' i ' and ' i + 1 ' a d j a c e n t t o t h e t r a i l i n g e d g e . The r e s u l t i n g f u l l K u t t a c i r c u l a t i o n ?o a b o u t t h e a i r f o i l c a n be c a l c u l a t e d f r o m t h e d e f i n i t i o n N = 4 V-d£ = I V As. J C i = l \ 1 (A8.h) To d e t e r m i n e t h e c i r c u l a t i o n r f r o m t h e m e a s u r e d l i f t , t h e K u t t a - J o u k o w s k y l a w e x p r e s s e s t h e m e a s u r e d l i f t f o r c e , L, o n t h e a i r f o i l a s L = PUP . (A8.5) The m e a s u r e d l i f t c o e f f i c i e n t C L i s d e f i n e d a s C T = T - V " ' U 8 - 6 ) p U 2 c 2 where ' c 1 i s t h e a i r f o i l c h o r d . T h e r e f o r e T = f U c C L - (A8.7) The l e f t . s i d e o f e q u a t i o n (A8-7) i s t h e r e f o r e w r i t t e n N N M N I V As. = I I B a + Y I A + UcosG I As. , i = l t i 1 i = l A j = l 3 1 3 k = l K l ^ 1 (A8.3) and (A3. 7) become; M C N r N r N ^ y B . . A S . • i 3 1 x a . + 3 I l i = l 2  A k i As . l J (A3.9) = -U N J c o s 6 . A s . i = l 1 1 + ^ C C L The l a s t row o f t h e m a t r i x C i n t h e s y s t e m o f e q u a t i o n s C ( o, y)-d, w i l l now be N 'j ,M+1 = I B ± A s i , j = l , 2 , . . . M , (A8.10) i = l N r N = y y A 'M+1, M+1 . L , L -. k i ' i = l <-k=l As. = c o n s t a n t , l ( A8. 1 1) and t h e l a s t c o m p o n e n t o f t h e r i g h t h a n d s i d e v e c t o r w i l l be M+1 -U N J c o s O . A s . i i i 1 x + | U C C L (A8. 12) The e x p r e s s i o n s ( A 8 . 1 0 ) , (A8. 11) and (A8.12) r e p l a c e t h e c o r r e s p o n d i n g e x p r e s s i o n s i n (4.2) and ( 4 . 3 ) . 116 ________ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i_______2_ __________ k i _ 2 _ _ _ _ i _ _ _ _ _ P r o f i l e F o r a t w o - d i m e n s i o n a l a i r f o i l p r o f i l e w h i c h c a n b e mapped c o n f o r m a l l y • o n t o a. c i r c l e o f r a d i u s R, t h e f u l l K u r t a c i r c u l a t i o n T 0 i s [ 3 7 ] (A9 1) F o = 4TTRU s i n ( a - c t o ) , where U, a and a 0 a r e t h e f l o w s p e e d , i n c i d e n c e , a n d z e r o - l i f t a n g l e r e s p e c t i v e l y . I f t h e c i r c u l a t i o n i s r e d u c e d t o a v a l u e k r 0 , t h e n T = k f 0 = 4 T r R U k s i n ( a - a 0 ) •= 4 T r R U s i n ( a - a 0 - A a ) . (A9.2) Hence Aa = ( a - a 0 ) - a r c s i n ( k s i n ( a - a 0 ) ) (A.9.3). i s t h e e f f e c t i v e r e d u c t i o n i n i n c i d e n c e r e q u i r e d t o r e d u c e t h e c i r c u l a t i o n (and h e n c e t h e l i f t ) t o t h e f r a c t i o n »k» o f t h e f u l l K u t t a v a l u e . I n ' o r d e r t o a c h i e v e a' r e d u c t i o n i n t h e " e f f e c t i v e c a m b e r , t h e p r o f i l e s h a p e i s m o d i f i e d b y " r a i s i n g " t h e t r a i l i n g - e d g e . P o i n t s on t h e p r o f i l e a r e r o t a t e d a b o u t t h e p r o f i l e l e a d i n g e d g e .through an a n g l e w h i c h i s p r o p o r t i o n a l t o t h e d i s t a n c e f r o m t h e l e a d i n g e d g e . T h e d i r e c t i o n o f r o t a t i o n i s s u c h t o r e d u c e t h e 117 e f f e c t i v e i n c i d e n c e o f t h e p r o f i l e . F o r an o r i g i n o f p r o f i l e c o o r d i n a t e s ( x,y) a t raid-chord, and f l o w f r o m l e f t t o r i g h t , t h e e x p r e s s i o n 0(x) = ^ - ( 1 + ~ ) , ( A 9 . 4 ) where * c ' i s t h e a i r f o i l c h o r d , w i l l a s s i g n a z e r o r o t a t i o n t o t h e p o i n t a t t h e l e a d i n g edge and t h e f u l l r o t a t i o n Act t o t h e p o i n t a t t h e t r a i l i n g e d g e . The m o d i f i e d p r o f i l e c o o r d i n a t e s a r e t h u s x' = xcos0(x) - y s i n 6 ( x ) (A9.5) yy = ycos9(x) + x s i n 9 ( x ) . The e f f e c t i v e r e d u c t i o n i n camber i s r o u g h l y p r o p o r t i o n a l t o . t h e amount o f l i f t ( o r c i r c u l a t i o n ) b e i n g d e v e l o p e d . The z e r o - l i f t a n g l e a 0 f o r t h e p r o f i l e w i l l n o t c h a n g e , s i n c e when a e q u a l s a 0 , A a w i l l be z e r o . 118 A p p e n d i x 10. The C o m p u t e r P r o g r a m f o r t h e E x a c t N u m e r i c a l T h e o r y The p r o g r a m c o n t a i n s a s u b r o u t i n e MAIN1, w h i c h c a l l s a l l o f t h e f o l l o w i n g s u b r o u t i n e s : CALCAB, ASSEMA, ASSEMB, ASSEMD, CPS, FORCES, a n d MODPRO. I n a d d i t i o n , s u b r o u t i n e s RE, WR, a n d WRD a r e r e q u i r e d f o r r e a d i n g a n d w r i t i n g m a t r i c i e s f r o m / i n t o p e r i p h e r a l s t o r a g e . The s y s t e m o f e q u a t i o n s i s s o l v e d b y t h e s u b r o u t i n e ATXB, d e s c r i b e d i n A p p e n d i x 3. The U.B.C. ( s y s t e m d e p e n d e n t ) s u b r o u t i n e s GSPACE a n d FSPACE a l l o c a t e a n d d e a l l o - c a t e , r e s p e c t i v e l y , b l o c k s o f r e a l memory r e q u i r e d f o r t h e m a t r i c i e s A, B, a n d C. The U.B.C. s u b r o u t i n e CALLER i s u s e d t o c a l l s u b r o u t i n e s w h i c h u s e m a t r i c e s t h a t h a v e memory . a l l o c a t e d b y GSPACE. A s u b r o u t i n e WALLCO i s u s e d t o c r e a t e t h e c o n t r o l . p o i n t c o o r d i n a t e s f o r a l l t e s t a i r f o i l s , f l a p s , s o l i d w a l l s e c t i o n s a n d w a l l s l a t s . The c o n t r o l p o i n t c o o r d i n a t e s XX and, YY, a l o n g w i t h DX, DY, DS, CS, a n d S I , a r e w r i t t e n i n t o p e r i p h e r a l s t o r a g e s o t h a t a l l c o o r d i n a t e s may be c h e c k e d b e f o r e f u r t h e r c a l c u l a t i o n . The d e f i n i t i o n s o f t h e v a r i a b l e s u s e d a r e d e s c r i b e d b y comment s t a t e m e n t s w i t h i n t h e s u b r o u t i n e s . The c o n t r o l p o i n t s (XSOLSL,YSOLSL) f o r a n a r b i t r a r y s o l i d s u r f a c e s u c h a s t h e p l e n u m b o u n d a r y a r e r e a d i n a t e x e c u t i o n 119 time by the program which c a l l s WALLCO. The c o n t r o l p o i n t c o o r d i n a t e s (XM,YM), and the v e l o c i t y VTI, on the s t r e a m l i n e r e p r e s e n t i n g the shear l a y e r a r e r e a d i n by the program which c a l l s MAIN1. T h i s program a l s o reads i n the c o o r d i n a t e s of the s l a t l e a d i n g edges (XLE), c e n t e r s ((XCENT,YCENT))., and t r a i l i n g edges (XTE), the flow angle t h a t each s l a t sees (ALF), and the f r a c t i o n o f the f u l l c i r c u l a t i o n (Kl) r e q u i r e d . As shown, t h i s program handles o n l y a s i n g l e t e s t a i r f o i l ; a s i m i l a r program i s used f o r a f l a p p e d a i r f o i l . The n o t a t i o n f o r the enumeration of the c o n t r o l p o i n t s i s shown i n F i g u r e A10.1. The l a y o u t o f the system of e q u a t i o n s to be s o l v e d i s shown i n Tab l e A10.1; the numbers i n p a r e n - . theses i n d i c a t e the p a r t i c u l a r DC—loop i n the program which assembles the c o r r e s p o n d i n g c o e f f i c i e n t s f o r the unknowns i n the e q u a t i o n s . The e q u a t i o n numbers (rows i n the m a t r i x C) a r e i n d i c a t e d by 'E*. Ac r o s s the bottom of T a b l e A10.1, the range of the index o f summation ( j , k, p o r q) f o r each column o f the m a t r i x C i s s p e c i f i e d . A complete sample run w i t h the r e q u i r e d c a l l i n g programs f o l l o w s . The sample i s shown f o r the C l a r k - Y a i r f o i l a t 20° 120 i n c i d e n c e , i n a 70% OAR TSUSL w a l l c o n f i g u r a t i o n . The shear l a y e r i s modelled by a s t r e a m l i n e which i s r e p r e s e n t e d by 20 c o n t r o l p o i n t s , t h a t i s 20 source and 20 v o r t e x elements. The a i r f o i l i s r e p r e s e n t e d by 50, the upper s o l i d e n d - w a l l s e c t i o n s each by 20, and the s o l i d lower w a l l by 80 c o n t r o l p o i n t s . There are 8 a i r f o i l - s h a p e d s l a t s , each r e p r e s e n t e d by 9 c o n t r o l p o i n t s . Thus t h e r e a re 262 c o n t r o l p o i n t s where the zero n o r m a l - v e l o c i t y boundary c o n d i t i o n i s s p e c i f i e d , and 20 c o n t r o l p o i n t s where the t a n g e n t i a l v e l o c i t i e s a r e p r e s c r i b e d . T h i s l e a d s t o 262 unknown source s t r e n g t h d e n s i t i e s and 29 unknown v o r t e x s t r e n g t h d e n s i t i e s . The r e s u l t i s 291 e q u a t i o n s i n 291 unknowns. The v e l o c i t y d i s t r i b u t i o n on t h i s s t r e a m l i n e r e p r e s e n t i n g the shear l a y e r i s s p e c i f i e d , and corresponds t o a p r e s s u r e c o e f f i c i e n t w i t h the average v a l u e o f -0.35, and i s s i m i l a r t o the mean v a r i a t i o n o f p r e s s u r e shown i n F i g u r e 8.7. The p r o f i l e s o f the w a l l s l a t s a re m o d i f i e d t o reduce t h e i r c i r c u l a t i o n t o 0.8 times t h e i r f u l l c i r c u l a t i o n . The output from the program.includes the p r e s s u r e . d i s t r i b u t i o n s on the w a l l s , the w a l l s l a t s , and the a i r f o i l . A l s o p r i n t e d are the l i f t , drag, and p i t c h i n g moment c o e f f i c i e n t s f o r the a i r f o i l and the w a l l s l a t s . MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) MAIN 10-22-75 12159113 PAGE P001 0001 000? 0003 OO01 0005 0006 0007 0008 0009 0010 00 11 0012 0013 0011 0015 0016 0017 0018 PC19 0030 0021 0022 0023 0021 0025 0026 0027 0028 0029 0030 0031 0032 0033 0031 0035 0036 0037 REAL XX(262),YY(262),DX(262),DY(262),DS(262),CSC262),SI(262) REAL SlG(291),VTT(262),CP(262),GAM(9),MU(20),GNU<20) REAL VTI(20),XM(21),YM(21) REAL AXX(2a2),AYY(242),ADXC2«2),A0Y(24?),AD3C242),ACS(242) REAL ASK2H2) C XM.YM •. PROFILE COORDS FOR STREAMLINE REPRESENTING SHEAR LAYER C XX,YY - CONTROL POINT COORD3> DX.OY.OS - ELEMENT LENGTHS C SIG - SOURCE STRENGTH OENSITlES (ALSO USED AS SOLUTION VECTOR IN C SYSTEM ) C (GAM,MU,GNU APE PART OF SIG) C VTT.CP - TANG VEL, PRESSURE COEFF'. REAL XQ(10),YQ<10),XH(10),YR(10) C XQ,YG,XR,YR - MODIFIED SLAT PROFILE COORDS REAL XCENT(fl),XLE(8),XTE(8),ALF(8 ) ,KH6) REAL YCcNT(ft),DTHICK(8) C AXX,AYY ETC - CONTROL PTS TO BE READ IN FROM FILE (PUT THERE BY C WALLCO) EQUIVALENCE (AXX,XX),(AYY,YY),(ADX,DX),CADY,DY),(ADS,OS) EQUIVALENCE (AC3.CS),(ASI.SI) INTEGER CALAH,CALCD,WRAR,WRCD,SDLV,GAUSS,ITER,CALCP,C»LCL,HSIG C O M M O N / e i / NW5,NSLAT,N3U1,NKA,NM2,MS V,NA,NSPS.NTEU,NTEL C O M M 0 M / A 2 / U,CH C0HM0M/B3/NUl,NWuT,NU3,NWU2,NLl,NKLl,NL3,NWL2iNS0Ll,NS0LSL,NP1, 1 NFLAT,N3PF,Ni1 C0MM0H/H4/CALAB,CALCD,WRAB,wRCD,S0l.V,GAU3S,ITER,CALCP>CALCL,HSIG C0Mn0H/n5> NPi,MP2,NL« REAL TITLE(20) REAI)C5,303) TITLE 303 FORMAT C20A/I) WRITEr6,304) 301 FORMAT(IH1) WRITE(h,305) TITLE 305 FOR MAT(iX,20A4) RE AD(5,25) CALAB,CALCD,WRAB,WRCO,SOLV,GAUSS,ITER,CALCP, CALCL » HSI G 25 Fn.7.MAT(20I«) WRITE(6,1) CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITCR,CALCP,CALCL,H3IG 1 FORMAT ( 'CALABr I ,I2,2X, ICALCO" 1 , 12, 2X, 'WRABa M 2 . 2 X , 'WRCO=M2,2X, { 'SOLVr',12,2x,'GAUSS=',I2.2X,'ITERai,I2,2X,'CALCP',I2,2X, 2 "CALCL"',12.2X,'HSIG=',12) RE A.i (5.30) NA,NWUl,NHU2,NwLi,NWL2,NS0LSL,NFLAT,NSPF,NSLAT,NSPS, t MSV 30 FORMAT(20I4) WRITE(6,31) NA,NWUl,NWU2,NKLl,NWL2iNS0LSL.NFLAT,NSPF,NSLAT,NSPS» 1 MSV 31 FORMAT<'NAs<,13,2X,•NWU1a•,13,2X,1NwU2a',13,2X,'NWL1a 1,13,2X, 1 INWL2=',IS,2X,'NSOLSLs',I3,2X,•NFLATt>,IS,2X,INSPFs',13,2X, 2 INSLAT=',I3,2X,iNSPSai,I3,2X,iMSVat,13) READ(5,32) NfEU,NTEL,CH,U 32 F0RKAT(2I1,2Ffl'.3) WRITE(6,400) NTEU,NTEL,CH,U 400 FORMAT ('NTEU= I ,I3,2X, 'NTELa' ,I3,2X, • CH= ', F8'. 3, 2X, 'U= ' ,F6,1) C READ COCROS FOR AIRFOIL, WALL & WALL SLATS FROM WALLCO FILE (WALLS & C SLATS HAVE YY = 0'.) READ(2) AXX,AYY,ADX,AOY,AOS,ACS,ASI YWal'8'. . 1.000 2.000 3.000 4.000 5,000 6.000 7.000 8,000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 15.000 16.000 17.000 18,000 19.000 20.000 21.0o0 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 12.000 43,000 44,000 45.000 46.000 47,000 48,000 49,000 50.000 51.000 52.000 • 53.000 MICHIGAN TERMINAL SYSTEM FORTRAN G(H336) MAIN I0-22-73 003B 0039 0010 OOH' COM2 0013 0011 eo-'i5 0016 0017 oo ie 0019 0050 0051 0052 0053 0051 0055 0056 0057 0058 0059 ooto 0061 0062 0063 0061 0065 0066 0067 0068 0 06? 0070 0071 0072 0073 0071 0075 101 33 31 35 C NUleNA+J NU3 = NuUNWUl MU'lsNu3 + NWU2-I WLUNU3 + NWU2 NL3=NLl+NWLl NL'l = NL3 + NwL2-i MS0LlrNL1+l IFCNSOLSL'.EO'.O) NSOLl=NLfl NF1=N|.1+N30LSL*1 I F C N F L A T ' . E O ' . O ; . N F I * N L4+N S O L S L NF1 « 1ST CON P T ON 1ST F L A T S L A T NSUi=NL1+NS0LSL*NFLAT*NSPF*i I F CN3LAT.E0'.0) NSU1=NLU+N30LSL*NFLAT*NSPF NSIin = NSUl + NSLAT*NSPS»l IF(NSLAT.EO'.O) NSU2=NSU1 NSU2 - L*ST CON PT ON L A S T S L A T Nil - 1ST CONTROL PT ON STREAMLINE FOR SHEAR LAYER Mil=NSU2+l I F CMSV.LQ'.O) NilsNSU2 WRITER,101) NU1,N';3,NL1,NL3»NS0L1.NF1,NSUI,NSU2,NI1 FORMATC'NU1««,I3,2X,INU3»',13,2X,"NL1"',I3,2X,• NLS»',13,2X, 1 iNSOLi"'#i3,2X,INF 1 = 1,I3,2X.'NSU1s',13,2X,'NSU2=1,13,2X,I Nils•, 2 13) NWALL - TOTAL * OF CON PTS ON ALL FLAT SOLID WALL SECTIONS NWALL=NWU1+NWU2+NWL1+NWL2 NW3 - TOTAL * CONTROL POINTS ON AIRFOIL SOLID WALL' SECTIONS & SLATS MWS=NA+NWALL+NSPS*NSLAT+NSOLSL+NSPF*NFLAT NKA « » OF EON FOR KUTTA ON TEST AIRFOIL NKA=NSU2+NSLAT*1 NSVT - TOTAL » SOURCE & VORTEX ELEMS NSVf=NWS+MSV NM2 «• » OF LAST EON FOR ZERO NORM VEL ON INNER EDGE OF S,L« NM2sNKA*M3V NUN - TOTAL « UNKNOWNS NUN=NtaS+2*MSV»NSLAT*l NFL - TOTAL « CON PTS ON FLAT SLATS ( N O K U T T A ) NFLsNSPF*NFLAT NSL - TOTAL * CON PTS ON A L L AIRFblLSHAPED S L A T S NSL=NSPS*N3LAT . WRITE (b.33) NA,NWALL,NSOLSL,NFL,NSL,NWS FORMAT('NA=',13,2X,'NWALL"',13,2X,INSOLSLo',13,2X,INFL"',13,2X, I iNSLs ' .njZX^NwSst.IS) WRITE(6,31) MSV,N3VT FORMAT('»3V=',l3,2X,'N3VT»",IS) WRITE(6,35) NKA,NM2,NUN FORMAT('NKA=',13,2X,'NM2s•,13,2X#'NUN«I,13) NA « a CON P T S ON SINGLE TEST AIRFOIL NlaNA+1 N2=NA+NWU1+NWU2 SET TESTSECTION WALL HEIGHT DO 2 I=Nl,N2 YW - Y-C00RO FOR UPPER AND LOWER W A L L S YY(i)=YW . N3=N2tl N«3N2*NwH + NKiL2 12159113 PAGE P002 51.000 55,000 56,000 57,000 58,000 59,000 60,000 61.000 62.000 63.000 61,000 65.000 66,000 67,000 68.000 69.000 70.000 71.000 72.000 73.000 71.000 75.000 76.000 77.000 78.000 79.000 80,000 81.000 82,000 83.000 81,000 85.000 86,000 87.000 88,000 89.000 90.000 91.000 92.000 93.000 91.000 95.000 96.000 97.000 98,000 99.000 100.000 101.000 102.000 103,000 101,000 105,000 106.000 107.000 108.000 MICHIGAN TERMINAL SY3TCM FORTRAN G(fll336) MAIN 0076 OO 3 i=N3,Nfl 0077 3 YY(I)=-YW 0078 IF(NSLAT.EO'.O) GO TO 500 0079 DO 1 I=NSU1,N3U2 0 0 8 0 fl YY(i)=YY(I)+YW 0081 5 0 0 CONTINUE 0032 IF(NSLAT'.EO'.O) GO TO 15 C XCENT.YCENT - CENTER OF SLATS 0083 REA0C5,16) (XCENT(K), Kai,NSLAT) OOSfl WRITEC6.308) 0085 308 FORMAT('XCENT') 0086 WRUE(6,1B) (XCENT(K), Kai,NSLAT) 0087 REA0C5,'i6) CYCfNT(K), K»l,NSLAT) 0 0 3 8 WRITEC6.309) C089 309 FORMAT('YCENT') 0090 KRITE(6,18) (YCENT(K), K«l,NSLAT) C MODIFY SLAT PROFILES FOR REDUCED CIRCULATION 009{ REAn(S,50) MOD 0092 50 FORMAT(12) 0093 wnifecft.si) MOD Oo9f | 51 FORMAT<'M0DPR0a',J2) C MOOiFY PROFILES.IF IMOD' NOT ZERO 0 0 9 5 IF(MOD,CO'.0> GO TO 15 C XLE.XTE - X-COOROS OF SLAT LEADING & TRAILING EDGES 0096 READ(5,16) (XLECK), K=l,NSLAT) 0097 16 F0RMAT(i3F6'.l) 0098 WRITE(6,306) 0099 306 FORMAT(* XLE1) 0100 WRITE(6,18) CXLE(K), Kn1.NSLAT) 0101 18 F0RMATUX,13F6'.l) 0 1 0 2 REA0(5,16) (XTE(K), Kai,NSLAT) 0103 WRITE(6,307) 0101 307 FORMAT( 'XTE *) 0105 WRlfEt6,1B) CXTECK), K=l,NSLAT) C ALF « FLOW ANGLE AT EACH SLAT 0!06 READ(5»16) (ALFCK3), KS=1,NSLAT) 0107 WRlfEC6,310) ClOB 310 FORMAT('ALFI) 0109 WRITEC6.18) CALF(K), Kai,NSLAT) C Ki « FRACTION OF CIRCULATION 011 0 REAO(5.17) (Ki'(K). Kai,NSLAT) 0111 «7 F0RMAT(16F5'.3) 0112 WRITE.6.311) CMS 311 FORMAT (' K 1 ') 0111 WRITE(6,19) (KICK), K»l,NSLAT) 0115 19 F0RMAT(1X,16F5.3) C DTHiCK . SYMMETRIC DISPLACEMENT THICKNESS 0116 READ(5.17) (DTHICK (K), Ka1,NSLAT) 0117 WRlfE(6,3l2) 0118 312 FORMAJCDTHICKI ) 0119 WRITE{6,19) CDTHICK(K), KoJ,NSLAT) 0120 NPS=((NSP3-l)/2)-i 0121 NSP=NSPS»1 0122 MPsNSPS+1 0123 DO 11 KS=1,NSLAT 12)59113 109.000 110,000 111,060 112.000 113.000 111.000 115.000 116.000 117.000 118.000 119.000 120.000 121.000 122.000 123.000 121.000 125.000 126.000 127,000 12e,000 129.000 130.000 131.000 132.000 133.000 131.000 135.000 136,000 137,000 138.000 139.000 110.000 111.000 112.000 113.000 111,0Q0 115.000 116.000 117.000 118,000 119.000 150,000 151,000 152.000 153.000 151.Opo 155.000 156,000 157.000 158,000 159.000 160,000 161.000 162.000 163.000 PAGE P003 h 00 M.CHIOAN Tt£RM_NAl. SY3TEM FORTRAN 6(11336) MAIN 10-22-75 12I59US PAGE P001 0121 0J25 0126 0127 oi'2B 0129 01*9 0131 0132 0133 0131 0133 0 136 0137 0138 0139 oi iq 0111 0112 0]13 0111 0115 0116 0117 0118 0119 ©ISO 0151 0152 0153 0.51 0155 0156 0157 0158 0159 0160 0161 0162 0163 0|61 0165 0166 0167 0168 0169 0170 11 15 16 17 18 15 C 22 23 36 NT»NSllUNSP3»(KS-l) 161.000 MTsNT+NSP 165,000 MlnMT.NPS 166.000 NLaNT+NPS 167,000 ALZ - ZfRO LIFT ANGLE FOR SLATS (0015) 168.000 ALZaO. t 169,000 CALL MODPRO(XX,Vy,DX,OY,03,CS,SI,NSVT,XR,YR,MP,NT,MT,ALF(KS),ALZ, 170.000 1 Kl(KS),0THICK(K3),XCENT(KS),YCENT(KS),ML,MT,NL,NT,XO,YQ,XLE(KS), 171.000 2. XfECKS)) 172. 000 CONTINUE 173.000 CONTINUE 171.000 MVS=MSV+1 175.000 iF(MSV.EQ'.O) GO TO 36 176.000 READ COORDS FOR STREAMLINE FOR SHEAR LAYER 177,000 READ(5»16) XM 178.000 REAOC5.16) YM 179.000 F0RMAT(l'2F6'.l) 180 . 000 WRITEC6.17) XM 181.000 WRITE'C6,18) YM 182,000 •F0RMAT('XMi,10F7",2:-' 183.000 FORMAT<'YMi,10F7.2) 181.000 DO 13 K=1,M3V 185.000 I=NwD+K 186,000 J= K+1 . . , 187.000 XX(I)a(XM(K)+XM(J))/2, 188.000 YYCJ)=(YMCK)+YM(J))/2. 189,000 DXCI)=XM(J).XM(K) 190.000 OY(i)=YH(J)«YM.K) 191.000 D3{n=S0RTlDX(I)*0X(I)*0Y'(I)*DY(I)) 192.000 CS(I)=BX(I>/nS(I) 193.00* Si(I)=DY(I)/DS(I) 191,000 M1,M2 * RANGE OF CONTROL PTS * I S ON SHEAR LAYER 195.000 MlaNW9+i 196.000 M2=NWS+MSV . 197,000 KRITE(6,7) 198,000 FORMAT('STREAMLINE FOR SHEAR LAYER•) 199,000 WRITE(6,6) 200.000 FORMAJ(7X,'XX',6X.'YY',6X,«0X',6X,<DY',6X»'D3',6X,'C3',6X,'SI') 201.000 WRITE(6,5) (XXCI),YY(I),DX(I),DY(I),DS(I),CS(I),SI(I), IaMl,M2) 202.000 F0RMATC1X,7F8'.3) 203.000 READ VELOCITY DISTRIBUTION ON STREAMLINE FOR S.L, 201,000 REA0C5/22) (VTI(I), Ial,M3Y) 205,000 FORMA7(i0F8'.3) 206.000 WRITE.6,23) VTI 207.000 F0RMAT('VTI',i0F8.3) 208.000 CONTINUE 209,000 NrNSVT. 210.000 MsNlJN 211,000 NG - a VORTEX STRENGTH DEN'S ON SLATS i TEST AIRFOIL 212,090 NG=NSLAT+1 213,000 NMaMSV 211,000 NLSsNSLAT 215,000 CALl. MAiNlCXX,YY,0X,DY,D3,CS,SI.N,SIG,M,VTT»CP,GAM,NG,MU#GNU,NM, 216.0 00 l" VTI.XCENT.YCENT.NLS) 217.000 STOP 218.000 MICHIGAN TERMINAL SYSTEM FORTRAN G'(HS36) M A I N 0171 END •OPTIONS IN EFFECT* 10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAD,NOMAP •OPTIONS IN EFFECT* NAME = MAIN , LINECNT = S7 •STATISTICS* SOURCE STATEMENTS a 171,PROGRAM SIZE n •STATISTICS* NO DiAGNoSTICS GENERATED NO ERRORS IN MAIN NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTiON TERMINATED SR "L0AD+0UJBSL3+ATXB 2 = FlL'E«<l7 3 = «A «o-B 9«*DUMMY* EXECUTION BEGINS 10-22-75 12I59U3 PAGE POOS 219.000 260 ft* CY A«20 N»50 70XTSUSL C/H".66 MSVs20 •** CALAH" t CALCDB 1 WRABa 0 WRCD" 0 SOLV" J GAUSS" 1 NAB 50 NWU1= 2 0 NWU2e 20 NWLl" 10 NWL2" 40 NS0L3L" N T T U » 2 5 NTELB 26 CH= 23 .940 U " l'.O N U 1 S 5 1 N U 3 = 71 NLt" 9 1 N L 3 * 1 J 1 NSOLlal70 N W A L L = 1 2 0 N S V T = J 6 2 N M 2 = 2 7 1 N S O L S L « SI.2 ie*. o i NUNI291 io'.2 8.0 • I'.B .1S'.8 •25*.fl i's'.o I'S'.O IB'.O 32.a 20.1 8.1 NA« 5 0 MSV = 20 NK A = 2 5 1 XCF.NJ 4 6 . 2 YCF/JJ 1 8.0 X L E , <i'i.<i XTE , i e . O 3 6 . 0 2 1 . 0 ALF -2.7 -4.8 -8.8 K> . • , . . . . O.BOOOjBOOO.8000.8000.8000.8000.8000,SOC D T H I C * o'.o o'.o o'.o o'.o o DEl.CPST = -u'.00943(RAD) 0Fi.EPST = -O^0l678C«AO) O E L C P S T = - 0 . 0 3 0 8 9 ( R A D ) N F L " 0 N S L " 7 2 N W S . 2 D 2 ITER" 0 CALCP 1 CALCl" 1 HSIGa 0 0 NFLATa 0 NSPF" 0 NSLATa 8 NSPSa 9 MSV" 2 0 NF1«IT0 NSUl«17i NSU2e2(|2 Nll"24S 12.0 15'. 2 -3'. 6 0.0 '15".0 -15.6 • 12*.0 27.6 21'.0 • S 7 . 8 IB'.O • 39*.6 -36*. 0 5.5 11.0 10. T D   - o ' . o 5 3 2 4 n ) O n . C P S T = o ' 0 1 9 2 4 ( R A O ) 0EL C P S T = 0 0 3 B 7 K R A O ) DELEPSTa O . 0 3 7 6 7 C R A O ) XM x« YM Y M Y " 00 00 -A.e'oo 22 60 18 . 0 0 STREAMLINE XX ' 49' 5 0 0 a'l 5 0 0 39 5 0 0 5 " ; 5 0 0 2 9 5 0 0 2 1 , 5 0 0 19 5 0 0 I'I,500 9 5 0 0 '1,500 - 0 , 5 0 0 . 5 , 5 0 0 .'JO,500 . 1 5 , 5 0 0 - 2 0 , 5 0 0 - 2 5 , 5 0 0 -30,500 -35,500 - 4 0 , 5 0 0 - 4 5 . 5 0 0 4 7 , 0 0 • 3 . 0 0 18 20 22.90 42,00 -8.00 0 0 0.0 -0 5 4 ( R E G > - 0 , 9 6 ( R E G ) - 1 , 7 7 ( D E G ) - 3 , 0 9 ( D E G ) - 3 , 0 5 ( D E G > 1 1 0 C 0 E G ) 2,22<l)EG) 2 . 1 6 C 0 E G ) 3 7 . 0 0 32*. - 1 3 . 0 0 - 1 8 . 0 0 0,0 Kao',80000 Kao',80000 K = o'.80000 K 0,80000 K=0.80000 Kao K = 0 Kao 00 B L O I S P L T HIC KB B L D I S P L T H I C KB B L O I S P L T H I C KB B L D I S P L T H I C KB B L D I S P L T H I C KB B L D I S P L T H I C KS B L D I S P L T H I C KB B L D I S P L T H I C KB 27̂ 00 22 00 17.00 -23.00 -28.00 -33,00 - ,80000 ,80000 ,80000 0,0 0.0 o;o 0,0 0,0 °f.° 0.0 00 00 18 40 22.90 18.70 22.60 2 2 . 1 0 19, 21'. 50 40 20,10 .70 20. 20.70 19,70 21,40 19.20 FOR SHEAR YY 100 300 550 900 300 800 400 050 750 350 750 900 750 350 750 050 200 450 850 250 L A Y E R DX , 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 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 ^ .000 , 0 0 0 , 0 0 0 , 0 0 0 -5.000 DY 0'.200 0'.200 0,300 0.400 0'.400 0'600 0.600 0'700 0.700 O'.SOO 0'300 0,0 .0.300 .0'.500 .0'.700 -0'.700 -f.000 -0',500 .0.700 -0'.500 DS 5,004 5,004 5,009 5.016 5,016 5.036 5.036 5.049 5.049 5.025 5,009 5.000 5,009 5,025 5.049 5.049 5.099 5.025 5.049 5.025 - 0 . 9 9 9 • o!.999 - 0 , 9 9 8 - 0 , 9 9 7 . 0 . 9 9 7 - O ' 9 9 3 - 0 . 9 9 3 »0'.990 -0J.990 . 0 . 9 9 5 -0',998 . 1 , 0 0 0 - 0 , 9 9 8 - 0 , 9 9 5 - 0 . 9 9 0 990 981 .0,995 •0,990 •0.995 SI 0*.040 0'. 0 4 0 0' 060 0.080 O'.OBO 0'. 119 0'. 119 0.139 0' 139 0.100 0'.060 O'.O .0,060 .0.100 -0' 139 -0.139 -0.196 -0'. 100 -0.139 -O'.lOO 7', 00 •43,00 22.10 18'.50 nj A J Aj A J A J —#rvj — - * A J o j —• — o o o o o o o o o o c o G o t i i t i i r t t i i t • • UlUJUJU-UJUIUJUJuJaJUJUJ Ul o- j cc o S K I o- o f f N ^ o o a o - a o o L n o - i r i r ^ co j ) c t C ' - K i r - N F « T L n J i n - 0 - — — =a ~ cr K I O J K 1 ^ 0 N M i T S Q D ^ O f \ J - £ 0 ' a j 3 o - G P - r v i n c - K i o - o o o o • • • I UJ tu UJ f\l o cc C M C =r =r oj o —• cr r-~ — — — r— o- (C — — Kl — ^1 O C T — — o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s i r * • » i » t i i i i i i r t i i t I I i • • UJ U l UJ UJ Ul DJ Ul U l Ul UJ U l Ul U l U l U l U l U l cc —* -O =3 ^ m ^ cr r \ i 1 , " , t \ j ^ ) - ' K ' i o £ ^ ) — N f \ J 3 u i - ' C ' C C ( r -o fu ( \ J - J D 3 0 J C 0 0 7 A J O ( M i a j 5 0 N N 3 0 J 3 « - * > ^ ' , - ' v f - ^ C l ! ' ' t ^ O ^ N ^ N - . . C i n K l — ' - O c J A j m c j CO =3' • ^ 0 ' J 5 N f V l t C N . 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I t I—< —< • » o o A l O Kl O o j C J OJ AJ 0 o 1 I W UJ .£> K l OJ IP O in oo K I in I P r j O K l OJ K l K l OJ OJ OJ CT - 4 CVI O J — ' - * O J O o o o o o • • • I I I Ul U l Ul U J Ul U J Kl C~ Kl r— Ul o =3 —« O Kl cr o o »— c O J — * r*- ry c o -o -o —• LJ-l O —• —« s£) o o < • I U! 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O U 7 0,050 0,051 0,052 0.053 0^053 0,051 0.055 - 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 -0.0 WALL VNOT - o j o - 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 -0,0 -0,0 - 0 , 0 - 0 , 0 - o . o WALL VNOT -l)'f0 - 0 , 0 - 0 , 0 -0,0 -0,0 - 0 , 0 - 0 , 0 - 0 , 0 - 0 , 0 -0,0 -0,0 - 0 , 0 - 0 , 0 -0,1) - 0 , 0 - 0 , 0 -0 0 -0 0 -0 , 0 -0,0 -0.0 000 0 0 0 ooo 000 0 0 0 ooo 0 0 0 000 000 000 000 000 0 0 0 000 000 VNT - 0 , 0 0 0 - 0 . 0 0 0 - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' o o o - 0 . 0 0 0 - o ' . o o o - o ' . o o o - o ' . o o o - o ' . o o o - o ' 0 0 0 - 0 . 0 0 0 - o ' . o o o VNT 0 . 0 0 0 6,000 0. 0 0 0 0' 0 00 0, 0 0 0 0.000 o ' . o o o 0' 0 00 0.000 0' 0 00 0. 0 0 0 0^,000 0 , 0 0 0 0 , 0 0 0 0 0 00 0.0 0 0 0'000. 0 0 00 0.0 0 0 0' 0 0 0 0. 0 0 0 •0.018 •0.017 -0.017 -0.017 •0.016 -0.016 -0.016 -0.017 •0.018 -0.050 -0.055 -0.068 -0.127 -0.563 -0.816 VTST • 0.130 -0.060 -0.037 -0.021 - o . o ; 6 -0.011 .-0.007 -0.001 • 0.002 -0,001 o . o o o 0.001 0.002 0.003 0.001 0.006 0.007 0.010 0.013 0,022 VTST 0.019 0.038 0.031 0.031 0.030 0,029 0.028 0.027 0.027 0.026 0.026 0.026 0.025 0.021 0.023 0,021 o . o i ' a 0.015 0.011 0.007 0.001 •0,019 .0,020 •0,021 -0,023 •0.021 - 0 l 0 2 6 •0,028 -0,030 -0,032 -0,035 -0,038 •0.010 - 0 ' O i l 0,855 0.803 VTVT -0.033 -o'032 -0,030 -0.029 -0J027 -0,025 -0.021 -0',023 -0,021 -0,020 -0,019 -0.018 •0.016 •0^015 - -0,011 - 0 , 0 1 1 -0,013 •0.012 VTVT - o ' . o o s - 0 ' 0 0 9 - 0 . 0 0 9 - o ' , 0 1 0 - 0 , 0 1 1 - 0 . 0 1 2 - 0 ' 0 1 3 - 0 . 0 1 5 - O ' 0 1 6 • 0 . 0 1 8 • o ] . 0 1 9 - 0 , 0 2 1 - 0 , 0 2 3 - 0 . 0 2 3 • 0 ' 0 2 7 - 0 . 0 2 9 •0.031 •0'.037 • o'.oio • 0'.013 . 1 , 0 0 0 • 1 , 0 0 0 • 1 . 0 0 0 . l'.ooo • r.ooo - l ' o o o - 1 . 0 0 0 • 1 , 0 0 0 - 1 , 0 0 0 - 1 , 0 0 0 - 1 , 0 0 0 - 1 . 0 0 0 .r.ooo r l'.OOO -r.ooo VTOT - r. o u o •r.ooo - r. o o o •r.ooo -r.ooo -r.ooo -r.ooo -l'.ooo • l' 0 0 0 -1 ooo - 1 . 0 0 0 - 1 , 0 0 0 - l . o o o -r.ooo - 1'. 0 0 0 r.ooo r.ooo -l'.ooo . 1 . 0 0 0 • l'.ooo VTOT r,ooo 1 , 0 0 0 1 . 0 0 0 r.ooo r.ooo l'.ooo r.ooo r.ooo r.ooo r.ooo r.ooo r.ooo r.ooo l'.ooo r.ooo r.ooo r.ooo r.ooo l!. o o o 1 . 0 0 0 r.ooo •1,066 -1,067 -1.068 -1,069 -l'.071 -1.072 -1,071 -l',077 -1.080 - l ' , 085 -l'.093 -1.108 -1.169 -0,709 -1.013 VTT -1.162 -1.093 • 1 .069 -1.055 - l ' . O l S -l',038 -1.032 •1.028 -1.025 •l',022 -l'.020 •1.017 •1-015 - l ' . o i i - 1.012 .1,010 -1,007 •1.001 • l'.OOO -0'.990 VTT l . O i l 1^030 1.025 1.021 1 0 1 9 1.0)6 1.011 •1.013 1.011 1.009 1,007 l'.oos 1.002 0.999 0'.996 0'.991 0'.987 0.981 0.971 0.967 0.958 2,591 2 , 1 6 7 2,317 2,231 2, 120 2,013 1,910 1 809 1,712 1,616 1,522 1,128 1,333 1,235 -1,828 1.119 -5.576 ASUK 1,165 1,261 1,356 1 , 1 1 " 1,511 1,631 1,728 1,821 1,923 2,026 2,132 2,176 2,600 2,727 2.858 2J.992 3.129 3'.266 ASUM 3".603 3,181 3,368 3,255 3,117 3,015 2,918 2,858 2,773 2,696 2,621 2.559 2,198 2,112 2,389 2,339 2,290 2,211 2,190 2,136 2.076 .8,131 -7'.711 -7,380 -7.013 -6',723 -6J118 -6.125 -5'813 -5 571 -5 312 -5.073 -l'.880 BSUH 0,610 2,566 3,111 1,039 1,511 1,927 5,307 5.669 6' 023 6,376 6.735 7.191 7'. 901 8'.356 8'. 865 9'.165 10,226 11,321 13.588 BSUM -13-770 - i l ' . 5 2 5 - i o ' . n a -9.712 -9.139 • a',662 .8'208 -7,878 -T.511 -7.232 .6'. 915 -6', 677 -6.127 • 6' 193 -5,971 -5!,769 -5,579 -5'.102 -5'.239 -5.088 -1'.918 8.00 8.00 8.00 8.00 8,00 8.00 a . o o 8.00 8.00 8.00 8.00 8.00 8,00 8.00 8,00 YV 6,00 8.00 8,00 e . o o 8.0 0 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8,00 8,00 8.00 8.00 8,00 8,00 8,00 8,00 YY 8.00 6.00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8,00 8.00 8.00 8,00 8,00 8.00 8,00 8.00 8.00 8,00 82.80 8 0 , 1 0 7 8 , 0 0 7 5 , 6 0 7 3 , 2 0 70,80 6 8 , 1 0 6 6 , 0 0 63,60 6 1 , 2 0 58,80 5 6 . 1 0 5 1 , 0 0 51,60 19 , 2 0 XX -19,20 -51,60 -51,00 - 5 6 , 1 0 -58,80 -61,20 -63,60 . 6 6 , 0 0 - 6 8 . 1 0 -70,80 - 7 3 . 2 0 •75.60 -78.00 - 8 0 , 1 0 -82.80 - 8 5 , 2 0 -87,60 - 9 0 , 0 0 - 9 2 , 1 0 -91,80 XX •91,80 - 9 2 , 1 0 - 9 0 , 0 0 -87,60 - 8 5 , 2 0 -82,80 - 8 0 , 1 0 - 7 8 , 0 0 - 75.60 - 7 3 . 2 0 -70,80 -68 , i|0 - 6 6 , 0 0 - 63.60 - 6 1 , 2 0 -5B.80 - 5 6 , 1 0 - 5 1 , 0 0 -51.60 - 1 9 , 2 0 - 16,80 •0.137 -0.139 -0.111 -0.113 -0,116 • 0.150 -0.15« -0.160 -0.167 -0.178 -0.195 -0.228 -0.366 0.197 -0.026 CP •0.350 •0.196 113 .113 0.092 077 0.066 0.057 '0,050 0.011 0.010 035 031 027 023 019 Oil 008 001 020 -0 -0 CP • 0.081 -0,060 •0.050 -0.013 -0.037 -0.033 -0.029 • 0.025 -0.022 • 0.018 -0.011 -0,009 -0.001 0.002 0.009 0.017 0.027 0.038 0.050 0.065 0.082 0.002 0.002 0.002 0,002 0.002 0.001 0.000 .0,001 -0.002 -0,005 -0,009 -0,018 -0,010 -0.110 0.002 SIG 0.032 0.023 0.019 0.016 0,011 0,012 0.011 0,010 0.009 0.008 0.007 0.0 07 0,006 0.006 0.005 0.005 0.005 0.005 0.005 O.OOU SIG -0.006 -0.006 -0.006 -0,006 -0.006. • 0,006 -0.007 -0,007 -0.007 • 0.008 -0,008 -0.008 -0,009 -0,009 • 0 , 0 1 0 -0,010 •0.011 -0.011 -0.012 -0.013 -0,013 to 130 ^ — < — H ~ - — - — « — - « ^ — - . — < — . 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M O O — —« O c o o i n i O M 0 3 i \ i c c C J . • O W n M N O W h O I I I . . I l l L T l A j - C e C c c r - - ' - — « m C L - 4 — r - o - y - K i A i n j — o ^ i n J J - O f i i n e j ' f f • H l s O ' K i r ' K M M O - H Q . 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K l c r A l O A l l T A J O C T f f O O A l O f f l T f f e C c r > O f f O 1— A I . - 4 ff1 — i n A T — ( - O ^ i n N K I O - T J O A J K l - 0 0 0 0 - - 0 » « M ffffnj^3in-^=Tr—ff K O « S r j - * K I * ' K t O r- ff o to ao —• • o r - o o o r - AJ • - A J K i r - f f A j c o - c r O — ' f f U l K t A J O f f O h - O f f f f f f O f f f f C O O O O f f f f c O O c O f f f f O ,»-. «. •- •- •- •• • > - * 0 0 0 0 0 0 0 — • > r * i O A J f f - O K l — ffCCff j - O f f f f f f O f f f f C O f f C O f f f f O O O C O f f f f f f O K I O C O K I O O O f f l 1— O ff ff ff O f f f f C O f f O O f f f f C C O c E f f f f f f c o m e r — K » r - f f K i J 3 cr ff• o* o o CD ff-cc ff O f f f f f f f f O t O f f f f f f co in ff —• K I P^- ff f — f f f f f f O O C O f f O f f f f f f f f O C O f f I— ftjcrocrsiAiNrjcr > O A J — » c r o r - c r A J c r > . » H O O O — 4 0 0 0 0 M D K I N t M O O O c O H O d f t l O I / I C C O w K l > c r c f c r K i * 4 K l K i f J ( \ J l ~ . •.*-•-•«-•-•-*-•-• > 0 0 0 0 0 0 0 0 0 I I t I O c O f f N ^ O c O f f O f - K i a - D a K i i n ~ c o r s »—. «. •-*-•-• : > o o o o o o o o o . B ft t • ft k - e X f f K l A J C O ^ J K I C O C T > f f c o f f o x A j i n i n i n h-» « . • . * . • „ • _ • . * » * . • > o o o — « o o o o o • I ft ft I ff AJ nJ AJ O —• AJ o cc — — t o c f f i n > r * * i - . c o i n M ^ T „ «w • > o o o o o o o • • • I I i n e o n j o o c j c o o r -h-; —., —• r j AT l- ec • r- i - 4 KI ( o « o c c - o c c r - ^ - o i n K - c c u o o c o f f o ; — * C O C J K l O r j C O C T - 4 - . K l O K l f l Cf C Aj4-«r-.CC • t— A I I —• i n i n eo K I ff cr t i t n r - > O N - ^ o r - M A J c r • O C C C I C O N C O N W • »— i n o- o K i e c K t e o f f i n COCO O n j K l c r c O K I - 4 C T . * i - o cc co m o o co in M o AI NO AI o o o o o o o o o o o o o o o o o o o • - o o o o o o o o o z . •. •- • > o o o o o o o o o o o o o o o o o o o o o o o o o o o * - 0 0 0 0 0 0 0 0 0 2 . « . « . • . • . • - • . • . • - • > 0 0 0 0 0 0 0 0 0 I I I o o o o o o o o o o o o o o o o o o I — o o o o o o o o o o o o o o o o o o o o o o o o o o o f - 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o f - o o o o o o o C C r - « 0 0 - » S f f O I - O K l -T - T O U l -T 1 K l o o - ^ o c r o — o — i o > © o o o — * o o o o • I I I ! N C C r - C O K l N ^ O •U— O PJ CT -1 O Iji . 7 in Cf 0 0 - 4 0 — o — o — o > o o o o — 4 0 0 0 0 • I. I ft I I r - r - r — — • o r - r - m o • H - r j ^ c f c r o u i - T O - o o o — « o c r o c r o — « o > o o o o — < o o o o r r i i i ff . T N t n o f l j r - K t n j U V f f C T K l O O ST ff? ff o o o o c r o — o — » o > 0 0 0 0 - 4 0 0 0 0 • I ft ft ft • co o ?— i n o AI r - H i n O J T K I O ^ c f o o — o c r o c r o z . •* s. v • > 0 0 0 0 - 4 0 0 I r t i • t c e c c c i n i n N c r K i ? r t- AJ K l CC ff m m c o r— o ff>CTffOO-4L-lOOCT • r- z . • .—* O O O O O O O O I » n o I ft I I I ffff-4=T-4r-ffr-£> I- | j ' KV N CC ff d =T K I T - CO > K > A J A J K 1 K 1 0 A J K 1 K > CO 2 . «• • „ • » * . • . »- • — > o o o o o o o o o C I ft * ft f J O 5 fO O -O N O PJ . | _ 2 3 C J i M N 3 O W a- z . • ..«.*.. . — : > o o o o o o o o o • ft I I I I i n i*- o f J " n ff m o I ^ C T C T C T C C K l K t O K l A J 0 > O K I K I O h N C - - 0 O 2 . • . « . * . •« A J 3 > 0 0 0 0 0 0 0 0 0 cr -< 3 N co in N t - s J U l f f U" -3 O IP L T - > P J C K l - L " i O C f • A J > 0 0 0 0 0 0 0 I I I »~ AJ O K I T t l O K l AI K l t c n C C O N - ' ' " « - 0 * M " f— > o e > o - 4 « - « o o — « o < t i l l CO O — « f f C - 4 - t O ( V i * 4 0 o —• -4 4\j cr o O ff'-* « ( n M « K i t o M - i \ i c e r > OOOti— o o o o < 1 I ft " _ J CO co c o K t r u c r o n j c r P j P i 5»^b_ AJ K l Ul O h - Kl O K l CO t— > O O O O — t O O O O < t I I - J CO ff cr O K t K V C X cr O CO AJ cr ff L I K I ff tn cr r— O W O C l M K l h - 4 S T O O f_ > O O O O W M O O O ft I ft ft O -< <H O O K M A: I - - * L l ej-r- O AJ o a c o A i i n K i A j i n — * m ^ . ». V. *- «. V • J— > C * 0 0 0 — - 4 0 < ft; ft t CO 0',0?7 -0,279 0.156 -0.247 SLAT »216 VNST -o',44t -0,463 -0,136 0-'130 1,251 0,703 0,268 0 101 0.397 221 VNVT 0',397 0,3o7 0.183 o',024 -0,251 -0,259 -0.221 -0'.228 -0'.386 0,183 0.091 VNOT 0* Oil 0,156 -0,0t7 -0 454 -1,000 -0,444 -0,047 0,126 .0.010 0,000 0.000 VNT • 0'. 000 • O'.OOO - o ' . o o o - o ' . o o o - o ' . o o o o ' . o o o o ' . o o o 0.000 o ' . o o o •0.078 .0.333 VTST 0.001 -0.315 -0.175 0.369 1 .I'll 0.825 0.331 -0.136 -0'.265 0,560 0.511 VTVT •01203 .0,150 -0.170 •ol190 -0.010 0',265 0,375 0,162 0.466 0.996 i',207 22.870 VTOT .0'.999 -0.988 .0'.999 .0'.891 0'.002 0.896 0' 999 0.992 I'.OOO VTT -1.201 -1.452 -f.343 -0.712 T.433 1.986 1.704 1.318 1.201 . ASUM 3,358 8,175 24.983 23̂ 049 20,332 20,432 21.680 -t ' 173 18,10 -1,26 •1.146 0^084 10.022 17.92 -0,18 •0,456 0,562 BSUM YY XX CP SIG -8' 002 18.03 -12.18 •0,442 0.599 1,149 17,88 -13,26 -1.108 0.099 7.307 17,76 -14,69 -0.804 0.110 i2'.89« 17.88 -15,40 0.493 0,172 8. 348 18.00 -15.58 -1.054 0.134 -5.410 18.12 -15,40 -2.945 .0.039 • fe', 44 1 18,23 -14,69 -1.905 .0,100 -3'.587 18.14 -13,27 -0.737 -0,134 7'. 7 01 18,03 -12,18 .0.442 .0,602 S L A T »225 VIJST -o',414 -0,3l'9 -0,004 0,468 0,969 0,424 0,1Q4 0,036 0.379 S L A T «234 V N S T -O'163 0,141 0,529 0,863 0,245. •0,057 0.134 STRCAMLINC VNST 0',089 0,830 0,476 0,397 0,342 0,328 0,156 0,074 -0,096 -0,274 -0,375 -0,460 .0,487 -0 421 -0,407 -0,331 -0.262 233 VNVT 0̂ 343 0.148 0,015 •0,057 -0.147 •0'.342 24 2 VNVT 0'.094 VNOT O'071 0,171 -0,047 -0,459 - 1 , 0 0 0 -0,439 -0,047 0,111 -0.038 V NOT O' 069 0,171 -0.047 ".0*, '158 -1,000 -0,439 -0,047 0,112 -0.036 •0,071 0,137 0.194 OJ105 0.014 • 0'.098 FOR SHCAR LAYF.R VNVT -0'.049 -0'.790 -0J416 -0.317 • 0',262 -0,209 -0.037 0,235 0,373 0.435 O',460 VNOT -0̂ ,040 -0,040 -0,060 -0,080 -0,080 -0,119 -0 119 -0 139 -0,139 1 0 0 0,019 0.087 0.322 0'.268 0J192 0,066 0,031 -0,158 »0'. 186 -0,060 -0,0 0,060 0,100 0,139 0 139 0 196 0,100 0,139 0.100 VNT VTST - o ' . o o o .0.255 - o ' . o o o -0.540 -0.000 -0.252 -0̂ 000 0.452 -0.000 1.910 o ' . o o o i. o ' . o o o 0 .484 0' 000 -0.099 0.000 -0.203 VNT VTST . o ' . o o o -0.326 . o ' . o o o -0.558 - o ' . o o o •0.454 -0,000 -0.113 -0.000 0.793 o ' . o o o 0.777 o ' . o o o 0.597 o ' . o o o 0.224 - o ' . o o o 0.035 I VNT VTST - o ' . o o o •0.849 o ' . o o o 0.298 • o ' . o o o 0.281 - o ' . o o o 0.204 - o ' . o o o 0.392 -0,000 0.457 - o . o o o 0.580 - o ' . o o o 0 .588 -0,000 0.582 0.000 0.5f'3 o ' . o o o 0.349 o ' . o o o 0.215 0' 000 0.059 o . o o o -0.071 o ' . o o o •0.115 0'000 •0.202 0 000 -0.273 -0.000 -0.269 - o ' . o o o -0.256 - o ' . o o o -0.171 VTVT 8* 188 0,209 0,199 0 J209 0,277 0,192 0,201 0.256 0'.268 VTVT 0,267 0,255 0,266 0,313 0,320 0.067 • o'000 0,006 0.022 VTVT •0,319 •0,309 -0.261 • 0'.479 -0̂ 573 -0,732 -0,781 -0.600 iO'470 -0.316 - o ' . i e s -0'. 136 -0'.033 0' 054 0,098 0,111 0.072 VTOT .0' 997 .0.985 .0".999 .0'889 0.002 0'.899 0.999 0".9?q 0.999 VTOT »0'.998 -0'.985 • 0'.999 .0.889 0' 002 0.898 0'.999 0'.994 0'.999 VTOT -0'.999 • 0'.999 • 0'.998 • 0'.997 • 0'.997 -0'.993 -0'.993 0'.808 -0.990 0'.759 -0'.995 • 0'.998 • I'.OOO -0.998 • 0'.995 »0'.990 • 0'.990 -0'.9B1 -0'.995 -0'.990 -0'.995 VTT •1.065 •1̂ 317 -1.052 -0'.228 2.190 2.219 1,684 1,151 1.065 VTT -f.057 -1.289 -1.187 -0.688 1.116 1.742 1.595 1.224 l'.057 VTT •1,015 •1.020 •1.026 -1.054 -1.084 • l ' . 109 -1.145 -1.183 -1.217 -l'.24I -T.249 -1.255 -1.255 -1.251 • l'.241 -l'.225 -T.200 - l ' . 166 -1.136 -1.095 ASUM 5̂ 666 4,791 5,430 10,191 25.486 21̂ 553 18,379 18 335 19.559 ASUM 6̂ 772 5,808 6,381 11,032 25.567 20),770 17,431 17,290 18.449 ASUM -1 f,696 •12,560 -6,478 -15.433 -7' 946 -ll,5(/j -11,110 -11,112 -13.140 -ll'.917 - 1 3 ; i i o •12.416 -12', 223 -12,'123 -10.616 -12', 381 -9 316 -10,026 -9 276 -5.801 BSUM .7̂ 097 2,175 7,784 12,464 6.486 -6,815 -7.185 .4'. 044 6'.869 BSUM -6̂ 934 2.417 7,799 12.113 5'.579 -7'.291 -7'.315 • 4'.009 6'.742 YY XX CP SIG 18,06 -24,18 -0.133 l'.068 17.89 -25, • 26 •0.734 0,172 17.76 -26, ,69 .0.107 0.164 17.87 -27 ,40 0.948 0.218 18.00 -27 ,58 -3.794 0,124 18.12 -27 ,40 -3.925 -0.099 18.23 -26 ,69 -1.835 •0.157 18,16 -25 • 27 -0.325 -0,204 18,06 -24 .18 -0.133 -1,070 YY 18,06 17.89 17.76 17.87 18.00 18.12 18.23 18.16 18,06 XX •36,18 •37,26 •38,69 •39,40 •39,58 •39,40 •38.69 '37,27 •36,18 CP •0.117 -0.661 •0.409 0.526 .0,245 -2.033 -1.545 •0.499 -0.117 SIG 0.666 0,095 0.086 0.138 0,121 -0.019 -0,076 -0.128 -0.689 BSUM YY XX CP SIG -4',866 16.10 49,50 -0,030 0.024 8,277 18,30 44,50 -0.040 .0,(106 -3.238 18.55 39,50 -0.053 -0.010 -3.127 18.90 34,50 •0.111 •0.016 •1.312 19,30 29,50 -0.175 -0.009 -4'.578 19.80 24.50 -0.230 0.010 0̂ 351 20,40 19,50 •0.311 0.018 -1,842 21.05 14,50 -0.399 0,011 0.351 21.75 9,50 •0.461 0.061 0,326 22.35 4,50 •0.540 0,020 0.467 22.75 -0.50 •0.560 0.197 t',591 22.90 -5,50 -0.575 0,080 0.869 22.75 -10.50 -0.575 0,0*>6 2.284 22.35 -15,50 •0.565 0.059 1.748 21.75 -20.50 -0.540 0.050 1̂ 846 21.05 -25,50 .0.501 0,028 3,213 20.20 -30.50 -0.440 0.005 -0.417 19.45 -35.50 -0.360 •0.019 7̂ 470 16,85 -40,50 -0.291 •0.052 2.238 18.25 -45,50 -0.199 -0.0>>8 CAM" 0'.23810E-Ol -0'.O26ilC -02-0'.1015hE-Ol -0'.16088E -0l.0', ,9:»665E-62 0 ' . 9 9 9 ( I 1 E " 0 2 0'.17789E"01 o ' . 1 1 4 3 0 E - GAM"» 0 ' . 6 1 i a 5 E - 0 1 Mu< ' ' MU: GNU = V<U=> 0.-71097 VKL = - 0 ' . 7 l 0 S 0 FORCES ON BODY # i , 5 0 CENTER AT C O'.O , o'.O ) CU 3 3',013I'1 COT= 0 ' . 0 6 5 9 9 CMOs 0 ' . 5 7 0 3 B CM« = CIRC« 37'.0638f CLCs 3 ' . o 9 6 4 0 PERIMs 49,298 FORCER ON BODY # 1 7 1 . 1 7 9 CENTER AT ( 4 6 ' . 2 0 , le'.OO) ilT- 2'.07780 C O r = O ' . 4 o 0 9 6 CMOs - 0 ' . 3 S 3 4 7 CM4 = Cl«C= l'.R8B9i CLCs l'.0'l911 PERIMs 7 ' . 3 7 7 5 FORCES ON HODY # i f lo,l8A CENTER AT ( 3 l ' . 2 0 , IB'.OO) CLT = -o',24827 COT= -0* . 0 o 7 5 0 CMOs - 0 : . 1 0 2 7 0 CM4* ClRCs - 0 ' . 4 4 0 1 3 CLCs - o ' . 2 4 4 5 2 PCRIMs 7'.3776 FORCES ON BO')Y #189, 197 CENTER AT ( 2 2 ' . 2 0 , 18 ' .00) CLT3 . o ' . f l a s f l o CDT = - o ' . 0 5 4 2 5 CMOs -0.20537 CM4 = ClRCs -0'.7112B CLCs -0'.111B2 PERIMs 7 ' . 3 7 7 6 FORCES ON IIPOY #198,206 CENTER AT ( 10 ' .20, l f l ' , 0 0 ) CLT= -0',7569l CRTs -o'.14861 CMOs -0'.36254 CM4 = ClRCs - 1 . 17703 CLCs -o'.65391 P E R I Ma 7'.3778 FORCES ON BUOY #207,215 CENTER AT ( -f.80, IB'.OO) CLT= -O',5066S CDT= - o ' . 0 2 9 4 4 CMOs -o'.23201 CM4s ClRCs - 0 . 6 * 2 1 3 CLC= -o'.37S96 PERIMs 7'.3778 ON tfnDY #210,224 CENTER AT C -13'.80, IB'.OO) 0 ' , 4 7 6 7 7 COT= - 0 " . 0 4 0 6 6 CMOs 0 ' . 2 4 9 3 3 CM4 = 0.72990 CICs 0.10550 PERIMs 7 ' . 3 7 7 6 ON BODY # 2 2 5 , 2 3 3 CENTER AT < -25'.80, IB'.OO) 0',7C."66 CDl's -O'. 15769 C MOs 0 ' . 3 5 7 3 5 CMIs 1'.28502 CLCs 0.71390 PERIMs 7 ' . 3 7 7 7 ON tinOY # 2 3 4 , 2 4 2 CENTER AT C -37'.80, l f l ' , 0 0 ) 0',50513 CPTs - o ' . 0 { 7 2 2 CMOs o'.20048 CM4» 0 ' . 7 8 5 l j CLCs 0 ' . 4 3 6 3 4 PERIMs 7 ' . 3 7 7 7 0 1 FORCES C L T = ClRCs FORCES f L T = ClRCs FORCES C L T = C I R C S - 0 ' . 18295 -1.02292 » 0 . 0 4 0 6 4 P0'.09392 - 0 ' . 1 7 3 3 2 • 0 ' . 1 0 S 3 7 0 . 1 3 0 1 4 0 ' . 1 6 1 4 4 0 ' . 0 7 4 1 9 I' EXECUTION TERMINATED isiG S LO xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx^ M I C H I G A N TERMINAL S Y 3 T E M FORTRAN 0 ( 4 1 3 3 6 ) MAIN1 10*22-75 10119136 PAGE POOl 0001 0002 0003 9001 0005 0006 0007 OOOB 0009 0010 0011 0012 0013 0014 0015 0016 SUBROUTINE MAiNlCXX.YY,DX,0Y,D3,CS,S!.N»S!G,M,VTT.CP,GAM,NG.MU, 1.000 i GNU,NM,VTI,XC,YC,NLS) 2.000 REAL XX'(N),YYiN),DX(N),DY'(N),DS(N),CS(M),SI(N) 3.000 XX,YY > CONTROL POINT COORDSf DX,DY,D3 » ELEMENT LENGTHS 4.000 CS,SI - COS,SIN OF ELEMENT INCLINATION 5.000 REAL XC(NLS),YC(NLS) 6.000 XC,YC - CENTERS OF WALL SLATS 7.000 N i TnTLA * CONTROL POINTS 8.000 REAL Slr,(M),VTT(N),CP(N),r.AM(NC),MU(NM),GNU(NM) • •• 9.000 SIG - SOURCE STRENGTH DENSITIES (ALSO USEO AS SOLUTION VECTOR I N 10.000 SYSTEM ) 10.000 (GAM,MU,GNU,' ARE PART OF SIG) 11.000 M- TOTAL * UNKNOWNS IN SYSTEM C*SIG=D 12.000 VTT,CP - TANG VEL, PRESSURE COEFF'. 13.000 GAM - VORTEX STRENGTH DENSITIES ON TEST AIRFOIL t S L A T S 14.000 MIJ.GNU - SOURCE 8 VORTEX STRENGTH DEN'S ON STREAMLINE REPRESENTING 15.000 S'.L'. 15.000 REAL V T I ( N M ) 16.000 VTI - PRESCRIBED TANG'L VEL ON SHEAR LAYER STREAMLINE 17.000 CALAB,CALCD,CALCP,CALCL - IF NONZERO CALCULATE A,B,C,D,CP,CL 18.000 K- R A B, w R C D - IF MONZr-IP WRITE A,B,C,D INTO FILES 19.000 SOLV - IF NONZERO SOLVE SYSTEM OF EONS C*SIG=0 . 20.000 GAUSS - IF NONZERO USE GAUSS-ELIMINATION 21.000 ITER - IF NONZERO USE ITERATIVE METHOD 22.000 HSIC - IF NONZERO ALREADY HAVE SIG IN FILE FROM PREVIOUS R U N 23.000 INTEGFR CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITER,CALCP,CALCL,H3IG 24.0 0 0 COMMON/81/ N»S,NSLAT,NSU1,NKA,N«<2,MSV,NA,NSPS,NTEU,NTEL 25.000 N*3 - TOTAL # CONTROL POINTS ON AIRFOIL SOLID WALL SECTIONS & S L A T S 26.000 NSLAT'.NSPS - aSLATS,((CONTROL PTS /SLAT 27. 000 flSU 1 - 1ST CONTROL ON 1ST SLAT 28.000 NKA - EON M FOR KUTTA CONO'N ON TEST AIRFOIL 29,000 MSV - » CONTROL PTS ON STREAMLINE FOR SHEAR LAYER ( M U , G N U ) 30.000 NA - ((CONTROL PTS ON SINGLE TEST AIRFOIL 31.000 NTEU, M T F L - CONTROL PT * » TEST AIRFOIL TU'. ( U ' . L ) 32.0Q0 NM2-- « OF LAST EON FOR Z E R O NORM VEL ON INNER EDGE OF S.L'. 33.000 C0MH0N/B2/ U,CH 34.000 U - UNIFORM ONSET STREAM SPEED 35.000 CH - SINGLE TEST AIRFOIL CHORO 36.000 COHMON/P.3/NU1,NWIII (NU3,MWU2,NL1.NWL1,NL3,NWL2»NSOL1,NSOLSL,NF1, 37.000 1 NFLAT,N3PF,NI1 38,000 NU1,NU3.NL1,NL3 - 1ST CON PT ON EACH FLAT SOLID WALL SECTION 39.000 NWIJ1,NWU2,NWL1 ,NWL2 - « CON PTS ON EACH F L A T 30LIO WALL SECTION 40 . 000 N30LSL - * CON PTS ON ARBITRARY SHAPED SOLID SURFACE E'.G', PLENUM 41.000 BOUNDARY 41,000 NSOL1 - 1ST CON PT ON ARBITRARY SOLID SURFACE 42.000 NFLAT,NSPF - « FLAT SLATS, « CON PTS OFLAT SLAT - NO K U T T A APPLIED .43.000 NF1 - 1ST CON PT ON 1ST FLAT SLAT 44.000 Nil - 1ST CON PT ON SHEAR LAYER 45.000 COMMON/B4/CALAB,CALCD,WRAB,KRCD,SOLV,GAUSS,ITER,CALCP.CALCL.HSIG 46.00 0 EXTERNAL CALCAB,ASSEMA,AS3EMB,ASSEM0,RE,WR,WRD,ATXB,CPS,FORCES 47.000 N3VT=N " 4 8.000 NUNsM 49.000 NDA=4*N3VT*NSVT 50.000 NDBsNDA 51.000 NDCs4*NUN*NUN 52.000 I—' 4s> MICHIGAN TERMINAL SY3TEM FORTRAN G(41336) ..AINl 10-22-73 tOt19136 PAGE P002 0017 CALL GSPACE(A,NDA.0,&30n 53.0Q0 o o i a CALL GSPACE(B,NDH,0,&302) 54.000 0019 iF(CALAB.Nc'.O) GO TO 200 5 5 . 0 0 0 0 0 2 0 LA = 3 56.000 0 021 MsNSVf 5 7 . 0 0 0 0 0 2 2 HaNSVT 58.000 0 0 2 3 CALL CALLER(RE,AfiPTR(N),IPTR(M),IPTR(LA)) 59.000 0021 LB = 4 60.000 0 0 2 3 N=NSVf - 61.000 0026 M=NSVT 62.000 0027 CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)J 63.000 o o 2 e GO TO 2ol 64,000 0029 200 N=NSVf 65.000 0030 MrNSVT 66.000 0031 CALL CALLER{CALCAB,A,B,IPTR'(N),IPTR(M),IPTR(XX),IPTR(YY5,IPTR(0X), 67.000 1 IPTR(0Y),IPTR(05), IPTR(CS),IPTR(SI» IF'(HSIG),NC,0) GO TO 201 68.0 00 0032 69,000 O033 IF(WRAB.NE'.O) GO TO 202 70.000 0034 IFCCALCD'.EO'.O) GO TO 2 0 3 71.000 0 0 3 5 GO T0 204 72,000 0036 202 MsNSVT. 73.000 0 0 37 M=NSVT 74.000 0 0 3 6 LA = 3 75.000 0039 CALL CALLER(WR,A,IPTR(N),IPTR(M),IPTR(LA)) 76.000 C 0 1 « N=NSVT 77.000 o o i i M=NSVT 78.000 0042 LB = 4 79.000 0 0 1 3 CALL CALLER(WR,B,iPTR(N),IPTR(M)»IPTR(LB)) 80.000 O O H IF(cALCO'.eo'.O) GO TO 205 81.000 0045 204 IFC^RAH'.NE.O) GO TO 206 82,000 0016 N=NSvf 83.000 0017 MsNSVT 84,000 0 0 1 6 LB = 4 85.000 0 0 1 9 CALL CALLER(WR, B, JPTR (N) , IPTR (M) , IPTR(LB)) 86.000 0050 206 CALL FSPACE(B,»303) 87.000 C051 CALL r,SPACE(C,NDC,0,&304) 88.000 0052 M=NSVT 89.000 0053 MBNIJN 90.000 0 0 5 1 CALL CAi LERitASSEMA,A,C,IPTR(N),iPTR(M)) 91.000 0055 IFCWRAR'.NE.O) GO TO 207 92.000 0056 LA-:: 93.000 0057 MsNSVT 94.000 0058 MzNSVT 95.000 O059 CALL CALLER(WR,A,iPTR(N),IPTR(M)fIPTR(LA)) 96.000 0060 CALL FSPACE(A,*305) 97.000 0061 N0B=4*"SVT*NSVT 98.000 0062 CALL GSPACEitB,NOB,0,8306) 99.000 0063 LB = 4 100.000 0061 N=NSVT 101.000 0065 MsNRVT 102.000 0066 CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)> 103,000 0067 207 N=N3VT 104.000 0 0 6 e M = NIJN 103,000 0069 CALL CALLER(A3SEMB,B,C,IPTR(N),IPTR(M)) 106.000 0070 ND0=4*NUN 107.000 Ul MICHIGAN TERMINAL SYStEM FORTRAN G(«13S6) MAIN1 10-22-73 10119136 0071 CALL r,SPACE(D,N0D.0,i307) 108.000 0072 NsNSVT 109.000 0073 M=NUN 110.000 0071 N i = M 51V 111.000 0075 IFtMSV.CO'.O) Nfal 112.000 0076 N2=NXE - 113.000 0077 IF(MXE.EQ'.O) N2 = l 111.000 0078 CALL CALLER(A.1SEMr),0,IPTR(M),IPTR(CS),IPTR(SI),IPTR(N),IPTR(VTn, 115.000 l" 1BTR(N1).IPTR(VT0),IPTR(N2)) 116.000 0079 iF(SOLV'.LQ.O) GO TO 208 1 17 .000 0080 IF(r,All33'.NE'.0) GO TO 209 l ie .000 0031 IF(ITER.NE.O) GO TO 210 119.000 0082 208 IF(WPCO.I:0',0) GO TO 21 1 120.000 0033 210 LC=7 121.000 0081 NsHi.iN -• 122.000 0085 MsNUN 123.000 0086 CALL CALLER(WR,C,iPTR(N),iPTR(M),IPTR(LO) 124.000 0087 LD=8 125.000 00?8 MrNUN 126.000 0089 CALL CALLERC<.Rn,D,IPTRCN),IPTR<LD)) 127.000 0090 IF (GAUSS.EO'.O) GO \»i 212 128.000 0091 209 LU=6 129.000 C092 MsNlIN 130.000 0093 CALL CALLER(ATXB,iPTR(M),C,iPTR(SIG),0,IPTR(LU)) 131.000 C091 KRITEJ9) SIG 132.000 0095 wniTE(6,900) M 133.000 0096 900 FORMATCSIGCIS.')') • 134.000 0097 wRITE(h,3) GIG 135.000 0098 CALL F5PACECC&308) 136.000 0̂ 99 IF(CALCP'.EO'.O) GO TO 216 137.000 0100 201 iF(HSlG'.L'O'.O) GO TO 214 138.000 010f READ(9) SIG 139.000 0102 WRITE(6,3) SIG 140.000 0103 3 FORMATdX, 10C12.5) I'll.000 0104 214 IF{HSlG>E.O) CO TO 215 112.000 0105 NOAs'l*NSVT*NSVT 143.000 0106 CALL r,SPACE(A,NDA,0,*309) 144.000 0107 LA=3 . 145.000 0108 NsNSVT . . . 146.000 0109 MsNSVf 147,000 0110 CALL CALLER(RE,A,iPTR(N),iPTR(M),IPTR(LA)) 118.000 C ! l { 215 NrNSVf 119.000 0112 MrNUN 150.000 ' 0113 KsNJLAT 151.000 C114 IF (NSLAT.EO'.O) KSJ - 152.0Q0 0115 LsMsV 153.000 0 1 1 6 iF(MSV.EO'.O) Lsl , 154 .000 0117 CALL CALLER(CPS,IPTR(CP),IPTR(VTT)nPTR{XX),IPTR(YY),IPTR(C3)» 155.000 1 IPTR(SI),JPTR(N),IPTRCSIG),IPTR(M),IPTR(GAM),IPYR(K),IPTR(MU)r 156.000 2 iPTR(GNU),jPTR(L),A,B) 157.000 Olie 213 IFCCALCL'.EO.O) GO TO 216 158.000 0J19. CALL FfiPACE(A,&310) 159.000 0120 CALL FSPACE(B,&311) 160,000 0121 . NsNSVT 161.000 0122 Nl=i 162.000 •liCHIGAN TERMINAL SYSTCM FORTRAN G<41336) MAIN! 10-22-75 10119136 PAGE P004 0123 0124 0125 0126 0127 0126 0129 0130 C 13 i 0132 0133 0131 N2SNA XA--0'. YA = o'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(DX).IPTR(DY), 1 IPTP(OS),IPTR(VTT),IPTR(N),IPTR(U),IPTR(CM),IPTR(NI),IPTR(NJ), 2 IPTR(XA),1PTR(YA)) IF(NaLAT'.Ed'.O) GO TO 2 00 { K = 1,NSL AT Nl = MSlll + NSPS*(K-l) N2=M1+NSFS-1 CHrj'.h xs=xc'rK) YS=i«'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(OX).IPTR(DY), 1 IPTR (IIS), IPTR (VTT), IPTR (N), IPTR (U), IPTR (CH), IPTR (NI), IPTR ( N 2 ) , 2 IPTR(XS),IPTR(YS)) 0135 2 CONTINUE 0136 IF(HRIG'.NE.O) GO TO 217 0137 216 CONTINUE 0138 GO TO I 99 0139 203 STOP 203 o i i o 205 3T0P 205 c i i i 211 STOP 211 0142 212 STOP 212 0113 217 STOP 217 0111 301 STOP 30 i 0145 302 3T0P 302 0146 303 STOP 303 0147 304 STOP 301 0148 305 STOP 305 0149 306 STOP 306 0150 307 STOP 307 0151 308 STOP 308 0152 309 STOP 309 0153 . 310 STOP 310 .0154 311 STOP 311 0155 99 RETURN 0156 END •OPTIONS •OPTIONS • STATISTICS •STATISTICS* IN IN EFFECT • IO,EnCDIC,SOURCII,N0LI3T,NODECK,LOAO,NOMAP EFFECT^ NAME = MAINl , LINECNT a . 37 3J . J , :LE STATEMENTS = 156,PROGRAM SIZE s NO DIAGNOSTICS GENERATED 69S2 163.000 164.000 165.000 166.000 167.000 168.000 169.000 170.000 171.000 .172.000 173.000 174.000 175.000 176.000 177.000 n e . o o o 179.000 180.000 181.000 182.000 183,000 184.000 185,000 186.000 187.000 188.000 189.000 190.000 191.000 192.000 193.000 194.000 195.000 196.000 197.000 198, 000 199.000 200,000 NO ERRORS IN MAINl LO MICHIGAN TERMINAL SYSTEM FORTRAN G(41536> CALCAB 10-22-73 0001 SUBROUTINE CALCAB(A,B,N,M,XX#YY,DX,DY,DS,C3,SI) C CALCAB CALCULATES MATRICES A, B OF INFLUENCE COEFFICIENTS 0002 REAL XXCN),YYCN),DX(N),DY(N),OS(N),CS(N),SI(N) 0005 REAL A(N,M),B(N,M) 0 0 01 COHMOM/Bl/ NWS,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS.NTEU,NTEL 0005 DO ? I = i,M 0006 DO 2 J*1,N 0007 IFCl '.EQ'.J) GO TO J 0008 OXJ=XX(I)-XX(J) 0009 DYJiYYd)-YY(J) C X . I . Y J - DIST. riCE OF 'II TO ' J ' IN 'J< COORD'. SYSTEM 0010 XJ = DX . !*CS (J)+DYJ*SI ( J ) 0011 YJ=DYJ*C5(J)-DXJ*3I(J> 0012 l)SJ2 = r>r>CJ)/2'. 00 1 S D3J4 = DS.I2*D5J2 0011 XJS = X.I*XJ 0015 YJS = Y.r*YJ 0016 XP = X.J + DSJ2 0017 XM=XJ-DSJ2 0016 XP3=XP*XP 0019 XMS=XM*XM C XJ IS ZERO IF ELEMENTS VERTICALLY ABOVE EACH OTHER 0020 IFCXJ'.EQ.O.) GO TO 140 C PHIX IS VELOCITY IN *IND DIRECTION 0021 PHIX=ALOG((XPS+YJS)/(XMS*YJS)) 0022 GO TO 141 0021 140 PHiXrO'. C Y J IS ZERO IF ELEMENTS ARE ON SAME FLAT HALL SECTION 0024 141 IFfYJ'.EO'.O'.) GO TO 142 C PHIY IS VELOCITY PERP*. TO WIND RIRN 0 025 PHIY=2.*ATAN2((0S(J)*YJ),(XJS+YJS-DSJ4)) 0026 GO TO 143 <027 142 PHlYao. 0028 143 IFCSI '(.D'.EO'.O'.) GO TO 144 0 029 31•J = Sl(i)«CS(J)-CSCI)*SI(J) 0030 coj=cs(i)*cs(j)tsi(i)*si(J) C A is NORMAL VEL IN 'I' COORD SYSTEM c 8 is TANG ' L VEL IN W COORD SYSTEM 0031 A(J,I)=PHIY«COJ-PHIX*SIJ 0032 B(J,I)=PHIX*COJ+PHIY*SIJ 0033 GO TO 2 0034 3 A(TJ,I) = 6'.283l05 0035 B(J,I) = o ' . 0036 GO TO 2 0037 144 3ij=SI(I)*CS(J) 0038 COJ=CSCI)*CS(J) 0039 A(J,I)=PHIY*C0.1-PHIX*SIJ 0040 B(J,I)=PHIX*COJ*PHIY*SIJ 0041 2 CONTINUE 0.042 RETURN 0013 END ': • OPTIONS I N EFFECT* ID,EBCDIC,SOURCE, NOLIST, NODECK»LOAD, NOM AP • OPTIONS I N EFFECTii NAME = CALCAB , LINECNT » 57 •STATISTICS* SOURCE STATEMENTS s 43,PROGRAM S I Z E B . 1700 •STATISTICS* NO DiAGNOSTTCS GENERATED NQ ERRORS IN CALCAB 10119137 PAGE P001 2 0 1 . 0 0 0 2 0 2 . 0 0 0 203.000 204.000 205.000 206.000 207.000 208.000 209.000 210.000 211.000 212.000 213.000 214.000 215.000 216.000 217,000 2 i e . o 6 o 219,000 220.000 221.000 222.000 223.000 224.000 225.000 226.000 227.000 22e,000 229,000 230.000 231.000 232.000 233.000 231,000 235.000 236.000 237.000 23e.000 239,000 240.000 241.000 242.000 243.000 244.000 245.000 246.000 247.000 248,000 249.000 250.000 251.000 U) CO MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) \SSEMA 1 0 - 2 2 - T 5 1 0 1 1 9 | 3 7 PAGE P 0 0 I 0 0 0 1 0002 0003 0001 0005 0 0 06 0007 0008 0009 0010 0011 0012 0013 0011 0015 0016 (.017 0018 0019 0020 0021 0022 0023 0021 0025 0 026 0027 0028 0029 003Q C031 0032 0 0 3 3 0 0 3 1 0 0 3 5 C 0 3 6 0 0 3 7 0 0 3 8 0 0 3 9 ooio 0 0 1 1 0 0 1 2 SUBROUTINE AS3EMA(A,C,N,M) c c is MATRTX FOR SYSTEM C * S I G » D C ASSC.MA ASSEMBLES THOSE PARTS OF C THAT DEPEND ON A'. REAL A(N#N).C(M,M) INTEGER i:,P.O C0MMON/B1/ NWS,NSLAT,NSUt,NKA,NM2,MSV,NA,NSPS,NTEU,NTEL • COMMON/B2/ U,CH NSPsNSPS-1 NN2=NM2+MSV NWSV=MWS+M8V C»**«« LOOP 1 - ASSEMBLE NORMAL VEL EQNS FOR ALL NWS CONTROL PTS ***** DO 19 1=1,NWS C E iS EOUATION « E = I C L00P2 - NORM VELS AT A L L NWS C, P'. DUE TO ALL NWS SOURCE ELEMS DO 2 .1=1, NWS 2 C(J,E)=A(J,i) iF(MSV.EB'.O) GO TO 19 C LOOP 8 - NORM vELS AT A L L NWS CON PTS DUE TO SOURCE ELEMS (MU) ON C S'.L DO 8 K=i,HSV J = N K A + K M=NWS+K 8 C(J,E)=ACM,i5 19 CONTINUE IFCNSL AT'.EO'.O) GO TO 12 C****«LOOP '[O - ASSEMBLE KUTTA EONS FOR AIRFOIL-SHAPED SLATS ***** DO 52 KS=1,NSLAT KL = NSUl+N3PS*(K3-'l) • KU=KL+NSP E = N*'S + KS C LOOP 13 - TANG VELS AT T.E*, 00 13 Ks'l, NSLAT J=NwS+K P = NSU'f*NSPS*<K-l) Q = P+NSP . . . SA = o'. DO 11 M=P,Q 11 SA=SA+A(M,KL)+ACM,KU) 13 C(J.E)=SA SA = o'. C LOOP 15 - TANG VELS AT T'.E', OF SLATS DUE TO VORTEX ELEMS ON TEST C AIRFOIL DO 15 K=1,NA 15 SA=3A+A(K,KL)+A(K,KU) C(NKA,E)=3A IF(MSV.EO'.O) GO TO 52 C LOOP l"8 - TANG VELS AT T'.E', OF SLATS DUE TO VORTEX ELEMS ON (GNU) C S'.L DO 18 Kil.MSV J=NM2+K M=NKS+K 18 CCJ,E)=A(M,KL)*A(M,KU) 52 CONTINUE 12 IFtNSLAT.EO'.O) GO TO 20 OF SLATS DUE TO VORTEX ELEMS ON SLAT3 252.000 253.000 251.000 255.000 256.000 257.000 258.000 259.000 260.000 261.000 262.000 263.000. 261.000 265.000 266 .000 267.000 268,000 269.000 270.000 270.000 271 272 000 000 273.000 271.000 275 276, 000 000 277,000 278.000 279.000 280.000 281.000 282.000 283.000 281.000 285.000 286,000 287.000 28e.000 289,000 290.000 2 9 1 .000 292.000 292.000 293.000 291.000 295.000 296.000 297.0Q0 297.000 298.000 299.000 300.000 3 0 1 ,000 302.000 303.000 CO MICHIGAN TERMJNAL SYSTEM FORTRAN GCH336) ASSEMA 10-22-T5 10ll9|37 PAGE P002 0013 0 0 1 1 0015 0 0 1 6 0 0 1 7 00 IB 0 0 '19 0 0 5 0 o o 5 i O032 0 053 0 0 5 1 0 0 5 5 0 0 5 6 0 0 5 7 0 0 5 8 0 0 5 9 0 0 6 0 0 061 0 0 6 2 0 0 6 3 0 0 6 1 0 0 6 5 0 0 6 6 0 0 6 7 0 0 6 6 0 0 6 9 0 0 7 0 0 0 7 1 C 0 7 2 0 0 7 3 0 0 7 1 0 0 7 5 0 0 7 6 0 0 7 7 C07B 0 0 7 9 0 0 8 0 0 0 8 i 0 0 8 2 0 0 8 3 22 21 20 C C 23 26 21 C*****ASSCMfiLC KUTTA EONS FOR TEST A I R F O I L ***** 301.000 C LOOP 2{ - TANG VELS AT TEST A I R F O I L T.E-. DIT'. VORTEX ELEMS ON S L A T S 3 0 5 . 0 0 0 00 21 Kai,NSLAT 3 0 6 . 0 0 0 P B U 3 U 1 + N S P S * ( K - 1 ) 3 0 7 . 0 0 0 OsPtNsP 3 0 8 , 0 0 0 JoNWS+K 3 0 9 , 0 0 0 3A = u'. 3 1 0 . 0 0 0 DO 22 MaP.Q 3 1 1 . 0 0 0 SA=SA+A(M,NTEU>+ACH,NTEL) 3 1 2 . O p O C(J,NKA)=SA 3 1 3 , 0 0 0 3A = o'. 3 1 1 . 0 0 0 LOOP 2 3 - TANG VELS AT TE3T A I R F O I L T'.E*. O'.T'. VORTEX ELEMS ON TEST 3 1 5 . O o O AIRFOIL 3 1 5 . 0 0 0 DO 2 3 Kil.NA 3 1 6 , 0 0 0 3A = SA + A'CK,NTEU)+A(K,NTEL) 3 1 7 . 0 0 0 C(NKA,HKA)=SA 3 1 8 , 0 0 0 iFCMSV'.EQ'.O) GO TO 2 1 3 1 9 . 0 Q 0 LOOP 26 - TANG VELS AT TEST A I R F O I L T.E*. o'.T*. VORTEX ELEMS (GNU) ON 3 2 0 . 0 0 0 DO 26 K = 1,MSV 3 2 1 . O O O J = M<2 + K 3 2 2 . 0 0 0 M=NwS+K 3 2 3 . 0 0 0 C(J,NKA)=A(M,NTEU)*A(M,NTEL) 3 2 1 . 0 0 0 I F ( M S V . E O ' . O ) GO TO 61 3 2 5 . 0 0 0 C«****ASSEMf)LE NORMAL VELOCITY EONS FOR MSV CON PTS ON S,L'. 3 2 6 . 0 0 0 DO 27 KM=1,MSV 3 2 7 . 0 0 0 IsMwS+KM 3 2 8 . 0 0 0 E=NKA+KM 3 2 9 . 0 0 0 C LOOP 28 - NORM V E L S AT MSV CON P T S O'.T'. A L L NWS SOURCE E L E M S 3 3 0 , 0 0 0 0 0 28 Jal,NWS 3 3 1 . 0 0 0 C ( J . E ) = A ( J , I ) 3 3 2 . 0 0 0 LOOP 33 - NORM VELS AT MSV CON PTS O'.T'. A L L MSV SOURCE ELEMS (MU) 3 3 3 . 0 0 0 ON INNER 3 3 3 . 0 0 0 DO 3 3 Kel.MSV . 3 3 1 . O g o J = Nk'AtK 3 3 5 . 0 0 0 MBNHS+K 3 3 6 . 0 0 0 C(J,E)=A(M , i ) 3 3 7 . 0 0 0 CONTINUE 3 3 e . « o o ***»*ASSCMBLC TANG'L V E L EONS FOR MSV CONTROL P O I N T S ON INNER EDGE OF 3 3 9 . 0 0 0 S IL*.***** 3 3 9 , 0 0 0 DO 35 KM=1,MSV 3 1 0 , 0 0 0 I=NWS+KM 3 1 1 , 0 0 0 E=NM2+KH 3 1 2 , 0 0 0 IFCMSLAT'.En'.O) GO TO 37 3 1 3 , 0 0 0 LOOP 38 - TANG VELS AT MSV CON PfS D'.T'.ALL VORTEX EiEMS ON S L A T S 3 1 1 . 0 0 0 DO 38 Kc1/NSLAT 3 1 5 , 0 0 0 PsMSUl+NSP3*CK-l) 3 1 6 . 0 0 0 OsPtNRP 3 1 7 . 0 0 0 J=NwS+K 3 1 8 . 0 0 0 3A = o'. 3 1 9 . 0 0 0 DO 39 M=P,0 3 5 0 . 0 0 0 3A = SA + A(M,I) • • - • 351 . 0 0 0 C(J,E )=3A 3 5 2 . 0 0 0 SA = o'. 3 5 3 . 0 0 0 LOOP 10 » TANG V E L S AT MSV CON PTS D.T'.ALL VORTEX CLEMS ON TEST 3 5 1 . 0 0 0 3 5 1 . 0 0 0 28 3 3 27 C C 39 38 37 C C AIRFOIL O MfCHICAN TERMINAL SY3TCM FORTRAN C ( H 3 3 6 ) A9SEMA t0-22"75 l o t 10137 PA6E POOS 0081 0035 0036 DO 40 K « 1 , N A 10 3AaSA + A ' ( K , n C(NKA,E)a3A LOOP 42 • TANO V E L S AT MSV CON PTS D'.T.ALL VORTEX E L E H S (GNU) ON C . C s L" 0097 * * 0 0 4 2 Kil.MSV 0 0 8 8 JeNM2+K 0 0 3 9 MsNwS+K OO'Q 1 2 C(J.E)=A(M,i) t'O'l 3 5 CONTINUE 0 0 9 2 61 CO'iTlNUE 0 0 9 3 RETURN 0094 END •0PTJOM3 I N EFFECT* ID , EHCDIC , SOURCE,NOLI3T,NODECK#t.OAD#NOMAP •OPTIONS IN EFFECT* NAME = ASSEMA , LINECNT • . 5 7 •STATISTICS* SOURCE STATEMENTS s 94,PROGRAM SIZE a •STATISTICS* NO DIAGNOSTICS GENERATED NO CRRORS I" ASSEMA 353 . 0 0 0 356 , 0 0 0 3 5 7 . 0 0 0 3 5 e . 0 0 O 3 5 8 . 0 0 0 3 5 9 . 0 0 0 3 6 0 , 000 3 6 1 . 0 0 0 3 6 2 . 0 0 0 3 6 2 . 5 0 0 3 6 3 . 0 0 0 3 6 4 . 0 0 0 3 6 5 . 0 0 0 2980 MICHIGAN TENMjNAL SYSTEM FORTRAN 0(11336) 'SSEMB 10»22»75 10U9I38 PAGE P001 0001 0002 0003 0001 0005 0006 0007 0008 0009 o o i o e c u 0012 0013 con O0I5 O0I6 0017 t i o i e 0019 0020 0021 0022 0023 0021 0025 C026 0027 0028 0029 0030 0031 0032 0C33 C031 0035 0036 0037 0038 0039 ooio 001 i 0012 0013 SUBROUTINE ASSEMBCB,C,N,M) 3 6 6 . 0 0 0 C A3SEMB ASSEMBLES THOSE PARTS OF C THAT DEPEND ON B*. 3 6 7 . 0 0 0 REAL B<N,N) ,C (M,M) 3 6 8 . 0 0 0 INTEGER C.P.O 3 6 9 . 0 0 0 COMMON/BIZ NW3,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS»NTEU,NTEL 3 7 0 . 0 0 0 COMMON/B2/ U,CH — 3 7 1 . 0 0 0 N S P = N S P S - 1 3 7 2 . 0 0 0 MN2=NM2+MSV 373.OQ0 NWSV=NHS+MSV 3 7 1 , 0 0 0 C*****ASSEMRLC NORMAL VEL EQNS FOR ALL NWS CON PTS ***** 375.OgO 00 i|9 1 = 1,NWS 3 7 6 . 0 0 0 E=I 3 7 7 . 0 0 0 IF (NSLAT'.EO'.O) GO TO 3 3 7 8 . 0 0 0 C LOOP i\ - NORM VELS » ALL NWS CON PTS D'.T'. VORTEX ELEMS ON SLATS 3 7 9 . 0 0 0 DO 1 K=i,NSLAT 3 8 0 . 0 0 0 J=NWS+K , 3 8 1 . 0 0 0 P=NSUi+NSRS*(K-l) 3 8 2 . 0 0 0 0=P+NSP - 3 8 3 . 0 0 0 3B = o'. 3 8 1 . 0 0 0 00 5 "=P,Q 3 8 5 . 0 0 0 5 Se=SS-B(M,I) 3 8 6 . 0 0 0 I C C J , E ) = S B 3 8 7 . 0 0 0 3 3B = o'. 3 8 8 . 0 0 0 C LOOP 6 - NORM VELS » A L L NW3 CON PTS o'.T'. VORTEX ELEMS ON TEST 3 8 9 . 0 0 0 C AIRFOIL 3 8 9 . 0 0 0 DO 6 K=i,NA 3 9 0 . 0 0 0 6 3B=SR-B(K,I) 3 9 1 , 0 0 0 C(NKA,E)=OB 3 9 2 . 0 0 0 IF(MSV.CQ'.O) GO TO 1 9 3 9 3 . 0 0 0 C LOOP 9 - NORM VELS • A L L NWS CON PTS D'.T'. VORTEX ELEMS (GNU) ON 3 9 1 .000 C S ' . L ' . 3 9 1 . 0 0 0 00 9 K = i",MSV 3 9 5 . 0 0 0 J=NM2+K 3 9 6 , 0 0 0 M=NWS+K 3 9 7 . 0 0 0 9 C(JiE)=-B(M,I) 3 9 8 . 0 0 0 1 9 CONTINUE 3 9 9 . 0 0 0 iF(NSLAT.t'O'.O) GO TO 12 1 0 0 . 0 0 0 C*****A3SrMP.LC KUTTA EONS FO AIRFOIL-SHAPED SLATS * * * * * 1 0 1 . 0 0 0 DO 5 2 KS=t,NSLAT -- - 1 0 2 . 0 0 0 KL = NSlll+N3PS*CKS-l) 1 0 3 . 0 0 0 KU-K L+NSP IOI.OOO E=NwS+KS 1 0 5 . 0 0 0 C LOOP i l - TANG VELS # T'.E'. OF SLATS D'.T', A L L NWS SOURCE ELEMS • 106.OOO 00 11 Jsl.NWS 1 0 7 . 0 0 0 I I C(J,E)=B(J,KL)+B(J,KU) • 1 0 8 . 0 0 0 IFCMSV.EO'.O) GO TO 5 2 1 0 9 , 0 0 0 C LOOP 1 7 - TANG VELS • T'.E', OF SLATS D'.T. MSV SOURCE ELEMS (MU) ON 1 1 0 , 0 0 0 C S'.L. 1 1 0 . 0 0 0 DO 1 7 K=1,MSV 1 1 1 . 0 0 0 J = NK A + K 1 1 2 . 0 0 0 MsNWStK 1 1 3 . 0 0 0 1 7 C(J,E)=B(M,KL)+B(M,KU) 1 1 1 . 0 0 0 5 2 CONTINUC 1 1 5 . 0 0 0 C LOOP 1 9 - TANG VELS « TEST AIRFOIL T'.E'. O'.T', ALL NWS SOURCE ELEMS 1 1 6 . 0 0 0 1 2 DO 1 9 Jal,NWS 1 1 7 . 0 0 0 t o M I C H I G A N T E R M I N A L S Y 3 T E M F O R T R A N G ( 4 1 3 3 6 ) A 3 3 E M B 10-22-75 10110)38 P A G E P 0 0 2 0 0 0 1 9 0 1 5 0016 0047 0048 004") 0 050 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0 062 0063 0064 0065 0066 0067 0068 0 0 69 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081* 0082 0083 0084 0085 •OPTIONS 1 9 C(J,NKA;aB ( J ,NTEU)+B ( J ,NTEL) IF (MSV.CO',0) GO TO 24 C LOOP 25 - TANG V E L S * T E S T A I R F O I L T . E * . o'.T. MSV S O U R C E E L E M S ( M U ) C ON INNER . 1)0 23 Ksl.MSV J s N K A + K M=NWS+K 25 C(J.NkA)nB(M,NTEU)+B(M,NTEL) 24 IF (MSV .Efl'.O) GO TO 61 C**«**ASSEMBLC NORMAL VEL E Q N S FOR MSV CON PT3 ON S H E A R L A Y E R * * * * * 00 27 KMaijMSV ISNKS+KM E=NKA+KM IF(NSLAT'.Eu'.O) GO TO 29 C - LOOP 30 - NORM V E L S • MSV CON P T S D , T ' . V O R T E X E L E M S ON S L A T S 00 30 Ks1,NSLAT JrNWS+K P=N3U1+NSPS*(K-1) 0=P+NSP SB = 0'. DO 31 M=P,0 31 3BsSB-B'(M,l) 30 C(J.E)»SB 29 3B = o'. ^. • C - LOOP 32 - NORM V E L S » MSV CON P T S D .T . V O R T E X E L E M S ON T E S T C A I R F O I L DO 32 K=l,NA. 32. 3B=SB-B(K,I) C(t«A,E) = SB C - LOOP 34 - NORM VEL3 • MSV C O N P T S O ' .T ' . MSV V O R T E X E L E M S ( G N U } ON C s' L ' DO 34 Ksl . M S V J=NM2,K M=NW8+K 34 C(J,E)=-B(M,I) 27 C O N T I N U E . . . C*****AS3CMRLC TANG'L V E L E O N S FOR MSV CON P T S ON S H E A R L A Y E R * * * * * DO 33 KM=1,MSV I=NW5*KM E=NM2+KM C LOOP 36 - TANG V E L S P MSV CON P T S D .T , A L L N W 3 S O U R C E E L E M S DO 36 Jrl.NWS 36 C S . L C(J,E)=B(J,I) LOOP 41 - TANG VELS » MSV CON P T S O'.T". MSV S O U R C E F! E M S ( M U ) ON DO 41 Ksl.MSV JrNKA-tK M=NwS+K 41 C(J.E)=H(M,i) 35 CONTINUE 61 CONTINUE 64 CONTINUE RETURN END IN EFFECT* ID,EBCDiCSOURCE,NOLIST,NODECK,LOAD,NOMAP 418.000 419.000 420.000 420.000 421.000 422.000 423.000 424.000 425.000 426.000 4 2 7 , 42e, ooo ooo 429.000 430.000 431.000 432.000 433.000 434.000 435.000 436.000 437.000 438.000 439.000 440.000 4 41.000 4 41,000 442.000 • 443.000 444,000 445,000. 445.000 446,000 447.000 448.000 449,000 450.000 451.000 452.000 453.000 454.000 455.000 456.000 457.000 458.000 458.000 459.000 460.000 461 .000 462,000 462.500 463.000 464.000 465.000 466.000 MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) SSEMO 10-22-75 10H9I3B PAGE PO01 0001 0002 OC03 oooo '0005 0006 0007 OOOB 0009 0010 0011 0012 00 13 C011 0015 0016 0017 0018 0019 0020 0021 0022 0023 0021 0025 0026 0027 J02C 0 029 *OPTJONS *OPTIONS • S T A T I S T I C S * • S T A T I S T I C S * SUBROUTINE A S S E M 0 C 0 , M / C 3 , S I , N , V T I . N 1 , V T 0 , N 2 ) 167.000 A3SEMD ASSEMBLES R ' . H ' . S . VECTOR FOR SYSTEM C * S I G « 0 16e.000 R E A L 0 < M ) , C S ( N ) , S l ( N ) , V T I ( N i ) , V T 0 ( N 2 ) 169.000 INTEGER P . O 170.000 C0MM0M/H1/ I J * S , N S L A T , N S U 1 ,NKA,NM2,MSV,NA,NSPS ,NTEU,NTEL 171.000 C0MM0N/P2/ U , C H 172.000 N S P = N S P S - 1 173.000 NWSV=MWS+MSV 171,000 NN2=NM2+H3V 175.000 L O O P 13 - NORMAL ONSET FLOW VEL AT ALL NWS CON PTS 176.000 00 13 1 = 1 ,NWS 177.000 niti)=u*stm i7e.o6o I F ( w S l.AT ' . E O . O ) GO TO 11 179.000 L O O P 15 - T A N G ' L ONSET FLOW VEL3 # T'.E*. OF SLATS 180.000 D O 13 K s t , N S L A T 181.000 I=NnS+K 182.000 P = l v S U l + M S P S * ( K - l ) 183.000 0=P+NRP 181,000 n ' ( I )a .U*CCS ' (P )+CS(D) ) 185.000 LOOP 1/| - T A N G ' L ONSET FLOW VELS • T ' . E . OF TEST AIRFOIL 186,000 D ( N K A ) = -0*(CS(NTEU)->CS(NTEL)) 187.000 I F ( M S V . C O ' . O ) G O TO 16 188.000 L O O P 17 - NORMAL ONSET FLOW VEL • ALL MSV CON PTS 6.N SHEAR l ' . 189.000 D O 17 K=1,M3V 190.000 JsNWS+K 191.000 I=NKA+K 192.000 D(I)=H*SrCJ) 193.000 L O O P 18 - TANG'L ONSET FLOW VEL • ALL MSV CON PTS ON SHEAR L' . 19a.000 * PRESCRIBED TAN ' L VEL THERE*. 195.000 DO 18 K=1,MSV . • • • 196.000 JshwS+K 197.000 I=NM2+K . 19e .00O D ( I ) = - U * C 3 ( J ) + V T I ( K ) 199.000 CONTINUE 500.000 RETURN 501.000 END 502.000 IN E F F E C T * ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOADiNOMAP IN E F F E C T * NAME = ASSEMD , LINECNT » 5T SOURCE STATEMENTS = 29,PROGRAM SIZE n 1191 - • NO DIAGNOSTICS GENERATED 13 15 C 11 17 C 4 8 16 u NO CRRORS IN ASSEMD 4 ^ MICHIGAN TERMINAL SYSTEM rORTRAN G(41336) CPS 10-22-.73 10119t 36 PAGE P001 o o o f SUBROUTINE CP3<CP,VTT,XX,YY.CS,SI,N1,3TC,N2,GAM,N3,MU.GNU,N0,A,B) 503.000 c " CPS CALCULATES VEL, PRESSURE » ALL CON PTS'. 504.000 o o o ? REAL A(Nl,Ni),H(Nl,Nl) 503.000 0003 REAL SIG(NJ),GAM(M3),MU(N4),GNU(N4) 506.000 0001 REAL VTT(Nn,CP<Nl),XX(Nl),YY(Nl),CS(Nl),SI(N15 507.000 0005 INTEGER P,0 ; 508.000 OC06 COPMOM/Hl / NWS,NS1.AT,NSU1,NKA,NM2,M3V,NA,NSPS,NTEU,NTEL 509.000 0007 coMMON/nn/ U,CH 510.000 OOOB C0MMOH/B3/NUl,NWUl,NU3,NWU2,NLl,NWLi,NL3,NHL2,NS0Ll.NSOLSL,Nri, i N F L A T , N 3 P F , N I I 511.000 512.000 0009 NN2sNM2tMSV 513.000 oo l o NWSV=M«3+MSV 514.000 9011 NVNsMwS+MSV 515.000 0012 NSU2=NSU1+MSP3*NSLAT-1 516.000 0013 NSP=NSPS-1 517.000 0011 iF(N3LAT'.En'.0) GO TO 2 518.000 0015 110 1 K = i,N3LAT 519.000 0016 I=NWS+K 520.000 C GAM . SLAT VORTE* STRENGTH DENSITIES 521.000 0017 1 nAM(K)=srGm 522.000 c GAMM - TEST AIRFO VORTEX STRENGTH DENSITY 523,000 0018 2 GAMMsSIG(NKA) IFCMSV.CQ'.O) GO TO 4 524.000 0019 525.000 ;>o2o 00 3 K=1,M8V 526.000 0 021 IsNKA+K 527.000 0022 J=NM2+K 528.000 c • MU - S'.L'. SOURCE STRENGTH DENSITIES 529.000 0023 MUCK)=SiG(I) 530.000 c GNU - VORTEX STRENGTH DENSITIES FOR SHEAR LAYER 531.000 0021 3 GNUCK)=SIG(J) 532.000 C025 4 LL»0 533.000 0026 DO } 2 lel.NVN 534.000 0327 li=I+N3P " 535.000 002B LK="SultLL*NSPS 536.000 0029 IFCI . C f " . ' , 1) WRITE(6,52) 537.000 0030 ' 52 FOIiMATClHl) . , • •• • • 538.000 C VNSf.vTST - TOTAL NORMAL & TANG'L VELS DUE TO SOURCE ELEMS 539,000 C'031 VN3f=0, 540.000 C032 VT3T=0. 541,000 0033 ASM=ol 542,000 0031 B3MiO. 543.000 0035 DO 5 .i=i#Nws • - 544.000 0036 VNSf=VNRT+A(J»I)*SIG(J) 545.000 0037 VT3T=VTST+BCJ,I)*SIG(J) 546.000 003B ASM=A3M+A(J,I) 547.000 0039 5 BSM=BSM+B£J,I) 548.000 0040 A3 = 0. 549.000 004 1 B3 = o'. 550.000 0012 IF CNSLAT.EO'.O) GO TO 8 551.000 •t 043 DO 6 K=ltNSLAT ' • . 552.000 eon • P=N3Ui'tNSPS*(K-l) 553.000 0015 QsP+NSP 554.000 C P,Q - {ST & LA3T CON PTS OS A SLAT 555.000 c AP.BP - NORM * TANG VELS DUE TO VORTEX ELEMS ON SLATS 556,000 0046 AP = o'. 557.000 MICHIGAN TERMINAL SYSTEM FORTRAN G(11336) "PS 10-22-75 10H9I38 PAGE P002 0017 0018 0019 OcSO 0 051 0052 0053 0051 5 055 0056 0057 0058 C059 0060 0 061 0062 0063 0061 Oo65 0066 0067 0068 0069 0070 0071 0072 0073 0071 0075 0076 0077 0 078 C079 0030 0031 0032 0083 OoSI 0085 9086 0C87 0088 0039 0C90 0091 0092 0093 BP = 0 . 00 7 M=P,Q AP=AP+A(M,I) 8PaflP + r>(M,I) AP.= AP*GAM(K) nP=BP*GAH(K) A3=AS+AP PS=ns+BP AT.BT - NORM AT = 0 . BT*o'. no •> . ! = ! i NA AT=AT+A(J,I) BT=BT+R(J,I) AT=AT*GAMM RTsHTftGAHM AM,B« - NORM & TANG VELS DUE TO VORTEX ELEMS ON TEST AIRFOIL t. TANG VELS DUE TO MSV SOURCE ELEMS ON INNER EDGE OF NORM & TANG VEL3 DUE TO MSV VORTEX ELEMS GNU ON INNER C •C S'-L'. AM = U'. BM = u'. C . AG,BO C EDGE OF S ' L ' . AG = O'. BG = o'. IFCMSV.EO'.O) GO TO 11 n o i o K=I,MSV J=NWS+K AG=ACH A'CJ,I)*GNUCK) flG=HGtB(J,I)*GNUtK) AM = AM+A(.J,I)*MU(K). 10 RM=RM+RCJ,I)*MU(K) 11 VNST=VNST+AM VTST=vTST+BM C VNVT.VTVT - TOTAL NORM t TANG VEL DUE TO ALL VORTEX ELEMS VNVTs-HS-BT-HG VTVT=AStAT+AG C VNOT,VTOT - NORM & TANG-VEL DUE TO UNIFORM STREAM U VNOf=-U*3ICI) VTOT=U*CSCI) VNT=VMnT+VNVT+VNOT VTT'ci) = VTST + VTVT + VTOT C VKL V K U - T E S T AIRFOIL T'.E*. KUTTA VELS IF(l'.r"'.NTEU) vKUrVTTCI) IF(l'.EQ'.NTEL) VKL = VTT'(I) CPCI) = 1'.-VTTCI)*VTT(I) iFti'.r's'.n GO TO ni IFUl'.EC.NUl)' AND*. CNWUl'NE',0)) GO TO 12 IFC'CI Ea'.NU3),ANDr(HWU2.NE',0)) GO TO 13 IF(CI.EO'.NLI) AND ( N W H . N E . O ) ) GO TO 11 tFCJl,EO.NL3)lAND.CNWL2.NE'.0)> G 0 TO 15 IFCCI EO'.N8OLI).AND'.<N80L3L'NE,0>) GO TO 46 IFC(I.L:(!.NF1)'.ANO'.(NFLAT'.NE.O)) GO TO IT IF((NSLAT,.EO'.0).OR".(l'.GE',NSU2)) GO TO 70 iFCCl'.Eo'.LKj'.ANn'.CLL'.LE'.NSLAT)) GO TO 18 70 CONTINUE 558.000 559.000 560.000 561 .000 562.000 563.000 561.000 565.000 566.000 567.000 56e,000 569.000 570.000 571.000 572.000 573.000 571.000 571.000 575.000 576.000 577.000 577.000 57e.000 579.000 580.000 581.000 582.000 583.000 581.0(10 585.000 586.000 587.000 588.000 589.000 590.000 591.000 592.000 593.000 591.000 595.000 596,000 597.000 598.000 59").000 600.000 601.000 602.000 603.000 601.000 605.000 606.000 607.000 608,000 609.000 610.000 M I C H I G A N T E R M I N A L S Y 3 T E M F O R T R A N G ( H 3 3 6 ) C P 9 10-22"T5 lOt t<9t3R P A G E P 0 0 3 009q 0095 CO'6 0097 00<"8 C099 0100 '0131 01 32 C 1 03 0 101 0105 0106 0107 010B 0109 0110 0 1 11 Ci 12 0 113 0114 01 15 0116 o i 17 0118 0119 0!20 0121 Cl'22 0123 9'24 0123 0126 0127 0123 0129 0130 0131 0132 0133 0134 C135 0136 0137 0138 0139 c i i o 0111 0142 0113 0144 •OPTIONS iF((I.EO.Nii).AND.(MSV'.HE.O)) GO TO 49 GO TO 12 00 F0RMAT.C4X, 'VNST' ,4X, l VNVT ', 4X, I VNOT t, 4Xi 'VNT ' »5X» 'vfST',4X,'VTVT•, 1 IX,'VTOT'»4X,'VTT',5X,'ASUM',4X,'BSUM',5X.'YY',5X,'XX',5X,'CP'i 2 6X,'SiG') 41 WnifE't6,30) 30 FORMATC'MAIN AIRFOIL') GO TO 51 42 WRlfE(6,31) 31 *̂ n.i.\,-('o.nv-.' Ri«.c-: cwilw M A L L ' ) GO f o 51 43 wRITEi'6,32) 32 FORMAT('UPPER LEFT SOLID MALL') GO fo 51 44 WRITE'c6,33) 33 FORMAT('LOWER LEFT SOLID WALL') GO TO 51 45 WRITE(6.34) -34 FOfiMATCLOUER RIGHT SOLID WALL 1 ) no fo s i 46 WRITE(6,35) 35 FORMAT('G0LIO STREAMLINE WALL') GO TO 51 47 WRlfE(6,36) .. - . 36 FORMAT('FLAT SLATS/NO KUTTA') GO TO 51 48 IFd'.ME'.NSUl) GO TO 67 wRlfE(6,37) 37 FORMATPUPPER SLATS') 67 LL=LLtl 69 IF(LL',GT'.NSLAT) GO TO 12 MRITE(6f68) I,II 68 FORMAT(/,'SLAT *',I3.2X,I3) GO TO 51 49 WRlfc'(6,38) 38 FORMAJ£'STREAMLINE FOR SHEAR LAYER") 51 WRITEC6.40) 12 WRI TEC'', 13) VNST, VNVT, VMOT, VNT, VTST, VTVT, VTOT. VTT'(I), ASM, BSM, YY(I),XXci),CP(i),SIG(I) 13 FORMATdX, 10F8.3,2F7.2,2F8,3) IF(wSLAT.CO'.O) GO TO 20 WRlTi : (6iH) GAM WRlfEC6,18) GAMM FORMAH'GAMMs • ,G12-.5) IF(MSV.EQ'.O) GO TO 19 WRlfE(6,15) «U KRITEfh,16) GNU FORMAT ('GAMs ', i0G12*.5) FORMAT('MU=',10G12.5) FORMAT('GNU=' , iOG12'.5) WRlfE(fc,17) VKU,VKL FORMAT (' VKU=' ,G12'.5,2X, ' V K L » ' , Gl2'.5) RETURN END I N EFFECT* 10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAO,N0MAP 20 18 14 15 16 19 17 6 1 1 . 0 0 0 6 1 2 . 0 0 0 6 1 3 . 0 0 0 6 1 4 , 0 0 0 6 1 5 . 0 0 0 6 1 6 . 0 0 0 6 1 7 . 0 0 0 6 1 8 . 0 0 0 6 1 9 , o r o 620 . 000 621 . 0 0 0 6 2 2 . 0 0 0 6 2 3 , 0 0 0 6 2 4 . 0 0 0 6 2 5 . 0 0 0 6 2 6 . 0 0 0 6 2 7 . 0 0 0 6 2 8 . 0 0 0 6 2 9 . 0 0 0 6 3 0 . 0 0 0 631 . 0 0 0 6 3 2 , 0 0 0 6 3 3 . 0 0 0 6 3 4 . 0 0 0 6 3 5 . 0 0 0 6 3 6 . 0 0 0 6 3 7 . 0 0 0 6 3 8 , 0 0 0 6 3 9 . 0 0 0 6 4 0 , 0 0 0 641 . 0 0 0 6 4 2 . 0 Q 0 6 4 3 . 0 0 0 6 4 4 , 0 0 0 6 4 5 . 0 0 0 6 4 6 . 0 0 0 6 4 7 . 0 0 0 6 4 8 . 0 0 0 6 4 9 . 0 0 0 6 5 0 . 0 0 0 651 . 0 0 0 6 5 2 . 0 0 0 6 5 3 . 0 0 0 6 5 4 . 0 0 0 6 5 5 . 0 0 0 6 5 6 . 0 0 0 6 5 7 . 0 0 0 6 5 8 , 0 0 0 6 5 9 . 0 0 0 6 6 0 . 0 0 0 6 6 1 . 0 0 0 6 6 2 . 0 0 0 6 6 3 . 0 0 0 6 6 4 . 0 0 0 MICHIGAN TERMINAL SYSTEM FORTRAN G(11336) i-ORCES 10-22-75 0001 SUBROUTINE FORCES(CP,XX,YY,DX,DY,OS,VTT,N.U.CH,Nl,N2,XC.YC) 0002 REAL C P C''0«X X ( N ) , Y Y(:N),D X C N ) , D Y C N),D S < N > , V T T C N ) C XCYC - CENTER OF BODY 0003 WRITEC6.5) N1.N2.XCYC • 0001 5 FORMAT (' FORCES ON BODY * • , IS,1,',13,3X,1 CENTER AT (',F7*,2, t >.>.F-:s,n'i 0005 C L T B O 0006 COTBO*, 0007 CM0=0. 000B CiRCeO. 0009 PER=0*. 0010 DO 1 I=N1,N2 0011 CLT=CLT-CP(i)*OX(i) 0012 CDTBCDT+CP(t)*DY(I) 0013 C1RC=CI«C+VTT(I)«0S(I) C P E R - ROOY PERIMETER CC11 PCRiPCfND3(i) 0015 1 C M O = C M O + C P C I ) * ( ( X X ( I ) - X C ) » D X C I ) * ( Y V { I ) . Y C ) * D Y ( I ) ) C CLT - TUNNEL LIFT COEFF'. 0016 CLT=CLT/CH C COT - TUNNEL DRAG COEFF*. (THEOR »Y ZERO) 0017 CDT=CDT/CH C C*0 - TUNNEL MIDCHORD PITCHING MOM, COEFF*. 0018 C M O i C M O / C H / C H C CM'I - TUNNEL OUARTERCHORD PITCHING MOM'. COEFF*. 0019 CMflsCMO-CLT/'1'. C ciRC - CIRCULATION ABOUT BODY C CLC- LIFT COEFF'. FROM CIRCULATION 0020 CLC=2,*CtRC/CH/U 0021 wRITEC6,2) CLT,CDT,CM0,CM1 0022 2 FORMATCCLTi'.Flo'.S^X.'CDTa'.Flo'.-S^X.'CMOs'.Fio'.-S^Xj'CMfl 1 F10'5). 0023 HRiTEf6,3) CIRC,CLC,PER 0021 3 FORMATCCIRCa'.FlO'.S^X.'CLCo'.FlO.SjZX.'PERIMBljGia'.S) 0025 RETURN 0026 , END •OPTJONS IN EFFECT* 10iEBCDIC,SOURCE,NOL1ST,NODECK.LOAD,NOMAP •OPTIONS IN EFFECT* NAME = FORCES , LINECNT " 57 •STATISTICS* SOURCE STATEMENTS s 26,PROGRAM SIZE B 1276 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN FORCCS 10H9I39 PAGE P001 665.000 666.000 667.000 666.000 669.000 670.000 671.000 672.000 673.000 671.000 675.000 676.000 677.000 67e.000 679.000 680.000 681.000 682.000 683.000 681,000 685.000 686.000 687.000 688.000 689.000 690.000 691.000 692.000 693.000 691.000 695.000 696.000 697.000 698.000 699.000 700.000 14 o o o o o o o o o o o o o o o o o o o o o o o o o o o o c - o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o - o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o J r " n a uioVJ tr*o- r j i n en* OVOJ c o - o to . • ^ b ^ o ^ « ( ^ o w f J l o c . ^ o ^ « ( ^ o * < ' , J ' M ^ r t n c ^ « « ^ o » ^ ^ J W ! ^ ^ n o o o o o o o o o - - « w - - -H r j r j r j r j r j r j r j r j r j r j i o ro M M M KI KI KI KI n c =r =r ~ — — — — - LO LfVLn LH LO i n ru <, ~ co _» •> ULJ« »JZ CO z •»-» o »to n c >- Lu » a. C i » - « ~ X » z z X •» w 2 O 3 >- >-* U D » >- —I » — >- z *-« e_ * Z X X X Z X X ^ u c x 0 U J - - or >- ̂ 01 » Z a. C h- X. O Z >- " a L J >- >- u » * L_J X '-x ^ Z » Z CL - r-4 a c - w I H U X ^ — . • — X X U- *J s < z O *-» CJ Lu » - (D D < < 3 LJ LJ <r> cr cr cc z. « L r - o c LO •—*- • • CT 1— C" If fl u u r- cr < L i t ^ IU II U_ cr ti. •--*-« 1 N O I I CT • -K X - O LO LO C U I ru ru r j r j »-N U J _ 4 - J ' -J C O a a . u o u 1; II n • 1- o cr u LJ G i <: t~ a o * W . I- C. G C LJ < _J 11 n » - £ c:. h- O C l - LJ UJ c: c •j-> I— v- I x : Z Z IU •» U U ' H • J '_> X < ;< 1 II 11 It o o —<; X C ( . • o r\! e: tu x . »-i _J * Z i-t (_) c: u_ H U C X >- + -« CO *- e 1- c ru a . w w CO z z LJ LU o o z z < < H H U U » I I ru ru cr.n +. x >- x >• I Q _ « — X >- II II j ro >- «!-«>- : o x » X X ru 1 o 11 '-> a. a. » x >- x >- CL w o Z LU J H ' XOO+.-C ^ ^ » » Z Z . x >• - • —• Lu '—I "_' ^ U . . • - . - . I I X . il- X < < 'J X J U II ^ * < * wv^t-i-k c C 1 H M H i ^ j H h i n i 1 r j . v « ~ ^ L". _ . C U O H h H H H . MX>. «̂  a o JO: c c.̂*'-' a 11 n w 11 11 n - w 11 ^ • COUU.li. I ' J O Z 11 o u. u. C X >• X • - • < U « M > - H M ^ o « " »-» • X X O X X o X >• L> •H s: x >- C3 r u f\ * r i n o e» o o o o o 0 0 0 o O O O O co o o - — • ro ^n *r i n o* r * - 0 0 o 0 0 — • — « — • —• ^ O G> O O O CT C> C> O O O O O C ' C ' ^ ' O O C ' C * O O O O o " —« a i KI c IP o r - c o* o • —< <\r KI cr tn o > r - co cr- • ru"»n er r u r u o j r u f t J P J f u r v j r u r u M KIKIKIMMMMK\ m c r c j c c cr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T> & cr> o cz> c> C C> C- t> i » O o c» -> o o o c - o c » o o O O ' O O O O M I C H I G A N TERMINAL SYSTEM FORTRAN G ( 4 1 3 3 6 ) HOOPRO 10-22-75 0045 M=MP-I C XX.YY - NOW M O D I F I E D CONTROL P O I N T COORDS. 0046 DO 5 I=L1,L2 0047 J=l-Lt+1 0048 K = .W1 0049 XXCl) = (XH(J) + XM(K))/2'f 0050 YY(I)=(YMCJ)+YM(K))/2. 0051 OX(I)=XM(K)-XM(J) 0 0 5 2 DYCI)=YM(K)-YM(J) 0 0 5 3 D3CI)=S0RTCDX(I)*0Xa)+DYCI)*DY'(I)) 0054 CS(i)sDXCI)/DS'(n 0C55 5 3i(i)=DY(I)/D3'CI) 0C56 RETURN 0057 END • O P T I O N S I N E F F E C T * 10,CMCDIC,SOURCE,NOLIST,N O O E C K.LOAD.NOMAP •OPTIONS IN E F F E C T * NAME = MOOPRO , L I N E C N T * S7 • S T A T I S T I C S * SOURCE STATEMENTS = 57.PROGRAM S I Z E s • S T A T I S T I C S * NO D I A G N O S T I C S GENERATED NO ERRORS I N MOOPRO 2700 10119t40 P A G E P002 756.000 757.000 758.000 759.000 760.000 761.000 762.000 763.000 764.000 765.000 766.000 767.000 768.000 769.000 MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) RE 10-22-75 0001* SUBROUTINE RE(A,N,M,LA) 0002 REAL A(N,M) OC03 READ(LA) A 0001 RETURN 0005 E N O •OPTlGNS IN EFFFCT* 10.EBCOIC,SOURCE,N0LIST,NOOECK,L0AO,NOMAP •OPTIONS IN EFFECT* NA«C = RE , LINECNT • 57 •STATISTICS* SOURCE STATEMENTS * 5.PROGRAM SIZE s 410 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN RE MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) WR 10-22-75 C00 1 SUBROUTINE WR(A,N,M,LA) 0002 REAL A(N,M) 0003 WRITE(LA) A 0001 RETURN 0005 END • OPTIONS IN EFFECT* In, EBCDIC, SOURCE , NOLIST, NODECK, LOAD, NOMAP •OPTIONS IN EFFECT* NAME = WR , LINECNT * 57 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o *STAT13T1C3» NO DIAGNOSTICS GENERATED NO CRRORS IN WR 440 MICHIGAN TERMINAL SY3TLM FORTRAN G(11336) WRO 10-22-T5 0001 0002 0003 OC01 0005 SUBROUTINE WPD(D,M,LD) REAL D(M) wRITE(LO) D RETURN END • OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOAD,NOMAP •OPTioNS IM EFFECT* NAME a wRD , LINECNT « 57 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN WRD 38B NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTION TERMINATED 13IG lOllRJll PAGE P001 770.000 771.000 772.000 773.000 771.000 lOllRHl PAGE P001 775.000 776.000 777.000 778.000 779.000 10ll9t11 PAGE P001 780.000 781.000 782.000 783.000 781.000 152 A p p e n d i x 1 1 . L i s t o f E q u i p m e n t U s e d I n s t r u m e n t B a r o c e l P r e s s u r e S e n s o r T y p e 511 (10mm Hg = 10 V o l t s ) B a r o c e l E l e c t r o n i c M a n o m e t e r T y p e 1018B B a r o c e l S i g n a l C o n d i t i o n e r T y p e 1015 D i s a D i g i t a l V o l t m e t e r T y p e 55D31 D i g i t e c D i g i t a l V o l t m e t e r M o d e l 2780 L e e d s & N o r t h r u p M i c r o v o l t I n d i c a t i n g A m p l i f i e r M o d e l 9835-B D r u c k p r e s s u r e t r a n s d u c e r M o d e l PDCR-22 D e s c r i p t i o n p r e s s u r e t r a n s d u c e r f o r p i t o t s t a t i c t u b e m e a s u r e m e n t s 4-1/2 d i g i t v o l t m e t e r f o r w i n d s p e e d f r o m p i t o t s t a t i c t u b e . a m p l i f i e r f o r B a r o c e l ' p r e s s u r e t r a n s d u c e r s . 3- 1/2 d i g i t v o l t m e t e r f o r w i n d t u n n e l b a l a n c e m e a s u r e - m e n t s . 4- 1/2 d i g i t v o l t m e t e r f o r w i n d t u n n e l b a l a n c e m e a s u r e - m e n t s a n d S c a n i v a l v e p r e s s u r e m e a s u r e m e n t s , s t r a i n g a u g e a m p l i f i e r f o r A e r o l a b w i n d t u n n e l b a l a n c e . f o r S c a n i v a l v e a i r f o i l s u r f a c e p r e s s u r e m e a s u r e m e n t s  154   157   160 161 -4- i ' 1 •14-4 1 1 i i_ ! I 1 i 1 1 l 1 ! i 1 I 1 1 ! i 1 1 j 1 1 i a F i g u r e 5 2 Comparison o f a i r f o i l p r e s s u r e ! 1 i 1 1 —l—:— i 1 1 ! ! ! 1 ! 1 i i 1 1 ] i ! : c o e f f lnifitir.s: Theory 1 1 i 1 i i i 1 1 ! 1 ! I i 1 1 > i i 1 i 1 1 > I 1 .1 _L 1 1 l : ! ! I 1 j TTT 1 > i 1 i i 1 1 i i 1 •< l l 1 1 1 1 ! i ! ! 1 1 I 1 I ! i i 1 l L . . L M i j 1 1 ! 1 1 1 i j i 1 i 1 ! 1 1 j j i - j - i i i r i 1 1 I ! ! 1 I ! 1 > 1 i ( i ! 1 1 I i i 1 i 1 i i i I I ! ; j I 1 i I t ! 1 [ t * i I 1 i 1 '< 1 l i 1 i 4 - 1 1 1 1 I 1 •H+ j j ! ! 1 1 j ' 1 - T r 1 ! i i 1 1 1 1 1 ! ! i i i i i ...1 1 1. I i 1 i ! 1 1 1 1 i i i < i I 1 1 1 i 1 1 M l 1 ! i 1 ! ! i -}- 1 1 i i i I 1 1 i ! ! 1 1 ! i 1 ! 1 1 1 1 ! 1 I ' ! 1 ! 1 i 1 i 1 I i t i i • 1 1 1 i ! 1 • ! ! '• 1 - H 1 - ! 1 _ . i H | i P 4-- 1 1 4 I j 1 i |! 4- 1 < X f f 4-4-- / • 4 i -\ i — I 4 - / c ¥- \ Hi 1 » 1 1 1 _ C L 1 1 1 1 1 i { 1 !' i \ HS -WW-—\-{& "-I- 4-f 1 • j - 1 C11 \ / - M M i. ..v J . 1 i i M M i i i i f>°A a t \- 1 I i 1 'v- -f j i 1 i i Qt/v K A . i i ! V P c r 1 1 1 -i.. i - H i 4 1 I ttl (TTT ~1 R" r- i 1 ill i 31! 1 I - 4 4 - 1 r> y a [ hfl Ii- n 6 \ t ! i i. >t _ ii.- M i l B | M i J> *T a i \ j i , i 1 1 ret r — - i M l . ! 1 1 1 -trb r-- •ff H- T U t i 1 i i i 1 i i i 1 j i'L 1 j 1 \\ 1 i i M i i i i l l 1 1 1 I ) ! H i l j i 1 i 1 1  ! 1 I ! ! 1 1 I I i i ~]—j— i ,'„. i ~ 4 " L V "1 i - j - i 1 . _ i i 1 I . i r T I i 1 I i i i i ( • i i — T ~ 1 1 1 l I 1 1 >̂  I 1 ! i i t! i } i 1 ! " i 1 i i V j 11 I | i 1 ! S N I— - f - ! 1 - p s i - f - - I i i r ! !, r" ! i I I i i i I ! >*. i - I -J 1 i i ! I i 1 1 i i ! i 1 i 1 i i i 1 1 1 i 1 ! ! i 1 —i—\— —j— 1 i 1 : i 1 i i • j I L 1 f - i 4 — | - -—j - 1 i i 1 i i i i — 1 ~ i • i i — r - 1 i 1 i i i i i i - 4 - r 1 I- i • I i • | -44- l I I 1 i 1 i I 1 1 \ 1 i I i i 1 t I i 1 - 4 " i T" i | " i i — r — 1 i 1 | i ! 1 i i i i 4- i I i TT"..._I u- - i i 1 I I M 1 1 M M M M i i i 1 1 | ; 1 i I i i i i 1 i -j- i i i _ 1 1 i i " T i I l - 1 4 - 4 , ._! - ' - L i . 1 ~~t~ 1 I i i i 1 ^ ; j ' -4+ 1 ! 1 _ 1 i i ~ F ~ T T i _ 4 i i T T s i i r 1 1 1 M '" , 1 • i i 1 i i i l i i i 1 I ' M i i 1 1 i i 1 1 M i i 1 1 1 l ! i l i i i i i I ! 1 ! I 1 i i l i - j — 1 | 1 —r~ -44 —r \ ! i i 1 ! 1 '. : -1 1 'i 1 1 1 1 . M M ! I .1. I i i i I i i i : i i 1 1 ! 1 ! 1 '• 1 ' | — T T ...1 1 % I -r _ i _ 1 • i i i i t | i r r 1 . . . . . I- - ! - ! -j * \ 4- ._!_ . ! _ !__ 1 j 1 1 i i i :< 1 [ -j \ N 1 .1 I j -|- i - U I i (1 f : R! ~ A / f I 4 •1 1 1 i X >: i i h [ ( } ii rj"l'.L t .„L. § ft 1 ! i 1 >' 3 i . 1 ( "T ~rr _L. 1 ; I t 1 >' * i ft j t i 1 ! i i \ h _ _-|_ ! 1 - !. f [ Id i / \ A _ - j . . I • • ^ r~ . - ( • r ! . ' i , l / i) \ i 1 t \ L , O H i r' "\ • t-1 -1- I.. [ "] 1 _ 11 •" - 4- • -1 " « 1 1 L 1 ! 1 y i / r ty j 1 | J "J Q O i 1 { i L'TJrL I 1 j •1_ I 1 I ; "T ] \ j .... j / & I j ~\~ " 4 \ ~\'~ ._.}. !... 1 r i u • i V _1_ t ) J t f Li ) } I l 1 f 1 N ""] ~| C f i I i j / 4 ] f « *• \ t' << i L V • j < u La e. m i. *. .-- - - * : i_ a . »• • - - - - - - - c B - - - - - -c - - - _ - * « ? - t- C t , ^ m ] I * > 1 <* « s T ! i < t r i i » i •! i i - t / c i: ( ft >v ( "I I J -1 i. oc U i 1 1 ,.;.L fh » j I ! i > 1 I 'ttt «* "~r I • I , i _ r ! 1 j i ! r . . . c V 1 I £ A -|_!_ : ! - - il * e B - - J L': U . . . _ - - - --- - L i . ... t sua c «< u 12S3S1 _l_ • -I.J.-r i ; i r t i _i_ ~ j _ el "/ "1" L I ) " " ! ' ' ! i • i ! _ V a r i a t i o n of along for - -- -tigure • J pressure c o e i i i c i e n t a s t i d i y n t correction uuunucix y - - -±t a two -•axmensxonax a x r i o i j . wx t n zero f rneoxy - _ - - 16.3m F i g u r e 6.1 U.B.C. Mechanical 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 windtunnel F i g u r e 6.2 V a r i a t i o n of mean windspeed i n two-dimensional t e s t s e c t i o n i n s e r t on v e r t i c a l p i t o t s t a t i c t r a v e r s e CTi  166 F i g u r e 6.4 E f f e c t of e n d p l a t e l o a d i n g s f o r two-dimensional a i r f o i l t e s t s 167 168 Figure 6.6 Error bars for the measured a i r f o i l rHi1 I i l i f t c o e f f i c i e n t s 169 - _ L L L - L i - F i g u r e 6.7 V a r i a t i o n of measured a i r f o i l l i f t c o e f f i c i e n t s on t h r e e c o n s e c u t i v e runs I'M ; 1 I M i J - .» > U 1 1 1 - i 1—1 • ' ! ! I_L < > ! ! ! ! M l fHT-H i F-H Lr i i j - t i J4 ^ 4 t : T T £ i i M i n i ' i i l i ' i l p T i . l : 4 H - T H + H - 4 T - 4 - - - h r + h - 4 + f - r - ^ - H - ^ = f T + "! h 4 H T t + h r H T S S I 1 T f c r T j S H 4 - _ u ^ -f—r-r-M4 444^-44—- r - - !4- - L - u L . _L I 4 i I ' M 1 i I M i ' i 4 ^ 1 : 4 4 ^ 1 ^ 4 - ! f f - p T Z M T i r x t i z z z ^ ^ ^ 4 4 - - g & 4 ^ g ^ ^ p f e 4 ^ - 1 ' 1 . | | 1 .'-V4- - r 1 I 1 1 1 1 1 / " ~ f - Ui-U-Ui- 1 1 1 r4--U... 1 ' M /I " I T T 44-I r f - 4 - r 1 + 1 4-TT-4-4 4 / - = MI 1 i 1 1 f-n 1 i/|Ar— 1—h T 4 r i ^ - 4 M X ^ 4 - r - - - T : T - - - T ^ :?=5z : = : : 170 i ! .' M i l ! 1 ; 1 1 ; III " 1 J M J M i M'T l i i - T 1 ' 11 1 i r. - -----MI T - t L 1" 1 d i / - A 4 C % - ^ - i /—' T T h r - T T T T — T M H - - t t / 1 " - L - T r h - j u 1 d ± l~ E S':--i4Zx^: V 8 ( ^ % [ :AR:: _, . / . . LJ-,... " 11 1 T " M L __4|_i_ MMJ L-4-L44 ^~ T i [ 1 H- 1 M i _ i I ' ' " T - , S3:—| 1 - ' ; 1 1 -4- M ' h . 1 i ' - T i _ 1 L 1 _I_| X J x : ± 4 M - - E x T - I ^ - - r r - - - N T _ -1 i 1 ___i - - , 1 1 , , 1 M i_ _ i _ i ^ - ^ - ^ 1 : 1 , 1 j 1 i z b f a : i _ J L," T 1 i -J 1 pOJ j L . 1 - T 1 ! 1 11 1 — 1 1 i 1 i 11 1 1 1 r - n 1 | 1 — J — - ; | • 1 -L-i 1—„ M—/ I 1 1 1 U | | 1 u l - z + p z ^ z z P z z ^ z z x z ^ z T - ^ - H — 4 - r r L - f f - J 1 1 —i- ~i — : 1 ! : '"• ' — r - - , ^ " i - T - ^ r ^ M I d G T T - T : 4 ^ 4 4 ^ I , I _ M I M ' • • -J i / M I M I"1" - M i 4_ X X • 7 M i i 1 - I - l i F ! M . _ l _ ' - -4H _ Z _ j _ i "TT" j _ ~ U i . _ i i A / Ii • - •  -Hr , _ j_ u T T 4 -^X—^-^-L-^-^-M-n- r X : ^ C T # L X - ^ : _ t t : _ i M M -4- !• • 1 i i i / i i i M M --H H — r — r ± H - t R T drxx d+q- £ z ± zzrzr ± 4 4 - 1 : 1 - 4 ^ T f e x E 1 ? 1 ^ i H j f f - i U - 4 f 4 ^ ^ ; 1111 i ! 1 111 11 -144 U —j U _ L M _ . L . ..I...I '/ 1 M l 1 M -: 1 _]_ _ M / L • 1 i i i M~ . L. 4. 1 1 / 1 1 1 i 1 1 - M l MM i i i ' imi / i 1 •• i M 1 T T r r r —rrr 11 MI 11 i t / M 1 1 MI -I----h —'Mr M 1 _j 1 1/ M 1 1 i r - M.I - J - U r — i - 1 — M — L i i _,_ 1 ' g i i M . 1 1 _ i • - -1 MI 1 -r' I T " L - i—4-- 4- Mm< MI- _ M 1 x T T T T 1 1 - ' -xr x " M 1 0 0 1 1 ' ' ' ' 1 1 1 > ' 1 M r , s - -i-L, ~rrH— T ± T 4 - x i + i m A o M H T t T ^ —H^e4Hff U ! TUl— 1 - 1. |T' 1 1,' U4-. Mil .... M - I 11 I-l 1 -H- h " f - 4 - r T - T T f 4 ^ L r - 4 4 4 L - 4 ^ _ 4 p T T T - - -j-j-H-- F i g u r e 7.1 V a r i a t i o n o f a i r f o i l l i f t cc ' - ~ r x !_J 1 , L J L t- —L ' 1 1 I - L i - . 4 - X ± E V X : X X : 4 ± I 4 ± - L - - T X ! T T 1 1 - j - •• 1 11 - - T i X 1 T T T - ^ - n U r t^~vf 1 1 i f f f , - T - E 1 1 M " : x : x 4 t T F i T ± : : : _ _ _ T T 4- 1 1 X ~ T T 1 j • - ~M 1 i 1 4 — J J ^ - A ^ j X X X T ^ T T - T — H r — i L +J L p - l f - L - ^ J 4 - X - T 1 1 , 1 ' T T T ± r ~ - ^ r f f - ^ r r - r L - . 1 T ! T f i T " "f^f M . . | I M l 1 I|J?S M l M' 1 - H — r r 4 4 - 4 - -H+4- 44--^ -TTT- - 4 ± + r i r44dio!4"| H 1 H p - 4 - 1 - T j - r C ^ r - M . , 1 . L T . H H 4 r r n — r T Izzzzzz. > e f f i c i e n t w i t h r s l o t t e d - w a l l o p e n - a r e a r a t i o : E x p e r i m e n t 171 172 o n r a t i o o f l i f t - c u r v e s l o p e s f o r d l X ' NACA-0015 a i r f o i l j± — H~h~r 173 F i g u r e 7.4 E f f e c t o f a i r f o i l s i z e o n l i f t - c u r v e s l o p e f o r NACA-0015 a i r f o i l : E x p e r i m e n t 174 1 I .U ! ! M i i . i M M ! - U _ ! .L M M 1 ! i 1 ! I t j i 1 i | i 1 ! | . i , . , .... r-7- 1 —r~ ! i i i i i _} 1 l _) j i 1 1 M M j [ i 1 i' I • i ! \ | 1 i i i • ; > ! 1 i I i 1 1 M i i i i ; i 1 ! 1 i l i ! 1 1 1 1 i ! I i M I ! 1 ! 1 i - _ | _ L i _ L. 1 ! | 1 1 1 1 ' 1 1 1 \ i I i i 1 I I 1 1 1 1 M M ! j 1 i 1 i I I t i ; . M M - | i ! j 1 L l L i i i i > 1 M M ' 1 1 i M M M I i 1 1 1 l l 1 i j • i 1 ', M i l i l l ! M l . i 1 i I i 1 1 1 ! j ! i : i 1 i j i 1 1 i 1 1 1 i l l ! 1 ! ' ! - - 4 " t-j 1 U - l l i i i 1 • 1 i "] 1 ! | 1 1 - I i i ! I ' 1 i 1 l L i 1 i 1 r n r 1 i ! 1 i I i i 1 I I j i 1 M i l M M i r i 1 i l i l i 1 : 1 1 i i i j 1 i i j | 1 ! " i T 1 1 ! I .1 i M l ! i i i M M \ - i 1 1 i M M 1 i 1 1 1 i i 1 1 1 l i j i i J i 1 1 1 i 4 . L' J j ._ _ L i Li L l 1 1 i 1 1 1 | j 1 1 1 1 - 1 i 1 T - P-\t ti~ <r V \ r f{ "V 1-F i i4. • f 1 r 4- 1 I 1 1 ! i 1 • • - i i l-y— M M -1 V-i U ! i i • 1 ~'~ -!- 1 1 —!— 1 i. t 1 —I— i I i ! 1 I 1 1 ' i ( C 1 I j _ L 1 1 1 1 I \ i I 1 i i | i 1 1 1 i (A _ t I r V j t I-1* 1 j 1 1 1 1 1 . ! I 1 i USL ta *" J: X Jj i 1 . ! ! 1 1 1 i 1 | j j 1 1 i 1 I i * 1 1 1 1 1 1 1 1 1 | 1 1 1 1 ; i i i i i I 1 1 1 1 i 1 i ! 1 l 1 i ; i . j I 1 1 i I 1 1 1 | ! i i 1 1 1 1 1 1 1 1 i 1 i j 1 i 1 i 1 ! | | I . 1 1 1 1 I 1 1 j 1 1 i 1 L i — 1 — I 1 1 + -t- s r< -w i ml r S lh f <- 1 - H - - 1 1 1 —r-| 1 1 J_"L M M fl J _ 1 111 i , VI 1 f 1 1 1 5r &\ -}— 1 1 ! 1 1 ! 1 1 1 i i 1 1 1 (j 1 I 1 1 1 I 1 pu 1 t 1 1 i 1 1 1 t 1 1 1 i 1 1 i 1 \ l •w-u I 1 1 1 I 1 1 > — '*! 1 O ! i 1 1 1 f 1 ! i 1 > * = t O j 1 . 1 1 1 I e a. •* 1 X n I -r. 1 j I ! ' 1 i 1 1 l 1 1 - C. 1 . i . l . U ( j | 1 1 l 1 I i ! —(— 1 1 1 | 1 | r .).. 1 l 1 ! i 1 i 1 I 1 I i i 1 j 1 i 1 1 1 1 | 1 v r - i I 1 / 1 c y i i >— \ l 1 1 I | j— i 1 i 1 i i i 1 i j | 1 1 1 1 1 1 1 i 1 l 4 1 i i 1 1 ; j ' 1 Y I ) ! u ! n 1 1 i u 1 L, i 1 1 1 | i " i i • 1 t ! i 1 i i 1 i i j 1 1 1 1 I i I 1 1 1 1 i 1 1 1 i 1 i 1 i 1 1 1 i 1 i i 1 i 1 | 1 I 1 1 [ . 1 1 3 1 1 1 i JUJU I 1 1 i i . 5 s, 1 ! 1 1 1 1 i i i ! i . 1 1 1 | 1 1 r i 1 ! 1 i s >. I 1 1 1 i i 1 1 1 | i ' i 1 1 1 j 1 1 i i I 1 I 1 1 1 I 1 •N - 1 1 i I 1 1 I 1 i i i 1 M "> 1 A I ! 1 1 ; 1 • 1 I i " I -1 1 t • 1 I | , — * _ l_ 1 O l f 1 | 1 1 1 i 1 1 «»» _ - I M 1 "** /a 1 t 1 i 1 1 sL 1 1 J l_L.i M i i I I 1 _ 1 t~ I I 1 3 - -j— .7!** 1 ! i • r % , i . --. "V . I t 1 — r f \ r. 1 1 •< i I 1 ; . i , - i i l l i i M i l 1 1 1 j i I 1 1 1 1 - ' J _ I 1 i I 1 1 1 1 i ' >v - tJVL/ AXI \ 1 1 j 1 i 1 1 1 1 i I | 1 1 1 1 1 I 1 1 i i J— i 1 1 ! i i [ i 1 I i ; I i I ; i i i 1 1 1 t i i 1 1 1 , i 1 ! I 1 1 i i I ! 1 1 L i i I 1 1 1 ! i 1 1 I i 1 1 1 1 i i 1 1 1 i 1 1 1 1 1 I i 1 1 . t— 1 . 1 i ! i 1 1 i 1 1 1 1 1 i 1 1 1 1 1 1 i ! ! i 1 1 1 i 1 ! 1 1 1 1 1 1 i I i 1 -4 p. 1 1 1 1 i I 1 1 1 ! 1 1 I 1 \ 2 \ > I 1 1 1 1 1 1 ' i I I I | 1 . 4 1 1 1 i 1 i ; 1 1 ! 1 ! | 1 1 t i i 1 1 | . 1 1 i i I 1 1 1 1 ! i 1 i i . i 1 1 1 i i 1 1 1 : ! 1 i ; i i 1 ; j 1 1 1 1 1 i 1 I 1 1 i 1 ! i 1 i | 1 i 1 i ...... 1 | j 1 1 i i 1 1 1 | i 1 ! 1 i i 1 ! 1 i 1 i 1 i 1 i ; 1 ! i i |. I i i 1 | i 1 1 1 I 1 ! 1 I 1 | 1 1 i ( 1 i 1 1 i r I ! 1 1 I i I i 1 i I ! ! 1 j i i i I I l i l t i i i I >< F 1 1 I 1 ! : l I 1 1 1 1 i | 1 1 1 1 ! . l i i i / L | 4 | I i | i i i 1 j_ i 1 i 1 ! i 1 ! 1 l H 1 1 j 1 ~r 1 i 1 i i 1 i | • • 1 1 i ! 1 ! i i IT I 1 i j ! 1 1 ! t i i i 1 ! 1 1 I • '• i i i i ' ' M i I ' l l i . i i 1 i I i 1 i « L M M II i ! i i ! t i i • i i 1 i i M M i 1 i C 1 1 i i 1 (A AS l ! ! r M I A ft 1 1 | i-U-TL f ! I 1 i J 1 i 1 I Zi 1 IV. o 1 j i irXJ- 1 + 1 I i I I . 1 i 1 I M i l l I I 1 i 1 I 1 " M - r r i I i i i i • 1 M M 1 ! 1 1 i 1 ! M U M 1 ! i i i 1 1 I M i l 1 j 1 i l | 1 1 1 1 1 : 1 i 1 1 ; 1 ! i 1 1 ' I 1 i i , 1 i i i 1 | 1 n 1 1 I I i ! I i i • i I 1 1 1 i p | | 1 i 1 1 i i _J I ! 1 1 i 1 i 1 1 I 1 i i ! i i ! 1 I [ I 1 1 i i 1 1 ; ; i 1 I 1 L _ 1 1 1 1 1 ! I 1 I 1 i 1 i i ! — t j- 1 i I i i 1 ! "i " 1 i ; i : ! 1 i ! 1 1 . 1 T M M 4 i T T i 1 i 1 i 1 - 4 - 1 44- i i i 4 4 - 4 4 j M 1 1 I ! -1444- i i i M M 1 | i i 1 - 4 - U 1 M M . 1 i. i 4 4 4 - I F i g u r e 7.5 E f f e c t o f a i r f o i l s i z e o n l i f t - c u r v e s l o p e f o r C l a r k - Y a i r f o i l : E x p e r i m e n t 175  177  179 i l l _ i i j lH-hi- ...LA I. ; | { i zhLtT _ — _ - i ! 1 {—•—-f — T - i _ L I _ i _ M M 1 1 -J4f 1 i 1 1 ! 1 1 1 i '!" 1 M M i j_ i t i M . f r i -i— i h f _ i | " i~ 1 M M | 1 M M 1 1 : T ~4 — — 1 r T 1 1 ! 1 T 1 1 1 ' 1 i I 1, 1 1 M M _! 1 "i 1 i ! ^ I 1 1 i i i ; 1 j ! ' 1 1 M M .L ] j 1 1 1 i 1 1 ! 1 I i 1 1 ! i ; ! ! 1 1 1 1 i i i i ! ! 1 111 _J_I - 1 1 ! 1 1 1 1 i 1 i i I 1 1 1 1 1 1 1 1 I l 1 t 1 1 1 • 1 1 1 — 4 ! 1 1 : 1 1 1 ; M M _ U ' i T : r r I ! | 1 ' 1 -L 1 i r ! 1 1 1 1 | 1 i i - - j - i_i I 1 I 1 1 I 4 1 i 1 4 1 1 T i 1 i i I 1 i i _Lj 1 1; ... 4- M-- I ! ! 1 ] 1 I i 1 1 i 1 i | 1 i 1 1 I"! 1 ! [ ! 1 ! I 1 . t I — 1 1 i I I i i 1 1 1 1 1 1 1 ! 1 M M 1 r r 1 1 : 1 ' i i V X 1 t i i .1 1 i 1 \ i 1 1 i 1 i 1 1 1 M _ 1 i I 1 i 1 1 1 1 1 1 : ; ! i id: i 1 1 | i i i I ! 1 -}- 1 j i — r 1 1 i I I ! 1 1 | - M 1 1 ' 1 1 | | | T 1 1 ~i—p" 1 ! ! i I I 1 1 1 I ; i ! i ! 1 i \ 1 i 1 ! 1 1 1 1 1 4+44 L 4 1 1 1 1 ! n 1 1 F i ~I 1 1 1 T i 1 i-S w < i . 1 M 1 I J 1.1 1 1 L i 1 I 1 1 i ! 1 1 v t\ _ A. C h 1 /\ iv J 1 ! 1 1 1 i 1 —:—I—I— r 1 \ /• V SCT: I \J M i rr u 1 1 1 i 1 i 1 1 1 I i ! _ l-H-i • h r 1 111 4TT 1 1 i I 1 — : — r i 1 M 1 T—1— — ! i t f ! | P h—4 •~W •Jfo.t - 1 1 1 1 1 i 1 1 1 I r -4-44 1 I] I - t - 1  1 V i 1 444- 1 ~ T 1 1 r _ t 1 h H - 1 1 c T U r f 1 1 1 i 1 1 i 1 1 I 1 _l_ 1 1 1 - 4-r • 1 i T • 1 1 ~i—1—r- 1 I 1 t • \ 1 1 p i I i t i 1 iitrr — ,.l | t 1 ! 1 1 I ! 1 t 1 1 r y u \ U TT: s 1 "1— 1 1 1 I 4—j— j 4 r i i -4 I 1 1 1 1 1 1 ' I ''• j 1 — 5 - 1V 1 1 r I 1 1 ! —1- f i 1 i L - U 1 \ i \* t ! 1 ! -~J~ t 1 - i 1 1 44 I 1 \ T* I i I i I 1 7- i l _ il 1 1 1 1 1 1 -p-L i i 4 4 - 1 1 • f i V •• 1 • n0r i n * 4 -r I T 1— l i i T 1 J C M \ 1 M M 1X2 LI - a LV r J -V J O 1 1 I M 1 1 1 I \ \ i l 1 1 M l . 1 1 1 1 1 1 M M 1 - i - i - 11 - t i -t =41 rr-11 1 ! n 4 4 n 4r-i ey f-t Y~ m r+~ 1 1 . . ., — t " J ii 1 1 I 1 1 r r LI ' 1 J J L ti - + 1 1 1 i 1 \ - ii^-1 1 1 t 1  III 1 1 1 ; 1 \ y 1 1 1 1 \ 1 ! ! i ~ j T t - - V 1 1 1 1 j 1 i 1 44—4- 1 i 1 J -V4-— 1 1 I 1 1 i • — — 1 1 1 1 1 i \ 1 ! 1 4 1 1 1 4 1 i 1 1 I i 1 1 1 1 i i t 1 1 1 \ l 1 1 ... 1 | 1 1 1 .1 1 I i ' 1 1 | 1 1 1 1 1 4 ! 1 i | t - V 1 1 1 . . 1 1 1 1 i !" v \ — L 1 1 I -)- | 1 rv \ ! i j -t— 1 1 1 i i > ( 1 1 1 1 | i i 1 1 1 1 | l 1 1 1 1 1 i 1 > * - - L . 1 | 1 i 1 1 -4 — r — 4 I 1 1 j s -> i 1 j 1 1 l 1 1 1 1 1 t 1 1 1 I >wS4_ ! 1 1 1 1 J _ ! 1 1 ^ I »-4 1 1 ! 1 1 1 1 4 " 1 -- 1 i I 1 | 1 1 « 1 1 - j - 1 1 1 I --  1 1 i i 1 i 1 1 i ! 1 1 1 " I T 1 i I 1 1 i ! j | J i 1 i 1 T \ 1 T T T i 1 1 1 ! 1 1 i 1  1 1 j j 1 1 i 1 | 1 1 11 1 • K I 1 1 1 1 1 i 1 1 |  1 1 1 1 1 1 1 1 1 • 1 1 - ! 1 1 1 1 " ! 1. i 1 1 1 1 ^ T 4 - 1 1 S > T ~ P 1 44 1 1 1" 1 —I— 4 1 —4 1 41- 1 1 1 1 i : 1 I 1 1 11 1 1 1 N 1 i 1 4 T i i i j i - f .!.. ..!., ; 1 1 1 1 1 i 1 |. - 4 . i 1 -i—ra 1 i > i 1 1 ! | t 1 1 t 1 4 - 4 4 1 j 1 1 1 1 V « S I I*" 4 4 4 - - 1 - r V 1 1 1 1 1 i ...| ; 1 1 1 1 1 1 J + t > ^& 1 i 1 n 1 i i i -1" r 1 I 1 1 1 t -4 1 i 1 1 1 1 -4 | 1 ! " ; J__ "S \m — — .{. | I u 44 i i 1 - r f t n "7*- i - ~ -1= 1 11 1 1 1 , v „,„,... I I J i_ 1 1 I i 4 i 1 1 i i J 4"^ T i 1 11 11 4 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 > 1 T i i 1 1 i ' ! i -14— i i. *> »• 1 I 1 : 1 __. 1 1 1 M M n i n •|" I i ! 1 1 I i 1 1 j | | 1 1 1 i i 1 M 1 v ! 1 ! *> i i 1 ; 1 1 ; 1 i ! 1' ! i i i r 1 1 ! i i 1 j •1 • 1 1 1 j ] I M " 1 1 1 1 ! ! i \ ! i 1 1 i 1 : 1 ; 1 1 1 1 i I I 11 M i 1 ! 1 - & i -j .... ; ' ! ' M M 1 1 i 1 1 1 • 1 1 I I I ! 4 4 - 4 4 i 1 1 1 1 w i 1 i 11 i 1 I 1 1 1 -7- 1 ! ! I 1 1 T + 1 I 1 1 1 i 1 1 l 1 j ! ! 1 11 i 1 ! 1 ; I j 11 1 1 | 1 I I 1 T I 1 ' i ! 1 lii i ' i 1 i 1 ! 1 "1 1 1 I T i I 1 1 ; 1 i 1 I i I -| i 1 1 I 1 1 I I ! i 1 1 1 ; i : 1 ! 1 i 1 1 [_ 1 I 1 I 1 1 i J_l 1 : 1 1 'i I i i f i , i ! i 1 1 1 1 1 \ ! i 1 1 i j ! 1 1 1 I 1 1 I 1 1  1 -J—J— 4 1 ! 4 f i i i T J i J +H- 4- 1 1 1 1 1 1 1 i 1 1 i 1 1 1 I ! i 1 1 1 1 4 1 1 — i — 1 1 • T~| 1 - . L . i j !.. i l l .1. i_ 1 F i g u r e 8.1 E f f e c t o f r e d u c e d c i r c u l a t i o n on a i r f o i l p r e s s u r e c o e f f i c i e n t : T h e o r y ( A p p e n d i x 8) F i g u r e 8.2 M o d i f i c a t i o n of a i r f o i l p r o f i l e to reduce t h e o r e t i c a l c i r c u l a t i o n to measured v a l u e : Theory (Appendix 9) 03 O 181 (a) Theory Figure 8.4 F i g u r e 8.5 The shear layer, i n the. plenum chamber surrounding the s l o t t e d w a l l CO 184 F i g u r e 8.6 E f f e c t o f d i f f e r e n t t y p e s o f w a l l b o u n d a r i e s o n r a t i o o f l i f t c o e f f i c i e n t s : T h e o r y 00 VJ1 186 4 1 T ~ T " - ! : : I I I ! M 1 i TTTT"' rr i r - > M i L ;~T -- ...... I ' M :4J ! 1 . j r-i 11 !' I ! 1 I I 1 1 1 1 I_I. ' 1 • 'i M M 4 , r ! 1 i i i i l l ! 1 M 1 r : T L K ! I i 1 1 1 1 1 11 1 1 i i -fl ! •• i ; _ ! M i l 'Ml 1 1 i i 1 1 TT i i i r i i T T 4 ! 1 1 ! i : i i I I 1 ! 1 i i ! ; 1 I I I ! M M , M i l . M M +t-h- i i i 1 I 1 1 1 1 ' 1 I I i 1 f t - i i i ! 1 i 1 i i i ! 1 i I 1 1 i I 1 i ! j 1 1 r 4 l 1 1 1 1 1 1 I i -TTT 1 !'" i / i f-j—1 1 i i i 1 ! -H ! 1 1 I i 1 4 i i ! 1 < 1 1 1 ! 1 1 1 1 -r- 1 1 1 ! ! te Ml r r 1 1 1 ! 1 ! ! : i ! 44 1 1 ! 1 1 1 1 1 i i ! i ; 1 : i 1 1 ' 1 1 1 J-4 1 \ 1 1 1 1 ! 4 M ! I 1 i i i i i i ' i i i i i i -rC ' 1 I i 1 ! i I Jk 1 1 WT i i t 1 1 1 | 1 ! 1 1 i I 1 +41 i i 1 1 1 M l ! 1 I 1 ! 1 44 L.....1 i i A i i i i i : I i I i i i < tit i i i i i i i I 1 i 1 ! 1 1 1 j 1 I 1 ^ i L L i I i 4 r , 1 ! 1 1 1 1 1 1 r4 1 1 1 1 i i ! 1J i S( 1 1 )°/ 1 i i i i i i i TA i R i i i i i i 44 i i i t X _ 1 ' 1 1 1 1 -4 i i 1 i i i i ' i M 1 1 i i i i _r i i i 4 1 r i i 4- i r fXf -hi k 11 i 1 1 i 1 t ; ' i i 1 1 1 1 1 t f 1 1 1 r f 1 - j— J f 1 i -4 i i I i 1 'I i • i • i- i i •• .... 4 i ! . - i i 1 1 i i 4 4- l ! I i 1 1 1 1 1 i j 44 } I ~r i 1 i I 1 ...!_ I 1 1 1 i —r 1 1 1 4 i • i i 1 Xt € T i . i i i 1 4 - i ! 1 1 f [4f n i . i i T - - -+ i i i i i i i i i i i i i 1 1 1 - 4 - i i 1 I i i i ! 1 1 i l -4-1- 4 T i i i i ! ! 1 -444 1 i i i ! 1 1 1 1 I 1 1 j ir i i i — "A // 1 1 1 1 1 1 1 1 1 — L 1 1 ' f i 1 1 r - 1 i v 4+ 1 A // i i 4-r̂ E > ^ i i --a V- 1 ftp a m 4^ 1 - I I 1 i 1 1 1 1 I i 1 m i I 1 1 4S —I—j Js« m _—_ I 1 I I I 1 44- 1 1 i i i i I 1 =u ::| 1 i i I i — . — L . ( 4— i— -, | | I 1 1 i i ti- i t 1 [— i — i ! i i 1 — | n Q-8€ )-* f f -i , . i 1 [ i i 1 i fr --)- 1 I 1 1 i 1 1 i t • i r/—- f r r \ - 4 t l "f l i . . j . . . ! 1 I I i i 1 1 i j — i i f L 4 —r — 1 — i i i 1 1 i i i — 1 \ 1 ! | i I 1 | i f ! | I | i i • ! 1 i i i I 11 _ | — i i i i 1 1 ! T 1 1 I 1 i 4 - i i I I i I ! I 1 T44- 1 1 i i X . 1. I 1 1 1 T T T - -4- i i *w i i i • i M l ' i i : T" —lT\ _ X J i ._ i i u _ 1 1 i i i 1 I i i 1111 i ! 1 1 | i H T - 1 - 1 I i 1 ! M i I i . J ! : i ! 1 1 i I r - JU 1 i \f\ I l l " 1 1 i TTTT- M l ! f\ | 1 - M M i i i • -r\ t 5 4 r - T 1 1 jr\ rrr. r 4rh r r r - - r "1 T T T i 1 1 10 1 I 1 I i 1 M i l W T - I 1 1 1 i y i . 1 1 1 1 1 ! i 1 m ; ' 1 4~ I - • i i I i ! 1 1 i —!—1—1 1 ! 1 1 I 1 M M M M 1 I ! !'•• ' i l l 1 1 1 t 1 i : 1 I i i t i l l !—j i — t - 1 1 i i 1 i i i 1 1 1 — ! i I M ' I • i 1 1 1 1 1 1 1 1 ' i 1 . M l M M M M i l l ! M M 1 i i 1 1 i J i i 1 -— ! 1 ! ) 1 1 : i ' i l l I ' M I M ! 1 ' —pr 1 I i i i i i 1 i i i ! 1 i i i 1 i L.| 1 1 1 1 M M M M M 1 | ' M l i F i g u r e 8.8 E f f e c t on a i r f o i l l i f t c o e f f i c i e n t s o f assumed p r e s s u r e c o e f f i c i e n t s on a s t r e a m l i n e r e p r e s e n t i n g the plenum shear l a y e r : Theory F i g u r e A l . l Geometry f o r i n t e g r a t i o n of a p o i n t source 188 Pfr.O.z) Axi-symmetric (b) *~ z Figure A5.1 The two-dimensional nozzle i n s e r t E-2 t h + S Constant-pressure boundary H il I 1 1 ) 1 1 I I I Figure A6.1 A l i f t i n g vortex between a s o l i d , a slotted, and a constant pressure boundary: Theory u> OJ Q Q + + ro cr cr I a~ ro cr I Q Q - a - - o - C T * - II Q ro cr ( D I <T> OJ o + cr OJ Q + cr W 1 <±) a + cr Q + cr Fiaure A6.2 The image system for a l i f t i n g vortex between a s o l i d and a constant pressure boundary: Theory NSLAT * NSPS NSOLI NU2 NUI U NTEL ^ NWLI NLI JH f i < r i i i > i ' i ' f » > > 11 1 r NWL2- NL2 NL3 NL4 i i i i r'lMiVi * ' '*' > i > > • i i i ) i r } i i Ji / i i it f > i' / i / t / i i i / >Hi Figure A10.1 Notation for the computer program of Appendix 10 192 Plate l b . The octagonal testsection i n the windtunnel 193 P l a t e 3. The w a l l s l a t s i n the s i d e w a l l frame  195 P l a t e 6. The 616mm NACA-0015 a i r f o i l i n the t e s t s e c t i o n . 196 T a b l e 1. A i r f o i l p r o f i l e c o o r d i n a t e s . 14% C l a r k - Y NACA-0015 J o u k o w s k y 1 1 % X Y U Y L . X + Y X : y U X Y L 0. 00 4. 19 4. 19 0. 00 0. 00 0. 00 3. 92 0. 05 3. 53 0. 32 5. 15 3. 15 0. 40 1. 37 0. 02 4. 16 0. 40 2. 90 0. 96 6. 15 2. 49 1. 00 2. 13 0. 35 4. 89 1. 09 2. 31 1. 92 7. 24 1. 98 1. 90 2. 88 0. 97 5. 41 2. 11 1. 77 . 3. 20 8. 35 1. 54 3. 20 3. 65 1. 91 6. 28 3. 45 1. 29 4. 80 9. 35 1. 15 4. 80 4. 37 3. 15 6. 96 5. 14 0. 88 6. 72 10. 26 0. 84 6. 70 5. 02 4. 74 7. 62 7. 10 0. 55 8. 96 1 1 . 14 0. 60 9. 00 5. 63 6. 63 8. 26 9. 41 0. 29 1 1 . 52 1 1 . 94 0. 38 1 1 . 50 6. 15 8. 77 8. 85 1 1 . 99 0. 12 14. 40 12. 65 0. 21 14. 40 6. 60 1 1 . 17 9. 38 14. 87 0. 02 17. 60 13. 25 0. 09 17. 60 6. 97 13. 86 9. 85 18. 05 0. 00 2 1 . 12 1 3 . 70 0. 02 2 1 . 10 7. 25 16. 76 10. 25 2 1 . 45 0. 07 24. 96 1 3 . 94 0. 00 25. 00 7. 43 19. 89 10. 58 2 5 . 10 0. 20 29. 12 14. 00 0. 00 29. 10 7. 50 23. 24 10. 83 2 8 . 95 0. 40 33. 60 1 3 . 95 0. 00 33. 60 7. 47 26. 74 10. 99 33. 02 0. 65 38. 40 1 3 . 74 0. 00 38. 40 7. 32 30. 44 1 1 . 07 37. 25 0. 96 43. 52 1 3 . 34 0. 00 43. 50 7. 07 34. 27 1 1 . 06 4 1 . 59 1. 29 48. 96 12. 73 0. 00 49. 00 6. 69 38. 22 10. 98 46. 06 1. 66 54. 72 1 1 . 85 0. 00 54. 70 6. 22 4 2 . 26 10. 80 5 0 . 58 2. 04 60. 80 10. 80 0. 00 60. 80 5. 62 46. 39 10. 55 5 5 . 12 2. 41 67. 20 9. 44 0. 00 67. 20 4. 91 5 1 . 15 10. 19 59. 69 2. 77 73. 92 7. 83 0. 00 73. 90 4. 09 54. 75 9. 86 64. 19 3. 10 80. 96 5. 92 0. 00 8 1 . 00 3. 14 58. 93 9. 43 68. 59 3. 40 88. 32 3. 86 0. 00 88. 30 2. 07 63. 05 8. 95 72. 86 3. 66 96. 00 1. 45 0. 00 96. 00 0. 84 67. 12 8. 44 76. 96 3. 86 100. 44 0. 00 0; 00 100. 59 0. 00 7 1 . 09 7. 92 80. 83 4. 01 74. 94 7. 38 84. 43 4. 11 78. 62 6. 85 87. 73 4. 16 82. 12 6. 34 90. 71 4. 17 85. 37 5. 85 93. 29 4. 14 88. 40 5. 40 9 5 . 50 4. 08 90. 71 5. 07 97. 27 4. 01 93. 29 4. 70 98. 61 3. 94 95. 50 4. 39 99. 50 3. 88 97. 27 4. 16 100. 0 3. 84 98. 61 4. 00 99. 50 3. 90 100. 0 3. 84 Table 1 c o n t 1 d . Main A i r f o i l 0.00 1.00 " 0.00 1.00 2.40 -1.10 2.50 3.61 -1.71 4.00 4.45 -2.10 7.00 5.65 -2.55 10.00 6.43 -2.92 15.00 7.19 -3.50 20.00 7.50 -3.97 25.00 7.60 -4.28 30.00 7.55 -4.46 35.00 7.43 -4.53 40.00 7.14 -4.43 45.00 6.80 -4.35 50.00 6.41 -4.17 55.00 6.00 -3.92 60.00 5.47 -3.65 65.00 4.95 -3.35 67.00 - -3.18 69.00 - -2.83 70.00 4.36 -2.51 71.00 - -1.98 72.32 - -1.02 74.57 - +0.67 75.00 3.78 77.82 - 2.30 80.00 3.08 2.67 82.70 2.64 2.64 196A NACA-23012 F l a p X y u X Y L 0.00 0.04 0.00 0.04 0.45 0.99 0.36 -0. 72 1.08 1.59 0.95 -1.00 2.11 2.27 1.74 -1.15 3.65 2.93 2.44 -1.21 5.17 3.33 3.44 -1.21 6.68 3.55 4.95 -1.15 7.69 3.57 6.45 -1.07 8.69 3.52 7.45 -1, 03 10.18 3. 32 8. 46 -0.99 12.66 2. 86 9.96 -0. 94 15.13 2.36 12.47 -0. 82 17.61 1. 85 14.98 -0.71 20.09 1. 35 17.49 -0.61 22.07 0.93 20.00 -0.46 24. 05 0.52 22.01 -0.33 25.54 0.21 24.02 -0.19 26.53 0.00 25.52 -0.08 26.53 0.00 O r i g i n o f f l a p c o o r d i n a t e s i s (78.87,-0.81) f o r 6=20°. T a b l e 2. F r e e a i r a i r f o i l c o e f f i c i e n t s : T h e o r y . NACA-0015 a 0 C L C M 0 / . d a 0 0.000 0.000 0.0000 0.1193 3 .365 .086 - .0050 .1229 5 .607 .143 - .0086 .1300 10 1.210 .282 - .0204 • 14% C l a r k - Y a 0 C L C M 0 C M c d C L d a - 6 . 3 0.000 -0.087 •4 -0.087 0.1206 -3 .401 .012 - .0883 .1208 0 .763 .101 - .0901 .1203 5 1.362 .244 - .0965 198 Ta b l e 3. A i r f o i l and w a l l c o n f i g u r a t i o n s examined t h e o r e t i c a l l y . A l l s o l i d w a l l s are 4.88m lo n g , w i t h MWUl = MWU2 = 20, and NWL1 = NWL2 = 4 0 . The s l o t t e d w a l l i s 2.44m l o n g , composed o f l a r g e (92mm) NACA- 0015 s l a t s w i t h NSPS = 9. A i r f o i l i s i n the c e n t e r of the t e s t s e c t i o n ; NA = 50 f o r C l a r k - Y and NACA-0015; NA = 81 (46 main and 35 f l a p ) f o r NACA-23012. F u r t h e r notes are found a t the end o f t h i s t a b l e . a CM 0 CMc 4 1. C l a r k - Y a i r f o i l a) Fre e a i r -8 -0.203 -0.137 -0.086 2 1.003 + .159 - .092 20 3.088 .603 - .169 b ) S o l i d w a l l s -8 -0.250 -0.150 -0.087 c/H = 0.53 -3 .444 + .014 - .097 2 1.140 .178 - .108 20 3.632 .711 - .197 c) 40% TSUSL -8 -0.270 -0.153 -0.086 NSLAT = 16 -3 + .376 -000 - .094 c/H = 0.53 + 2 1.012 .149 - .104 20 3.179 .610 -. .185 d) 60% TSUSL -8 -0.260 -0.151 -0.086 NSLAT = 1 0 -3 + .377 + .003 - .092 c/H =0.53 + 2 .992 .148 - .100 20 2.986 .572 - .175 199 T a b l e 3 ( c o n t ' d ) . a C M 0 C M c 4 c/H C l a r k - Y ( c o n t ' d ) ( 6 0 % ) 20 3.061 .579 - .186 0.66 70% TSUSL 20 2.929 0.570 -0 .162 0.25 NSLAT = 8 20 2.907 . 562 - .164 .39 20 2.923 .560 - .170 .53 20 2.970 .563 - .180 .66 20 3.084 .573 - .198 .86 20 3.200 .587 - .213 1.0 -8 - .256 - .149 - .085 .53 -3 + .381 + .005 - .091 .53 + 2 .989 .149 - .098 .53 80% TSUSL -8 -0.250 -0.14.8 -0 .086 0.53 NSLAT = 5 -3 + .388 + .007 - .090 .53 + 2 .994 .152 - .097 .53 20 2.888 .554 - .168 .53 NACA-0015 a i r f o i l F r e e a i r -2 -0.243 -0.058 + 0 .003 + 8 + .970 + .227 - .015 15 1.803 .412 - .038 20 2.382 .530 - .065 S o l i d w a l l s -2 -0.305 -0.068 + 0 .008 0.67 + 8 +1.223 + .272 - .034 .67 10 1.510 .333 - .045 .67 20 3.074 .675 .093 .67 + 3 .371 .087 - .006 .17 3 .388 .090 _ . 007 . 34 200 Tabl e 3 ( c o n t ' d ) - — a CM 0 4 c/H 2. NACA-0015 (cont'd) b) S o l i d w a l l s (cont'd) 3 .416 .094 - .010 .51 3 .453 .100 - .013 .67 3 . 546 .116 - .021 1.0 c) 40% TSUSL -2 -0.327 -0.072 +0.010 0. 67 NSLAT = 1 6 + 3 + .364 + .081 - .010 . 67 8 1.039 . 227 - .033 .67 20 2.592 . 550 - .098 .67 d) 60% TSUSL- -2 -0.320 -0.070 +0.010 0. 67 NSLAT = 10 + 3 + .359 + .081 - .009 .67 8 1.006 .222 - .230 .67 20 2. 421 .514 - .092 .67 e) 70% TSUSL +3 0. 355 0. 084 -0.005 0.17 NSLAT = 8 3 . 356 .083 - .006 .34 3 .358 .083 - .007 .51 3 . 361 .082 - .008 .67 3 .365 .082 - .009 1.0 f) 80% TSUSL -2 -0.311 -0.068 +0.010 0.67 NSLAT = 5 + 3 + .367 + .084 - .007 .67 8 1.000 .224 - .026: .67 20 2.335 . 500 - .086 .67 3. NACA-23012 a i r f o i l a) F r e e a i r + 8 2.442 0. 320 -0.291 201 Table 3 (cont'd) a V CMc 4 c/H NACA-23012 (cont'd) 70% TSUSL +8 2.415 0. 308 -0.295 0.2 NSLAT = 8 8 2. 305 .290 - .286 ' .4. . . . . . . 8 2.318 .283 - .296 .6 8 2.296 . 280 - . 312 -8 8 2.442 .280 - .331 1.0 C l a r k - Y a i r f o i l Compare C (C ) w i t h C T ( T ) , J-i p Li w i t h NA = 110 • and a = 20 o L p CM 0 4 c L ( D i ) F r e e a i r 3.117 0.548 -0.231 3.114 i i ) S o l i d w a l l s , c/H=.66 4.165 .744 - .298 3.742 C i r c u l a t i o n on w a l l s l a t s reduced by m o d i f y i n g s l a t p r o f i l e s , NSPS = 1 5 , a = 20°, c/H = 0.66. k C L CMc 4 70% TSUSL 1.0 3.010 0.569 -0.184 NSLAT = 8 .8 2.935 .556 - .178 • 7 2.610 .502 - .150 Shear l a y e r r e p r e s e n t a t i o n . MSV = 2 0, a = 20°, c/H = 0.66, V. = / ( l - C ). t p ' C P C L CM 0 CMc 4 i ) 60% TSUSL 0.0 2.420 0. 451 -0.155 NSLAT = 1 0 -.12 2.686 .502 - .170 T.28 3. 305 . 572 - .187 202 Tabl e 3 (cont'd) 4c) C l a r k - Y - Shear l a y e r r e p r e s e n t a t i o n ( c o n t ' d ) . C P CMc 4 i ) 60% TSUSL ( c o n f d)-.35 3.188 .600 - .197 NSLAT = 10 -.44 3.390 .640 - .207 i i ) 7 0 % TSUSL 0. 2. 321 .433 - .147 NSLAT = 8 -.12 2.591 .486 - .162 -.28 2.951 .558 - .180 -.35 3.101 .586 - .190 -.44 3. 308 .626 - .200 5. NACA-0015 a i r f o i l w i t h reduced c i r c u l a t i o n , a = 10°, s o l i d w a l l s ; a reduced a i r f o i l c i r c u l a t i o n determined. C T C M C M L Mo Mc _4 i ) from measured l i f t (k = 0.741) 1.120 .335 +0.0549 i i ) by m o d i f y i n g the p r o f i l e (k=0.724) 1.120 .280 - .0003 Notes: The p o s i t i o n s o f the w a l l s l a t s c o r r e s p o n d e x a c t l y t o those i n the e x p e r i m e n t a l setup. The s l a t s are spaced u n i f o r m l y , b e g i n n i n g w i t h a s l o t opening a t the upstream end on the s i d e w a l l . The number of l a r g e s l a t s r e q u i r e d f o r 40, 50, 60, 70 and 80% OAR i s 16, 13, 10, 8, and 5 r e s p e c t i v e l y . The e f f e c t o f i n c r e a s i n g the number of c o n t r o l p o i n t s on the t e s t a i r f o i l i s seen i n 1(a) and 4 ( a ) ( i ) ; o f i n c r e a s i n g the number of c o n t r o l p o i n t s on the w a l l s l a t s i n 1(e) and 4(b) (k=l). 203 T a b l e 4. A i r f o i l a n d e n d p l a t e l o a d i n g s . J o u k o w s k y R e = . 5 ( 1 0 ) 6 S o l i d W a l l s Ct C L C* Mc 4 Mc 4 c* C D -7 -.350 -.345 -.0719 - . 0 7 2 3 .0330 . 0194 -6 -.245 - . 2 4 1 -.0727 -.0728 .0317 . 0180 -5 -.137 -.134 -.0728 -.0727 .0309 .0171 -4 -.025 -.024 -.0727 -.0723 .0308 . 0168 -3 .086 .086 -.0724 -.0717 .0314 .0172 -2 . 194 .192 -.0716 -.0706 .0323 ' .0179 -1 .299 .296 -.0712 -.0699 .0340 .0193 0 .408 .403 -.0705 -.0689 .0361 .0212 1 .512 .506 -.0693 - . 0 6 7 3 .0384 . 0233 2 .617 .610 -.0669 - . 0 6 4 5 .0410 . 0259 3 . .721 .713 -.0652 - . 0 6 2 5 .0441 . 0289 4 . 820 .811 -.0639 -.0607 .0482 . 0326 5 .917 .907 -.0628 - . 0 5 9 1 .0530 .0370 6 1. 009 .998 -.0617 -.0574 .0592 .0425 7 1.103 1. 091 -.0600 - . 0 5 5 1 . 0652 .0478 8 1.192 1.179 - . 0 5 8 1 -.0524 .0719 .0534 9 1.277 1.265 -.0557 - . 0 4 9 1 .0790 .0594 10 1. 340 1. 326 -.0536 -.0466 .0869 . 0672 11 1.382 1.368 -.0540 - . 0 4 6 5 .0972 .0773 12 1.408 1. 394 -.0620 -.0546 .1161 .0959 13 1.199 1.186 - . 1 6 1 1 -.1538 .2551 .2345 C L ' C 5 c ' C D l o a d i n g o n a i r f o i l p l u s two e n d p l a t e s C M c ' C D l o a d i n g o n a i r f o i l o n l y 4 Table 5. Windtunnel balance r e s u l t s - 0.25-Clark-Y Pt = ' . «5(lO )6 SOLID WALLS ALF CL CD CM0 CMC/'I -10 ' . -0'.3'l5 0'.0260 -0.0916 -.«»'. -0 .251 0'.0?35 -o'. w.o -0 .0907 -o'. 151 0'.0221 -0.133 r-0 . 0906 -T. -0 ' . 057 0'.0215 -0.107 -0.0910 -6 ' . O'.O'II 0'. 0 2 0 1 —0.080 -0.0908 -5". 0'. I l l C '. 0 2 0 7 -0'.053 -0 .0911 - 1 ' . o'.2'U 0'. 0205 -0 .026 -0 .0908 0'.313 0'.02() '1 0-002 -0 .0908 " 2 r (>'.13S 0'.02?1 o'.032 -0.0871 o'.553 0'.0237 0.060 -0.0905 0'. (i'.665 0 -.0256 0'.037 -0.0942 i ' . 0 .753 o'.0278 0'. 1 11 -0.0920 2'. 0-815 0'.0302 0. 110 -0.0905 i'. 0'.938 0'.0330 0'. 168 -0 .0889 l ' . T.030 0'. 0373 0.193 -0.0880 5'. '1'. 1 oa O'.03(i7 0'.2 18 - 0 . o i a s 6'. l ' . 185 0'.0133 0.211 -0 .0820 l'.256 o'.oi87 0.260 -0,0782 B'. l'.321 0'.0532 0 -,286 -0.0736 r .3^6 0'.05R8 0.305 - 0 . 0 7 U 16'. r . ' i27 0-0650 0.321 -0,0655 i r . r . ' i35 0'.0716 0'.331 -0.0572 12'. 0'. 0788 0.333 -0.0513 13'. r . ' i26 0'. 1 03'l 0.322 -0.0638 11'. f. 388 0 -. 1218 0'.308 -0 .0682 -Y a i r f o i l s CL ARKnY Rti = *.'15(10)6. 'I0ZSS+P TSU AI.F CL CO CMO CMC/1 10'. -o',330 0'. 0237 -0 ' . 187 ^•0.0963 -9' -0,237 0'. 0218 -0 .160 -0'. 0916 -0 ' . I'll 0'.0195 -0-131 -0 .0915 -7' -o'. 016 0 ' . 0 1 « 9 -0'.107 -0 .0938 -6' 0'. 0 •'! 8 0'.0177 -0 .0 82 -0 .0912 •*5' 0. I ' l l 0'. 0 17 6 -0-055 -0,0911 0'.2'I0 0 0 1 7 '1 -0 .027 -0 .0921 — > 0'. 5'40 0'.0176 0.000 -0.0933 - 2 ' ()'.'I3 0 O". 01R5 0'.0 29 -0 ,0896 - i ' 0 .533 0'. 020 1 0'.055 -0 ,09 16 o' 0'.6'I3 0'.0231 0,081 -0 .0968 r . ()'.729 o'.0262 0.105 -0 .0958 z'r 0^8 15 0'. 0 285 0.130 -0 .0915 0',90l" o'.0315 0.155 -0 ,0933 0'.986 0-0316 0*. 181 -0.0911 5 » f.062 0'.0389 0.200 -0 .0885 6 1.133 0'. 0'I26 0 .227 -0 ,0852 7 1.201 0'.0'I62 0.2'19 -0 ,0826 8. l'.?68 0'. 0119 0-269 -0 ,0795 ?'. l'.323 0'.0550 0.288 -0,0761 10 * l ' . 359 0', 0 6 0 6 0.30a - 0 . 06.85 11. l'.377 O'. 0675 0.316 -0 .0609 12 m r .562 0'. 0765 0.319 ^0,0589 13 • l ' . 366 0'.09;i 3 0.309 -0 . 0636 11 • 1-333 0'. 1 162 0 ' . 2 9 « -0 ,0701 15. f .293 0'.1378 0'.280 -0 ,0718 CLARK-Y RC=.15(l"0)6 10%L5+P TSUSL ALK -10'. -< -8'. -7'. - 6 ; -5'. -1*. -z: -?-'. - 1 . 0". 1'. 2\ 3 1 5 6 7 e 9 10 *h 12. 13'. 15. Cl, -(>'. 310 • 0'. 2 'i 5 •O'. 117 -()'. 052 O', 012 O'. 133 0'.237 0'.335 O-. 126 0*.52H o'.6'll 0'.725 0'.812 0'.89 7 0'. 9 31 l ' . 053 132 ,203 ,267 .361 I'. 377 1-379 l'.365 1/.329 I'.293 cn 0 0 219 O', 0222 0'.02()0 O'. 0 196 0'.0181 0'.0182 0'.0185 0'.0192 0'.0196 ,0213 ,0212 ,0265 ,0201 0'.0319 0'.0319 03^5 0128 . 016 6 ,0502 ,0550 ,06o'l ,0666 ,0761 ,09i|8 .1165 0.1379 CHO -0 ' . 185 - 0 . 159 -0.132 -0.107 -0.081 -0.055 -0.028 -0.000 0'.023 0.051 O',073 0.103 0'. 128 0. 151 O'. 178 0'.201 0.221 0'.216 0.266 0.23 6 0'.302 0.312 0'.315 0'.307 0.290 0'.277 CMC/4 -0,0928 r-0.0921 -0.0917 -0.0921 -0,0921 -0.0919 -0.0923 ~0 . 0909 -0,0875 -0,0391 -0.0951 -0.0937 ^0.0927. -0 , 0899 -0.0888 -0.0353 -0.0827 -0.0796 .-0,0758 *0 ,0715 0̂ ,0618 -0.0586 -0.0551 rO .0591 -0.0676 -0,0715 205 Table 5 - 0.25-Clark-Y Ri: = '.45Cf0)6 50XL.S + P TSUSL CLARK-Y Rl! = '. '15 (10) 6 70ZL3+P TSUSL ALF CL CO CMO CMC/4 ALF CL CD CMO CMC/4 10'. -o'.333 0'.0239 -O'. 181 -0.0929 -IO'. -0'.324 0'.0273 -0'.133 -0.0925 -9'. -0.24U 0'.0210 -.0.158 -0.0915 -9'. -0'.228 0'.023'l - 0 ' . 157 -0,0929 -8. -0.113 O'.02o5 -o'.131 -0,0916 -8'. -O'. 132 0'.0222 -0.130 -0.0924 -7'. -0'. 019 O'. 0131 -.0.106 -0.0915 -7'. -O'. 038 0'.0212 -0 .103 -0.0922 -6'. O'. 015 0'. 0185 -0.080 -0.0915 0'.057 O'. 0 196 -0'.077 -0.0920 -5'. 0'. 110 0'.'173 -0.051 -0.09)6 ~sf. 0.153 0-.0191 -0.051 -0.0926 "If 0'.236 0'.0175 -0'.027 -0,0910 -4'. o'.246 O'.O 199 -0.021 -0,0919 *" • 0'.335 0'. 0 1 3 I 0.001 -0.0906 -3". 0'.34 2 0'.0198 o'.ooi -0,0914 0'.'I22 o'.oiaa 0'.029 -0.0870 0'.4 27 0'.0202 0'.032 -0,0372 0'.525 O'.02o7 • o'.osi -0.0898 o'.533 0'.0219 0'.058 -0,0910 0'. o'.63'l 0'.0230 0.080 -0.0912 0'.639 0'.0 239 0.083 -0.0952 t. 0'.720 0'.0251 0'. 105 -0.0932 f'. 0.722 0'.0263 0.108 -0,0939 2'. 0'.801 0'.0277 0.129 -0.0917 2'. 0'.807 0'.0284 0'.133 -0.0926 3'. 0'.S91 0'. 0 295 0.155 -0.0903 3'. 0'.892 0'.0293 0.158 -0,0913 4'. 0'.978 0'.0320 O-. 180 -0.0386 '4'. ()'.974 0'.0320 0.132 -0.0895 5'. T.051 0'.0373 0'.203 -0.0861 5'. V.05 0 0'.0354 o'.205 -0,0877 6'. .1.127 0'.0102 0.226 -0.0835 6'. 1'. 1 17 0-.0379 0.226 -0.0349 7'. l' . 195 0'.0137 0.217 -0,0301 7'. l' . 185 0'.0117 0'.24 7 -0.0326 8'. T.257 0'.0474 0'.268 -0.0762 8'. l'.243 0'.0462 0'.268 -0.0778 l ' . 1'. 311 0'.0521 0.236 -0,0724 1. i'.297 0'. 0 5 1 0 0'.236 -0,0741 10'. 1.35 0 0'.0577 0'.301 -0,0671 IO'. T.330 0'.0563 0.300 -0.0637 11'. f.372 0'.0613 0.311 -0.0623 .11. 1.339 0'.0628 0'.309 -0.0613 12'. T.370 0'. 0739 0.315 -0.0575 12'. l'.34 0 0'.0732 0.310 -0,0603 13'. l'.354 0'.09;>5 0.301 -0,0639 13'. T.323 0'.0928 0'.293 -0,0679 11'. 1'. 311 0'. 1156 0.286 -0,0718 14'. T.285 O'. 1 150 0'.233 -0.0734 15'. 1.282 0'. 1352 0.275 -0,0749 15'. i',246 0'.1337 0.268 -0,0773 CLARK-Y RE = '.'I5(T0}6 60%L3+P TSUSL . C\ ARKrY Rlt = *. 45 C {0) 6 80XL3+P TSUSL ALF CL CD CMO CMC/4 ALF -10'. CL CD CMO CMC/4 -10'. -0'.328 0'.0236 -0 .183 -0.0931 -0',313 0'.0258 -O'.l 74 -0.0881 -9'. -o'.23'l O'. 0210 -0 .158 -0.0933 -9 . -0.214 0'.023'l -0.118 -0 ,0886 O * -0.137 0'.0191 -0 .131 -0.0929 -8'. -0 ' . 123 o'.02?6 -0.122 -0,0874 -7 ' . -0'.013 O'.O 175 -0 .105 -0.0930 - 7 . -0.030 O'.02l0 -0'.095 -0.0863 -6'. 0'.051 0'.0161 -0 .079 -0,0921 -6'. 0'.063 O'.O 19 1 -0.073 -0.0900 -5'. 0'. 146 O'.O 151 -0 .053 -0.0925 -5'. O'. 155 O'.O 191 -0,048 -0,0396 -4'̂  0'.2'll O'.O 158 -0 .0 26 -0.0915 - l ' . 0.251 O'.O 138 -0'.021 -0.0893 -3:. 0'.336 0'.0161 0 .001 -0.091 1 - 3 ' . 0'.343 O'.O 191 0.006 -0,0880 -2'. 0'.425 o'.0166 0 .029 -0.0878 -?-'. 0'.131 o'.oiao 0 .034 -0.0843 - i ' . 0'.530 o'.oiao 0 '.055 -0,0907 "1". 0'.53S 0'.0202 0'.060 -0,0080 o:. 0.637 0'.02()7 0 .080 -0.0955 O'. o'.oio 0'.0219 0.035 -0,0910 1'. 0'.722 0'.0229 0 .a 0 4 -0.0946 i'. 0'.725 0'.0234 o'.l 10 -0.0898 2' 0'.307 0'.0254 0 .130 -0,0927 2: 0',806 0'.0215 0.135 -0,0873 z'. 0'.893 0'.0277 0 .155 -0.0909 3'. ()'.890 0'.0270 0.160 -0,0850 1'. 0'.978 0'.0302 0 .180 -0,0895 4'. 0'.967 0'.0296 O'. 135 -0.0313 5'. 1.055 0'.0352 0 .203 -0.0873 5'. l'.042 0'.0336 0.206 -0,0300 6. 1.127 0'.0381 0 .226 -0,0810 6'. r.107 0'.0366 0 -,228 -0,0761 7'. I'. 193 0'.0116 0 .247 -0.0807 7'. l' . 167 0'.0H3 o'.218 -0.0716 8'. l'.258 0'.0461 0 .267 -0.0780 8. l'.230 O'.O'ISI 0.268 -0,0692 9'. 1.310 O'. 0509 0 .286 -0,0727 <?'. l'.28l" 0'.0500 0.236 -0,0643 10'. l'.346 O'0562 0 .300 -0.0680 10". T.298 0'. 0557 0.300 -0.0548 1 r. l',359 0.0624 0 '.310 -0,0610 11'. t'. 315 O'. 0629 0.308 -0.0508 12' l'.359 0'.0726 0 .314 -0,0568 12'. 1.310 0'.0743 0'.306 -0,0510 13. 1 .345 0'.09i'l 0 .302 -0,0653 13'. i'.28 4 0'.0911 0.291 -0.0596 14'. T.307 O'.l 125 0 '.285 -0.0717 14'. i'.246 O'.l 122 0.275 -0,0650 15'. l',270 o'.1313 0 .274 -0,0739 15'. f,207 0'.1278 0.262 -0,0677 ft— ft— ft— i—* ft— I 1 i 1 I I i I J ^* IX. L-J ru ft— o o CO • c - in i i L I ro ft— o ft— ru 14 J l ~u CD o •r- n ' • ' -ft •• r~ 1 1 I 3} *— ;ft— ift— :•— 'ft— Ift— ' • - * ; » - * •>•— i C o o o o o o o c o o * ' ' 'ft -'ft -ft " ' • - • • • 'ft. -ft ' •ft. ' ' • 'ft 'ft • • ' * • '» " •« • • - n I ru ru L-l l - i 14 14 r<j ft— . • — o •a Co -4 '-4 o I i L-l r j l O o (-* ro L-l r -< c - o LNI X i i i L J O i i CD -— X i ~j Co -0 .—• 14 r j ro 14 X i i i J l J i L-4 L-J r j ru -o — CD ta OI * * • 03 i—» . - 4 ft* -0. CO — - j — CO — 43 ru ,L-1 ru o o o O o O o o C o o o o o c o c o o o o o o C o c n n II t—* »— o O o o o o o o o o o o o o o o o o o o o o o o •3 » * 1-J ••— •o - J c- J l tn i i I-: L-! 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