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

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cl A THEORY FOR HIND TUNNEL WALL CORRECTIONS. BY CHRISTOPHER D. WILLIAMS B.A.Sc, University of B r i t i s h Columbia, 1967. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF H&STER OF APPLIED SCIENCE i n the Department of Bechanical Engineering We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1973 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department' o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h C o l u m bia Vancouver 8, Canada Date A p r i l 16, 1973 A b s t r a c t Wall c o r r e c t i o n t h e o r i e s f o r two-dimensional a i r f o i l s i n wind t u n n e l s with p a r t l y open w a l l s are examined. Conventional w a l l c o r r e c t i o n t h e o r i e s are l i n e a r i z e d t h e o r i e s , v a l i d only f o r s m a l l t h i n models of s l i g h t camber a t low angles of a t t a c k . Such t h e o r i e s are shown to be u s e l e s s f o r the p r e d i c t i o n of the r e q u i r e d w a l l c o r r e c t i o n s f o r l a r g e models, models at high angles o f a t t a c k , or models d e v e l o p i n g high l i f t . An e x a c t numerical theory i s presented i n which i t i s not necessary t o make thes e assumptions. The a i r f o i l and any s o l i d w a l l s e c t i o n s are r e p r e s e n t e d by s u r f a c e source and vortex s i n g u l a r i t i e s as i n the method of A. l i . 0.Smith, aerodynamic l i f t i s determined by n u m e r i c a l i n t e g r a t i o n of the c a l c u l a t e d pressure d i s t r i b u t i o n s around the a i r f o i l contour. The theory i n d i c a t e s t h a t c e r t a i n w a l l c o n f i g u r a t i o n s w i l l r e q u i r e s m a l l or n e g l i g i b l e w a l l c o r r e c t i o n s f o r t e s t s on l i f t i n g a i r f o i l s . Supervisor i i T A B L E O F C O N T E N T S E M S I I n t r o d u c t i o n 1 I I C o n v e n t i o n a l T h e o r i e s f o r Wall C o r r e c t i o n s . 2 I I I P r e v i o u s E x p e r i m e n t a l Work. 4 IV F o r m u l a t i o n o f an Exact Theory. 6 V Numerical S o l u t i o n . 12 VI R e s u l t s . 15 VII C o n c l u s i o n s . 20 References 45 i i i LIST OF FIGURES AND TABLES Page Figure 1. A i r f o i l i n a wind tunnel. 21 Figure 2. Comparison of l o n g i t u d i n a l s l o t t e d wall l i f t theory with data for Clark-Y a i r f o i l . 22 Figure 3. Comparison of porous-wall l i f t theory with data f o r Clark-Y a i r f o i l tested between l o n g i t u d i n a l l y s l o t t e d walls. 23 Figure 4. Comparison of porous-wall l i f t theory with data for Clark-Y a i r f o i l with s l o t t e d f l a p tested between l o n g i t u d i n a l l y s l o t t e d walls. 24 Figure 5. Porosity parameter as a function of open-area r a t i o . 25 Figure 6 . Geometry and notation f o r Smith's method. 26 Figure 7 . Source and vortex d i s t r i b u t i o n s for a two-dimensional a i r f o i l between s o l i d walls. 27 Figure 8. Pressure d i s t r i b u t i o n s f o r Clark-Y a i r f o i l . 28 Figure 9. Pressure d i s t r i b u t i o n s for NACA 23012 a i r f o i l with f l a p . 29 Figure 10. L i f t c o e f f i c i e n t for Clark-Y a i r f o i l between s o l i d walls. 30 Figure 11. Ratio of l i f t c o e f f i c i e n t s for Clark-Y a i r f o i l between s o l i d walls. Figure 12. Streamlines for a set of multiple a i r f o i l s i n an i n f i n i t e stream. Figure 13. Streamlines for an a i r f o i l between transversely s l o t t e d upper and s o l i d lower walls. Figure 14. Variation of l i f t c o e f f i c i e n t r a t i o with wall geometry f o r Clark-Y a i r f o i l . Figure 15. Variation of l i f t c o e f f i c i e n t r a t i o with upper wall open-area r a t i o for Clark-Y a i r f o i l . Figure 16. Pressure d i s t r i b u t i o n for Clark-Y a i r f o i l i n c o r rection-free l i f t t est configuration. Figure 17. Variation* of l i f t c o e f f i c i e n t r a t i o with wall geometry for Clark-Y a i r f o i l . Figure 18. Variation of l i f t c o e f f i c i e n t r a t i o with wall geometry for NACA 23012 a i r f o i l with s l o t t e d f l a p . Figure 19. Relative error i n l i f t c o e f f i c i e n t for 10% upper wall open-area r a t i o . Table 1. Configurations tested. V ACKNOWLEDGEMENTS The author would l i k e t o thank Dr. G. V. Parkinson f o r h i s ad v i c e and guidance i n the course o f t h i s r e s e a r c h . The computing f a c i l i t i e s of the Computing Center of the U n i v e r s i t y of B r i t i s h Columbia were used to do the c a l c u l a t i o n s c o n t a i n e d h e r e i n . 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. 1 I I n t r o d u c t i o n The p r o b l e m o f d e t e r m i n i n g t h e e f f e c t o f t h e w a l l s o f a w i n d t u n n e l on a m o d e l u n d e r t e s t i s an o l d o n e , w h i c h has o c c u p i e d r e s e a r c h e r s f o r more t h a n f o r t y y e a r s . The p r o b l e m i s e x t r e m e l y c o m p l e x s i n c e t h e e f f e c t o f t h e t u n n e l w a l l s i s t o a l t e r t h e f l o w c o n d i t i o n s ( s p e e d a n d d i r e c t i o n ) n e a r t h e m o d e l f r o m t h o s e t h a t w o u l d e x i s t i n a c o r r e s p o n d i n g t e s t i n f r e e a i r . At a g i v e n p o i n t i n t h e f l o w f i e l d t h e s e e f f e c t s a r e f u n c t i o n s o f m o d e l s i z e and s h a p e . Any m e a s u r e d q u a n t i t i e s , s u c h a s f l o w s p e e d o r a e r o d y n a m i c f o r c e s a n d moments, must be c o r r e c t e d f o r t h e s e w a l l e f f e c t s . I d e a l l y s u c h c o r r e c t i o n s c o u l d be a p p l i e d d i r e c t l y t o t h e s e m e a s u r e d q u a n t i t i e s t o p r e d i c t t h e i r v a l u e s a s t h e y w o u l d be m e a s u r e d i n a f r e e - a i r t e s t e n v i r o n m e n t . 2 I I C o n v e n t i o n a l T h e o r i e s f o r H a l l C o r r e c t i q n s G l a u e r t ( 3 ) , G o l d s t e i n ( 4 ) , and A l l e n and V i n c e n t i { 1 ) used s i n g u l a r i t y image techniques to model the a i r f o i l - w a l l i n t e r a c t i o n s . 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 , 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 a s s o c i a t e d with d i s t r i b u t i o n s of v o r t i c e s , d o u b l e t s , and sources r e s p e c t i v e l y . For each type o f s i n g u l a r i t y , a s u i t a b l e system of "images" i s found i n the tu n n e l w a l l s , t o s a t i s f y the r e q u i r e d flow c o n d i t i o n s at the w a l l s ; f o r example, a s o l i d w a l l boundary r e q u i r e s zero f l o w normal to the boundary, w h i l e an open j e t boundary r e q u i r e s a c o n s t a n t p r e s s u r e along the boundary. I t i s very u s e f u l i f the induced e f f e c t o f each system of images at the a i r f o i l can be assumed t o be a d d i t i v e ; t h a t i s , these i n d i v i d u a l e f f e c t s can be superimposed. In terms of the co 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 , t h i s r e q u i r e s a l i n e a r i z e d t h e o r y , v a l i d o n l y f o r s m a l l w a l l e f f e c t s , that i s , f o r 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 s at low angles o f a t t a c k . Woods ( 1 2 ) uses conformal mappings t o tr a n s f o r m the a i r f o i l p l u s w a l l s i n t o a g e o m e t r i c a l l y simple boundary value problem f o r which a s o l u t i o n i s known; t h a t i s , a f u n c t i o n of a complex v a r i a b l e whose r e a l and/or imaginary p a r t s are p r e s c r i b e d on t h i s boundary. Woods* technique r e s u l t s i n i n t e g r a l e q uations f o r t h e mapping f u n c t i o n ; u s u a l l y numerical techniques 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 . E i t h e r approach can be formulated i n terms of the d i s t u r b a n c e flow v e l o c i t y p o t e n t i a l j , which f o r i n c o m p r e s s i b l e flow s a t i s f i e s L a p l a c e ' s e q u a t i o n , which i s l i n e a r . 3 For a s o l i d boundary which i s p a r a l l e l to the undisturbed flow at upstream i n f i n i t y there i s zero flow normal to the boundary; hence i s zero there. dn For an open j e t the l i n e a r i z e d condition of constant pressure at the j e t boundary can be expressed v i a B e r n o u l l i ' s equation as re q u i r i n g zero streamwise disturbance v e l o c i t y ; i s zero there. hence dx For s l o t t e d or perforated walls where an "equivalent homogeneous boundary condition" can be s t i p u l a t e d , (Baldwin et a l ( 2 ) ) , a l i n e a r combination of the boundary conditions f o r s o l i d walls and for an open j e t i s used. In such equivalent homogeneous boundary conditions a l l d e t a i l s of the s l o t or perforation 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 (longitudinal or transverse). Only bulk properties such as the porosity or open area r a t i o (OAR) are preserved. Wood (11) shows, f o r example, that the OAR f o r lo n g i t u d i n a l s l o t s would need to be les s than 1 % to achieve a boundary condition appreciably d i f f e r e n t from the open j e t case. In practice, at such a small 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 cross-flow would be i n v a l i d . Moreover, Wood's analysis of t h i s boundary condition indicates that only cross-flow v e l o c i t i e s of order l e s than .5% of the mean flow would be i n keeping with the l i n e a r i z a t i o n s involved in the derivation of t h i s boundary condition. 4 III Previous Experimental Work The r e s u l t s of conventional wall correction theories are shown compared with experiment (Lim{6)) on a 14% thick Clark-Y a i r f o i l , of four d i f f e r e n t sizes, in the presence of l o n g i t u d i n a l l y (streamwise) s l o t t e d walls. In these and subsequent f i g u r e s , C i s the a i r f o i l chord and H i s the wind tunnel test section height, or equivalent height, for two-dimensional testing (Figure 1). Figures 2,3,4,5 are taken from Parkinson and Lim(7). Figure 2 shows the r a t i o of measured l i f t curve slope m to i t s value m„ i n free a i r , as a function of model siz e C/H f o r l o n g i t u d i n a l l y s l o t t e d walls of OAR 0.,5.6,11.1,18.5 and 100%. The agreement with the theory f o r l o n g i t u d i n a l s l o t s i s good only f o r s o l i d walls; that i s , for zero OAR. A l l other walls were predicted to behave as i f they were almost completely open. Thus the theory f o r l o n g i t u d i n a l l y s l o t t e d walls i s useless f o r predicting the behaviour of a model under actual test . The same measured r a t i o of l i f t curve slopes i s compared i n Figure 3 with that predicted by porous wall theory (Woods (12)). This shows that the porosity parameter P (defined in Ref. 7) for a given wall OAR can be chosen to produce good agreement with the test data. This r e s u l t appears i n Figure 4 to be true also for an a i r f o i l with s l o t t e d f l a p , but the values of porosity parameter so obtained were not the same as for the basic a i r f o i l , for the same wall configurations. 5 F i g u r e 5 p o r t r a y s the p o r o s i t y parameter as a f u n c t i o n of the w a l l OAR, which a l s o depends on the model under t e s t , an i m p o s 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 porous w a l l t h e o r y . 6 17 Formulation of an Exact Theory What then i s needed i s an exact (rather than a linearized) theory, where the net corrections to measured forces and moments are achieved d i r e c t l y , rather than a d d i t i v e l y . The method used i n t h i s i n v e s t i g a t i o n i s an extension of the surface s i n g u l a r i t y d i s t r i b u t i o n theory of A.M.O. Smith(8). In Smith's method, the a i r f o i l i s represented by a d i s t r i b u t i o n of sources and vor t i c e s around i t s perimeter. The v e l o c i t i e s at points i n the flow f i e l d due to a l l such sources and v o r t i c e s are calculated d i r e c t l y . The usual flow boundary condition of zero flow through the a i r f o i l surface i s applied and a f i n i t e - v e l o c i t y Kutta condition i s applied at the t r a i l i n g edge. Again <j> i s the disturbance v e l o c i t y p o t e n t i a l which s a t i s f i e s Laplace's eguation, vanishes at i n f i n i t y , and s a t i s f i e s the appropriate boundary conditions on the a i r f o i l . The p o t e n t i a l at a point P due to a single three-dimensional source s i n g u l a r i t y at a point Q i s <£> --m J_ p 4TT rRQ ( D where m i s the volume flow rate of f l u i d emitted by the source, and IT PQ i s the distance between the points 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 an arbit r a r y surface S i s 7 p JCJ rPQ (2) where cr \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 / 4 7 T J , of the source element a t Q. Sin c e the d i s t u r b a n c e v e l o c i t y i s the g r a d i e n t o f <f>\, 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 (3) where ix i s the outward s u r f a c e normal, V» [the u n d i s t u r b e d flow at upstream i n f i n i t y , and F 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 . F i s z e r o f o r a s o l i d (impermeable) boundary, but nonzero f o r s u c t i o n or blowing t h e r e . A n a l y s i s (Smith (8)) shows t h a t the normal v e l o c i t y c o n t r i b u t i o n s at P on S by such source elements dS c o n s i s t o f a " l o c a l " term, ' 27TOr ( P ) (4) due to the source element a t P , p l u s a " f a r f i e l d " term (5) due t o t h e summation of a l l o t h e r source elements Q on S. The r e s u l t i n g boundary c o n d i t i o n 27ro-(P) -fffa(j—) a (Q)ds =-V»°n + F (6) comprises a Fredholm i n t e g r a l e q u a t i o n of the second kind, f o r the unknown conti n u o u s 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 8 funct ion o~|(P) . Existence and uniqueness theorems for such equations may be found in Kellogg (5). The surface S may be d i s j o i n t , but the outward normal vector must be a continuous function of p o s i t i o n . In pract ice an approximating polygonal surface i s used to represent a three-dimensional body so that the continuous d i s t r i b u t i o n of sources becomes a succession of N f i n i t e d i s t r i b u t e d source elements. The normal-flow boundary condi t ion i s appl ied at a s ing le " c o n t r o l po in t " on•each source element. Thus the r e s u l t i n g exact i n t e g r a l equation reduces to a set of H simultaneous l i n e a r a lgebra ic equations for the strengths of N f i n i t e surface source elements. Defining the l i n e a r operator ( 7 ) the boundary condi t ion appl ied at the i - t h c o n t r o l point ^Aj jo^ -V^rii + Fj-ind ica te s A (0) to be the normal v e l o c i t y induced at c o n t r o l point • i ' by a uni t strength source element at * j * . Hence Ajj for a l l i = 1 , 2 , . . . N . i s 2'TT For a two-dimensional a i r f o i l t h i s approximating surface becomes a polygonal c y l i n d e r . A Kutta cond i t ion i s applied at the two c o n t r o l points adjacent to the t r a i l i n g edge - th i s f i x e s the net c i r c u l a t i o n about the a i r f o i l . With respect to Cartesian axes x and Vj f ixed to the j - th 9 source element (Figure 6), the integration of a two-dimensional l i n e source i n t o a two-dimensional d i s t r i b u t e d source element produces at a point ' i ' v e l o c i t y components and v a x r - t ( x . _ ¥ j ) 2 + y / yi L x f +yji-(f f (9) (10) where Xj and yj are the distances from the j-th to the i - t h element; the j-th element has length ASj With reference to Cartesian "wind axes" X and Y, (X i s i n the wind d i r e c t i o n ) , the j-th source element i s i n c l i n e d at an angle 9] j to the X-axis. Thus A}i=Vy.cosie.-op -Vx.sirKtffflj) (11) j J and i t s orthogonal partner are the normal and tangential disturbance v e l o c i t i e s respectively at control point *i» due to a unit strength source element at ' j Corresponding expressions for the ve l o c i t y components due to a d i s t r i b u t e d vortex element of c i r c u l a t i o n strength density X(Q) can be obtained. Then the normal and tangential v e l o c i t i e s at control point ' i * due to a l l N source and vortex elements and the uniform approach flow U are 10 N N V l A ^ - l B w j i - U s i n f l j=l 1 k=l ( 1 3 ) and N N V . Z B j i o j +JjAkirk+Ucos£, ( 1 4 ) Hence the normal-flow boundary condition becomes, for zero V n j = 0 ( 1 5 ) at a l l N c o n t r o l points, while the f i n i t e - v e l o c i t y Kutta condition becomes V* =-V* •upper "lower ( 1 6 ) at the two control points adjacent to the a i r f o i l t r a i l i n g edge. In p r a c t i c e , a l l vortex elements on a closed body requiring a Kutta condition are chosen of equal strength Y , so the flow about an N-sided polygonal a i r f o i l requires the solution of N+1 equations i n the N+1 unknowns O j , a j l , . . . Oj^jand ^ j . Usually the s o l u t i o n i s obtained d i r e c t l y by the method of Gauss-elimination, although i n d i r e c t i t e r a t i v e procedures are also possible. . In the present i n v e s t i g a t i o n , the above method of d i s t r i b u t e d s i n g u l a r i t i e s i s extended to include multiple-element a i r f o i l s i n the presence of s o l i d or slotted wind tunnel walls. On s o l i d wall sections, only the zero norma 1-velocity condition applies, hence only source elements are applied there. On a i r f o i l - s h a p e d bodies with Kutta conditions applied at t h e i r 11 i n d i v i d u a l t r a i l i n g edges, both source and vortex elements are r e q u i r e d . Thus f o r the two-dimensional c o n f i g u r a t i o n of F i g u r e 7 , the system of N+M e q u a t i o n s to be s o l v e d i s composed of the N e q u a t i o n s N M R(k) l A j i o r - I ^ i B m i - U s i n f l i = i , 2 , - N j=l k=l m=l p r e s c r i b i n g zero normal v e l o c i t i e s , and the M e q u a t i o n s N M R(k) ? ( B J u s + B J L s H + ? r k ? ( A m U s + A ^ L s ) = - U(COS0 u +C0S(9L ) (17) k=l m=l s=l,2,-M (18) f o r the M b o d i e s with K u t t a c o n d i t i o n s at t h e i r t r a i l i n g edges. The s u b s c r i p t s D and L i n d i c a t e the c o n t r o l p o i n t s on the upper and lower s u r f a c e s of an a i r f o i l s e c t i o n , a t the t r a i l i n g edge. Thus there a r e : - a t o t a l o f H source elements d i s t r i b u t e d over the a i r f o i l s and s o l i d w a l l s . -a t o t a l o f 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 ]^ R(k) v o r t e x elements d i s t r i b u t e d over the k=l a i r f o i l s ; the k-th such body has R (k) source elements and R(k) e q u a l - s t r e n g t h v ortex elements d i s t r i b u t e d over i t . -N unknown source s t r e n g t h d e n s i t i e s OJ j. v 1 -H unknown vortex s t r e n g t h d e n s i t i e s \ . -N+M equations i n the N+M unknowns Of Utr 2 r • • • n 12 V Numerical S o l u t i o n A program written in FORTRAN for use on the UBC IBM 360/67 given computer i s used to construct the matrices Ajj and Bj the coordinates, lengths and orientations of the source and vortex elements on the a i r f o i l and walls. These matrices are then used to assemble the c o e f f i c i e n t s of the unknowns in the H+H equations. T y p i c a l l y N+M i s of order 300 to 100, hence the matrices Ajjj and Bji each contain more than 100,000 nonzero, nonsymmetric ent r i e s . The t o t a l computational capacity of the 360/67 i s only 250,000 en t r i e s , hence such large matrices must be partitioned f o r computation and temporary storage on au x i l i a r y devices; for example, on magnetic tapes or dis c s . The matrices and Bji describe the r e l a t i v e geometry of the source and vortex elements; that i s , t h e i r r e l a t i v e position and ori e n t a t i o n . These matrices must be recalculated for each change i n r e l a t i v e geometry. The system of equations to be solved i s written + CN+I,tf + '*+CN+M,I^M C i j 2oj +C 2 ) 2°2 + *" + C N,2 a N +CN+I,2)I + " + c N+M l2 r M = d2 C l ,N a i , + C2 , N a2 + " + C N , N a N + C N H , N ^ + " " + CN+M,N^M cl,N+lCTi + C2,N+I°2 + " " + C N , N + l A N + C N+I ,N* I^ + - " + C N + M , N / M = d N where the m a t r i x CjiJ.and the column v e c t o r a r e a s s e m b l e d from the m a t r i c e s Ajj (19) C l^+M^ + C2^+M a2 + ' " + C N^+M C T N + CN+l,N+M^ + ' " + CN+M,N+M^M = dN+M j in the system and Bji by means of 13 equations (17) and (18) ; that i s , AH RM ~XB m j m=l B J U S + B J L S j=l,2,-N; i = l,2,-N k=l,2,-M;i=N+k;i=l,2,-N j = l,2,- N;s=l,2,-Mv i=N+s Z ( A M U +Am, ) k=l,2,-M-,i=N*k;s = l,2,-M;i=Nfs and U s i n S j i=i,2,-N -UC0So\j -UCOS0 L s=lJ2,-M ;i=N+s The summations and N' M Rjk) Vn-lAjioj - Z r ^ B m i - U s i n S , j=l k=l m=! N M R(k) Vt. = l B j j 0 1 + Z r k l A m i + U c o s S i Ti j=| J J k=l Km=l (20) (21) (22) (23) provide the net normal and tangential v e l o c i t i e s at control points ' i * due to a l l sources and vortices ' j ' . At a l l control points on s o l i d surfaces, Vn j c o e f f i c i e n t i s zero, and the l o c a l pressure i s calculated from Vf, The r e s u l t i n g values o f C r (25) may be integrated numerically around the a i r f o i l contour to determine the l i f t , drag, and 1 4 pitching momant c o e f f i c i e n t s , from the expressions C L T=-£ZC P lAXi CD,=-c-£CpjAy, . 1 AXj = ASjCOS^i , Ay, =ASj sinc?j (26) where (27) and summations are performed clockwise around the polygonal contours. Resultant v e l o c i t i e s may also be calculated at points i n the flow f i e l d not on the a i r f o i l or walls, so that streamlines throughout the flow f i e l d may be drawn as i s o c l i n e s . 15 VI Results The agreement of pressure d i s t r i b u t i o n s calculated by Smith's method, for a i r f o i l s i n free a i r , with two-dimensional tests i s well established (Smith (8)).. Figure 8 shows a comparison of calculated pressure d i s t r i b u t i o n s for a 14% thick Clark-Y a i r f o i l i n free a i r and i n the presence of s o l i d walls. The walls, for t h i s s i z e model, produce 30% higher l i f t . Figure 9 shows a comparison of an experimentally determined pressure d i s t r i b u t i o n f o r an NACA 23012 a i r f o i l with f l a p (Wenzinger et al{10)), with calculated pressure d i s t r i b u t i o n s f o r the same a i r f o i l i n free a i r and i n the presence of s o l i d walls. Experimentally, the p o t e n t i a l flow f r e e - a i r pressures are not achieved because of boundary layer e f f e c t s . For both the above a i r f o i l s , an important observation i s that the undersurface pressure does not change much in the presence of s o l i d ^walls; hence the upper or suction surface provides most of the increased l i f t . The l i f t c o e f f i c i e n t s are reported i n terms of the l i f t c o e f f i c i e n t developed by the a i r f o i l i n the tunnel and C | _ F developed i n free a i r . with angle of attack Figure 10 shows the v a r i a t i o n of a ; f o r models of d i f f e r i n g C/H. For small models, the l i f t curves are concave downward as i s usual, while for large models they are concave upward. The r a t i o of l i f t c o e f f i c i e n t s i s shown i n Figure 11 as a 16 function of model s i z e , for three angles of attack. The corresponding prediction of conventional s o l i d wall theory (Woods (12)), which i s independent of angle of attack, agrees well with the test data (Lim (6)) . The very large corrections for large a i r f o i l s developing high l i f t , shown i n the above r e s u l t s , prompted a search for a wall configuration that would exhibit the known c a n c e l l i n g e f f e c t s of partly open, partly closed walls, and which would therefore provide n e g l i g i b l e or small wall corrections. The f i r s t configuration investigated was a set of multiple a i r f o i l s i n an otherwise uniform stream of i n f i n i t e extent (Figure 12). This configuration might be used to represent a two-dimensional tunnel with transversely (spanwise) s l o t t e d walls. With a i r f o i l - s h a p e d transverse s l a t s , no flow separation would occur at the s l a t s , as each winglet would be operating i n an unstalled condition, with a Kutta condition applied at i t s t r a i l i n g edge. Consider the l i m i t i n g lower streamline AB, which enters the tunnel near the entrance to the test section. Physically, t h i s lower streamline i s a shear layer, i d e a l i z e d as a free streamline, at constant zero reference pressure, which brings turbulent mixing into the tunnel, close to the a i r f o i l , an undesirable e f f e c t . But the corresponding streamline in the multiple a i r f o i l - i n f i n i t e stream representation i s not a free streamline, but merely one of the i n f i n i t e stream. Thus, i n t h i s representation, the pressure i s not zero on t h i s lower streamline, and errors would be introduced i n representing the 17 flow i n t h i s manner; in p a r t i c u l a r , close to the underside of the a i r f o i l . S i m i l a r l y the upper l i m i t i n g streamline cannot be c o r r e c t l y represented e a s i l y , but since t h i s streamline i s separated from the a i r f o i l by the intervening s l a t s with t h e i r boundary conditions impressed on the flow, the errors i n incorrect pressure and l o c a t i o n in the representation of t h i s streamline should have only secondary e f f e c t s on the main a i r f o i l . With only one s l o t t e d wall, Figure 13, i t should s t i l l be possible to produce the c a n c e l l i n g e f f e c t s of partly open, partly closed walls, since the upper slotted wall i s adjacent to the suction side of the a i r f o i l , where most of the increased l i f t i s developed. Hence a combination of a transversely s l o t t e d upper wall with a s o l i d lower wall was envisaged as an e f f e c t i v e l y correction-free test configuration, for a l i f t i n g a i r f o i l . This configuration should be accurately represented by two-dimensional p o t e n t i a l flow theory, since the flow angles at the wall s l a t s should be small enough that these winglets w i l l be unstalled, and the upper shear layer w i l l re-enter the tunnel only well downstream of the a i r f o i l . An i n v e s t i g a t i o n of t h i s configuration followed. of 20° , Figure 14 shows For the Clark-Y, at an angle abase the r a t i o of l i f t c o e f f i c i e n t s as a function of model size for a 50% and 75% OAR upper wall. Also shown are the curves for two s o l i d walls, and for the a i r f o i l i n ground e f f e c t . 18 For the same a i r f o i l a t the same angle of a t t a c k , the r a t i o of l i f t c o e f f i c i e n t s i n F i g u r e 15 i s shown as a f u n c t i o n of upper w a l l OAR, f o r a range of model s i z e s . An upper w a l l of zero OAR corresponds t o two s o l i d w a l l s ; 100% OAR corresponds to the a i r f o i l i n ground e f f e c t . Where the r a t i o of l i f t c o e f f i c i e n t s i s u n i t y , t h e r e i s zer o net w a l l c o r r e c t i o n ; t h i s o c c u r s f o r t h i s a i r f o i l at approximately 70% OAR. The p r e s s u r e d i s t r i b u t i o n f o r the Clark-Y a t an ^boge of 20° f o r such a zero c o r r e c t i o n c o n f i g u r a t i o n of 70% OAR, with model s i z e C/H of .72, appears i n F i g u r e 16 al o n g with the c o r r e s p o n d i n g p r e s s u r e d i s t r i b u t i o n f o r f r e e a i r . The net l i f t i s the same i n both c a s e s . Although t h e r e i s l e s s s u c t i o n over t h e forward upper s u r f a c e , t h e r e i s i n c r e a s e d s u c t i o n over the rearward p o r t i o n o f the a i r f o i l . S i m i l a r r e s u l t s are shown i n F i g u r e 17 f o r the same a i r f o i l but at a d i f f e r e n t ^ Q^ase of 12 . Again a s l o t t e d upper w a l l of 70% OAR sh o u l d p r o v i d e a r e l a t i v e l y 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 . Comparison i s a l s o made with the theory f o r a c i r c u l a r a r c a i r f o i l i n ground e f f e c t {Tomotika e t a l ( 9 ) ) of | s i m i l a r 5.3% camber, but at an a c h o nj jof 5 ° . The agreement i s ! f a v o u r a b l e . R e s u l t s o b t a i n e d f o r the NACA 230 12 with f l a p were s i m i l a r . F i g u r e 18 i n d i c a t e s t h at 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 of 70% OAR with a s o l i d lower w a l l p r o v i d e s a r e l a t i v e l y 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 . The r e l a t i v e e r r o r i n C|_ f o r a 70% upper w a l l OAR i s shown 19 i n F i g u r e 19 f o r the Clark-Y a i r f o i l a t f o u r a n g l e s of a t t a c k and the NACA 23012 a i r f o i l with f l a p . The r e l a t i v e e r r o r a t t h i s OAR i s l e s s than 3%, except f o r extremely l a r g e models. Table 1 o u t l i n e s the d e t a i l s of a l l c o n f i g u r a t i o n s t e s t e d . 20 VII C o n c l u s i o n An e x t e n s i o n of the two-dimensional p o t e n t i a l flow thoery based on the s u r f a c e s i n g u l a r i t y d i s t r i b u t i o n procedure has shown t h a t a r e l a t i v e l y c o r r e c t i o n - f r e e wind t u n n e l t e s t c o n f i g u r a t i o n f o r l i f t i n g a i r f o i l s can be achieved f o r a wide range i n model s i z e s by u t i l i z i n g a 70% open area r a t i o t r a n s v e r s e l y (spanwise) s l o t t e d upper w a l l , i n c o n j u n c t i o n with a s o l i d lower w a l l . Small or n e g l i g i b l e c o r r e c t i o n s can be achieved f o r a wide range of angles of a t t a c k , and f o r d i f f e r e n t models, t h a t i s , s i n g l e a i r f o i l s or a i r f o i l - f l a p combinations. A program of experimental v e r i f i c a t i o n o f these r e s u l t s should be undertaken. Where the w a l l c o r r e c t i o n s are not n e g l i g i b l e but s m a l l , a l i n e a r i z e d p e r t u r b a t i o n theory based on t h i s c o n f i g u r a t i o n might be developed. 21 cu c c 3 4-1 •o c - H rt c o t-l CU 3 OC ZD 22 .6 .8 1.0 Figure 2. Comparison of 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 l i f t theory wit h data f o r Clark-Y a i r f o i l . SOLID C L A R K - Y 14% L O N G I T U D I N A L S L O T S THEORY OAR% P EXPT 1 5.6 .46 A 2 II.1 .83 n 3 18.5 1.25 V 4 100 OD na .2 .4 .6 .8 c/, H Figure 3. Comparison of porous-wall l i f t theory w i t h data f o r Clark-Y a i r f o i l t ested between 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 . CLARK-Y 14% • S L O T • S L O T T E D F L A P LONGITUDINAL S L O T S O • T H E O R Y I OAR % II. 1 P .33 E X P T A 2 18.5 .67 • 3 29.6 1.54 V 4 100 00 na .2 .6 .8 1.0 H Figure 4. Comparison of porous-wall l i f t theory w i t h data f o r Clark-Y a i r f o i l w i t h s l o t t e d f l a p tested between 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 . O A R Figure 5. P o r o s i t y parameter as a fu n c t i o n of open-area r a t i o . AIRFOIL v WITH FLAP THEORY X F i g u r e 6. Geometry and n o t a t i o n f o r Smith's method. Figure 7. Source and vortex distributions for a two-dimensional a i r f o i l between solid walls. X \ \ \ \ CLARK-Y 14% a = 20° C/H=.72 XSOLID X WALLS FREE AIR Figure 8. Pressure d i s t r i b u t i o n s f o r Clark-Y a i r f o i l Figure 9. Pressure d i s t r i b u t i o n s t o r NACA 23012 a i r f o i l . F i g u r e 10. L i f t c o e f f i c i e n t f o r C l a r k - Y a i r f o i l b e t w e e n s o l i d w a l l s . t I t I • t CLARK-Y 14% / SOLID WALLS / 2 .4 .6 .8 1.0 f i g u r e 1 1 . R a t i o o f l i f t c o e f f i c i e n t s f o r C l a r k - Y a i r f o i l b e t w e e n s o l i d w a l l s . 32 C L A R K - Y 1 4 % a = 2 0 ° SLOTTED UPPER SOLID LOWER 8 .4 .6 .8 2 0 4 0 6 0 8 0 100 O A R - % F i g u r e 1 5 . V a r i a t i o n o f l i f t c o e f f i c i e n t r a t i o w i t h u p p e r w a l l o p e n - a r e a r a t i o f o r C l a r k - Y a i r f o i l . CLARK-Y 14% i Figure 16. Pressure d i s t r i b u t i o n f o r Clark-Y a i r f o i l i n c o r r e c t i o n - f r e e l i f t t e s t c o n f i g u r a t i o n . NACA 23012 a=8° 8 = 20° SLOTTED UPPER SOLID LOWER , S O L I D 7 0 % O A R G R O U N D E F F E C T 6 .8 1.0 H F i g u r e - .13. V a r i a t i o n o f l i f t c o e f f i c i e n t r a t i o w i t h w a l l g e o m e t r y f o r NACA 23012 a i r f o i l w i t h s l o t t e d -2 ' .8 1.0 9 7 H F i g u r e 19. R e l a t i v e e r r o r i n l i f t c o e f f i c i e n t f o r 70% upper w a l l open-area r a t i o . Table 1. Configurations Tested AF a NSA WALL CONFIGURATION NSU NSL NSLAT c/C t/c NSS OAR NWC C/H CLF CLT CLT/CLF C-Y 20 50 Solid 50 50 - - - - - 6.3 1.0 3.091 5.360 1.734 C-Y 20 50 Solid 50 50 - - - - - 6.3 .8 3.091 4.416 1.428 C-Y 20 -50 Solid 50 50 - - - - - 6.3 .72 3.091 4.121 1.333 C-Y 20 50 Solid 50 50 - - - - - 6.3 .6 3.091 3.785 1.224 C-Y 20 50 Solid 50 50 - - - - - 6.3 .4 3.091 3.378 1.092 C-Y 20 50 Solid 50 50 - - - - 6.3 .3 3.091 3.242 1.049 C-Y 20 50 Solid 50 50 - - - - -• 6.3 .2 3.091 3.149 1.020 C-Y 20 50 Solid *• 50 50 - - - - - 6.3 .1 3.091 3.100 1.003 C-Y 12 50 Solid 100 100 - - - - - 6 1.0 2.188 3.475 1.585 C-Y 12 50 Solid 100 100 - - - - - 6 .8 2.188 2.982 1.360 C-Y 12 50 Solid 100 100 - - - - - 6 .6 2.188 2.622 1.195 C-Y 12 50 Solid 100 100 - - - - - 6 .4 2.188 2.372 1.080 C-Y 12 50 Solid 100 100 - - - - - 6 .2 2.188 2.225 1.015 C-Y 10 50 Solid 50 50 - - - - - 6.3 1.0 1.955 3.065 1.567 C-Y 10 50 Solid 50 50 - - - - - 6.3 .8 1.955 2.648 1.355 C-Y 10 50 Solid 50 50 - - - - - 6.3 .6 1.955 2.337 1.195 C-Y 10 50 Solid 50 50 - - - - — 6.3 .4 1.955 2.119 1.083 C-Y 10 50 Solid 50 50 - - - - - 6.3 .2 1.955 1.990 1.018 C-Y 0 50 Solid 50 50 - - - - - 6.3 1.0 .7635 1.115 1.460 C-Y 0 50 Solid 50 50 - - - - - 6.3 .8 .7635 .988 1.293 C-Y 0 50 Solid 50 50 - - - - - 6.3 .6 .7635 .889 1.163 C-Y 0 50 Solid 50 50 - - - - - 6.3 .4 .7635 .818 1.071 C-Y 0 50 Solid 50 50 — — _ — — 6.3 .2 .7635 .775 1.013 AF a NSA WALL CONFIGURATION NSU NSL NSL C-Y -6.2 50 So l i d 50 50 -C-Y -6.2 50 So l i d 50 50 -C-Y -6.2 50 So l i d 50 50 -C-Y -6.2 50 So l i d 50 50 -C-Y -6.2 50' S o l i d 50 50 -C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. * - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 G.E. - 100 -C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. - 80 15 C-Y 20 50 T.S.U.S.L. 80 15 c/C t/c NSS OAR NWC C/H CLF CLT CLT/CLF - - - - 6.3 1.0 .0415 -.020 -1.39 - - - - 6.3 .8 .0415 -.013 -.903 - - - - 6.3 .6 .0415 -.004 -.290 - - - - 6.3 .4 .0415 .0049 .338 - - - - 6.3 .2 .0415 .0116 .80 .12 .33 9 .5 3.4 1.0 3.091 3.768 1.219 .12 .33 9 .5 3.4 .8 3.091 3.524 1.140 .12 .33 9 .5 3.4 .72 3.091 3.465 1.121 .12 .33 9 .5 3.4 .6 3.091 3.348 1.083 .12 .33 9 .5 3.4 .4 3.091 3.218 1.041 .12 .33 9 .5 3.4 .2 3.091 3.131 1.013 .06 .33 9 . 75 3.4 1.0 3.091 3.062 .991 .06 .33 9 . 75 3.4 .8 3.091 3.037 .983 .06 .33 9 . 75 3.4 .72 3.091 3.033 .981 .06 .33 9 . 75 3.4 .6 3.091 3.035 .982 .06 .33 9 . 75 3.4 .4 3.091 3.063 .991 - - - - 6 1.0 3.091 2.676 .866 - - - 6 .8 3.091 2.733 .884 - - - - 6 .72 3.091 2.762 .894 - - - - 6 .6 3.091 2.811 .909 - - - - 6 .4 3.091 2.919 .944 - - - - 6. .3 3.091 2.983 .965 - - - - 6 .2 3.091 3.045 .985 - - - - 6 .1 3.091 3.085 .998 .072 .33 9 .7 3.4 1.0 3.091 3.170 1.026 .072 .33 9 .7 3.4 .72 3.091 3.092 1.0 .072 .33 9 .7 3.4 .55 3.091 3.078 .996 .072 .33 9 .7 3.4 .39 3.091 3.087 .999 .072 .33 9 .7 3.4 .19 3.091 3.098 1.002 AF a NSA WALL CONFIGURATION NSU NSL NSLi C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. • - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 T.S.U.S.L. - 80 15 C-Y 12 50 G.E. - 100 -C-Y 12 50 G.E. - 100 -C-Y 12 50 G.E. - 100 -C-Y •12 50 G.E. - 100 -C-Y 12 50 G.E. — 100 -c/C t/c NSS OAR NWC C/H CLF CLT CLT/CLF .12 .33 9 .5 3.4 1.0 2.188 2.617 1.195 .12 .33 9 .5 3.4 .9 2.188 2.537 1.158 .12 .33 9 .5 3.4 .8 2.188 2.463 1.124 .12 .33 9 .5 3.4 .6 2.188 2.343 1.070 .12 .33 9 .5 3.4 .45 2.188 2.276 1.040 .12 .33 9 .5 3.4 .26 2.188 2.221 1.013 .12 .33 9 .5 3.4 .16 2.188 2.199 1.003 .06 .33 9 .75 3.4 1.0 2.188 2.234 1.021 .06 .33 9 .75 3.4 .6 2.188 2.179 .996 .06 .33 9 .75 3.4 .45 2.188 2.178 .994 .06 .33 9 .75 3.4 .26 2.188 2.187 .998 .06 .33 9 .75 3.4 .16 2.188 2.189 1.001 .06 .33 9 .75 3.4 .09 2.188 2.188 1.000 .072 .33 9 .7 3.4 1.0 2.188 2.284 1.044 .072 .33 9 .7 3.4 .8 2.188 2.233 1.021 .072 .33 9 .7 3.4 .6 2.188 2.200 1.006 .072 .33 9 .7 3.4 .45 2.188 2.190 1.001 .072 .33 9 .7 3.4 .26 2.188 2.192 1.002 .072 .33 9 .7 3.4 .16 2.188 2.190 1.001 .072 .33 9 .7 3.4 .09 2.188 2.188 1.000 - - - - 6 1.0 2.188 2.060 .940 - - - - 6 .8 2.188 2.068 .945 - - - - 6 .6 2.188 2.087 .954 - - - - 6 .4 2.188 2.121 .968 _ _ 6 .2 2.188 2.170 .990 A I' C-Y U 6 NSA WAL 50 L CONFIGURATION T.S.U.S.L. NSU NSL 80 NSLAT 15 c/C .072 t/c .33 NSS 9 OAR . 7 NWC 3.4 C/ll 1.0 CLF 1.483 CLT 1.557 CLT/CLF 1.050 C-Y 6 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .6 1.483 1.495 1.008 C-Y 6 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .45 1.483 1.485 1.002 C-Y 6 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .26 1.483 1.484 1.001 C-Y 6 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .16 1.483 1.483 1.000 C-Y 6 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .09 1.483 1.482 1.000 C-Y 0 50 T.S.U.S.L.' - 80 15 .072 .33 9 .7 3.4 1.0 .7535 .772 1.001 C-Y 0 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 .3-4 .6 .7635 .758 .992 C-Y 0 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .45 .7635 .757 .992 C-Y 0 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .26 .7635 .761 .996 C-Y 0 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .16 .7635 .762 .998 C-Y 0 50 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .09 .7635 . 762 .998 23012 8 -20 81=46+35 S o l i d 100 100 - - - - - 6 1.0 2.442 3.300 1.392 23012 8 -20 81 S o l i d 100 100 - - - - - 6 .8 2.442 3.009 1.233 2 3012 8 -20 81 S o l i d 100 100 - - - - 6 .6 2.442 2. 770 1.135 23012 8 -20 81 S o l i d 100 100 - - - - - 6 .4 2.442 2.587 1.061 23012 8 -20 81 S o l i d 100 100 - - - - 6 .2 2.442 2.472 1.012 23012 8 -20 81 G.E. - 100 - - - - - 6 1.0 2.44 2 2.176 .892 23012 8 -20 81 G.E. - 100 - - • - - 6 .8 2.442 2.212 .905 23012 8 -20 81 G.E. - 100 - - - - - 6 .6 2.442 2.261 .926 2 3012 8 -20 Kl. G. I'. - 100 - - . - - - 6 .4 2.442 2. 330 .955 2 301 ? 8 -20 81 c;. If.. - 100 _ _ _ _ v 6 9 2.442 2.4.1.3 .988 AF a NSA WALL CONFIGURATION NSU NSL NSLAT c/C t/c NSS OAR NWC C/H CLF CLT CLT/CLF 23012 8-20 81 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 1.0 2.442 2.413 .988 23012 8-20 81 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .8 2.442 2.402 .984 23012 8-20 81 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .45 2.442 2.416 .989 23012 8-20 81 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .26 2.442 2.439 .999 23012 8-20 81 T.S.U.S.L. - 80 15 .072 .33 9 .7 3.4 .19 2.442 2.443 1.000 23012 8-20 81 T.S.U.S.L.t — 80 15 .072 .33 9 .7 3.4 .09 2.442 2.442 1.000 AF - A i r f o i l configuration, a - Angle of attack-degrees. NSA - Number of source and vortex elements on a i r f o i l . T.S.U.S.L. - Transversely s l o t t e d upper and s o l i d lower w a l l s . G.E. - A i r f o i l i n ground e f f e c t . NSU - Number of source elements on upper s o l i d w a l l . NSL - Number of source elements on lower s o l i d w a l l . NSLAT - Number of a i r f o i l - s h a p e d s l a t s . c/C - Slat chord : a i r f o i l chord r a t i o . t/c - Slat thickness : chord r a t i o . NSS - Number of source and vortex elements per s l a t . NWC - Tot a l extent of w a l l i n a i r f o i l chords. 1. A l l e n , H . J . Vincenti,W. G. 45 REFERENCES "W a l l I n t e r f e r e n c e i n a Two-Dimensional Flow Wind Tunnel, with C o n s i d e r a t i o n of the E f f e c t of C o m p r e s s i b i l i t y " . NACA TR 782, 1944 2. Baldwin,B.S. Turner, J . B. Knechtel,E.D. "Wall I n t e r f e r e n c e i n Wind Tunnels with S l o t t e d and Porous Boundaries at Subsonic Speeds". NACA TN 3176, 1954. 3. G l a u e r t , H . "Wind Tunnel I n t e r f e r e n c e on Wings, Bodies and A i r s c r e w s " . RSM 1566, B r i t i s h ARC 1933 4. G o l d s t e i n , S . "Two-Dimensional Wind-Tunnel I n t e r f e r e n c e , P a r t I I . " RSM 1902, B r i t i s h ARC 1942. 5. Kellogg,O.D. "Foundations of P o t e n t i a l Theory". Dover 6. Lim,A.K. " E f f e c t s of Porous Tunnel Walls on High L i f t A i r f o i l T e s t i n g " . T h e s i s , UBC 1970. 7. Parkinson,G.V. Lim,A.K. 8. Smith,A.M.O. Hess, J . L. 9. Tomotika,S. Tamada,K. U memoto,H. 10. Wenzinger,C.J. Delano, J . B. 11. Wood,W.W. 12. Woods,L.C. 46 "On the Ose of S l o t t e d Walls i n Two-Dimensional T e s t i n g of Low-Speed A i r f o i l s " . CASI Trans. 4, Sept. 1971. " C a l c u l a t i o n of P o t e n t i a l Flow about A r b i t r a r y Bodies". P r o g r e s s i n A e r o n a u t i c a l S c i e n c e s Vol.8, 1967. "The L i f t and Moment on a C i r c u l a r - A r c A i r f o i l i n a Stream Bounded by a Plane Wall". QJMAH Vol.4, 1950. "Pr e s s u r e D i s t r i b u t i o n over an NACA 23012 A i r f o i l with a S l o t t e d and a P l a i n F l a p " . NACA TR633, 1938. "Tunnel I n t e r f e r e n c e from S l o t t e d Walls". QJMAM Vol.17, P a r t 2, 1964. "The Theory of Subsonic Plane Flow". Cambridge, 1961. 

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