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Potential flow theory for airfoil spoilers Jandali, Tarek 1970

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A POTENTIAL FLOW THEORY FOR AIRFOIL SPOILERS by TAREK JANDALI B.A.Sc. University of B r i t i s h Columbia, I964 M.A.Sc. University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1970 In presenting t h i s thesis i n p a r t i a l fulfilment of the require-ments for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely a v a i l -able for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his repre-sentatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Tarek Jandali Department of Mechanical Engineering The University of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia, Canada i i ABSTRACT A theory i s presented for the calc u l a t i o n of the pressure d i s t r i b u t i o n and l i f t for arbitrary thick a i r f o i l s f i t t e d with normal upper surface spo i l e r s i n two dimensional incompressible flow. A i r f o i l shape and angle of attack and spoiler l o c a t i o n and height are arbit r a r y and unrestricted. The theory uses a se-quence of conformal transformations from a basic flow past a c i r c l e , with one or two sources on that part of the c i r c l e cor-responding to the surface of the a i r f o i l and spoiler exposed to the wake. The flow inside the separating streamlines i s ignored, and the upper surface pressure downstream of the spoiler is,-, taken as an empirical parameter, assumed constant. The sources i n the wake permit s a t i s f a c t i o n of Kutta conditions with the desired pressure at the spoiler t i p and a i r f o i l t r a i l i n g edge. Features of the theory include good prediction of loading d i s t r i b u t i o n , a f i n i t e wake width and a pressure d i s t r i b u t i o n on the separating streamlines decreasing asymptotically towards the free stream value at i n f i n i t y . The theoretical predictions are compared with l i f t and pressure measurements on a Joukowsky a i r f o i l of 11% thickness and 2.L\% camber, and with l i f t measurements on a ~lLf/0 thick Clark Y a i r f o i l . Both a i r f o i l s were tested through a range of angle of attack with spoilers of 5 and 10% chord height,- each at several locations. Good agreement i s found. i i i TABLE OF CONTENTS Page I INTRODUCTION 1 II THEORY 4 2.1 Theory for Joukowsky A i r f o i l 4 2.1.1 Transformations 4 2.1.2 Mathematical Flow Model 12 2.1.3 Boundary Conditions 16 2.2 Theory for Arbitrary A i r f o i l s 22 2.2.1 Transformations 23 2.2.2 Boundary Conditions 34 2.3 Calculations 39 2.3-1 Additional Boundary Condition 39 2.3.2 Method of Solution 45 III EXPERIMENTS . 51 3.1 Joukowsky A i r f o i l 51 3.2 Clark Y A i r f o i l 54 IV RESULTS AND COMPARISONS 57 4.1 Joukowsky A i r f o i l . . 57 4.2 Clark Y A i r f o i l . 80 V CONCLUSIONS 101 REFERENCES 103 i v LIST OF FIGURES Page l a Complex Transform Planes 5 l b Complex Transform Planes 6 2 Spoiler Geometry. 3 3 Spoiler Geometry . . . . 9 4 S i n g u l a r i t i e s i n Plane 14 5 Clark Y A i r f o i l with Spoiler .24 6 Complex Transform Planes 25 7 Spoiler Geometry 31 ) 8 Spoiler Geometry. . .32 9 Joukowsky A i r f o i l with 10% Spoiler 5 2 10 Pressure D i s t r i b u t i o n for Basic Joukowsky A i r f o i l . .58 11 L i f t Coefficient for Basic Joukowsky A i r f o i l 59 12 Experimental L i f t C oefficient for Joukowsky i A i r f o i l with Spoiler 60 13 L i f t C oefficient for Joukowsky A i r f o i l with ; Spoiler 61 14 L i f t C o e f f i c i e n t for Joukowsky A i r f o i l with Spoiler . . . . . . . . . . . . . . .62 15 L i f t C oefficient for Joukowsky A i r f o i l with Spoiler . . .63 16 Experimental L i f t C o e f f i c i e n t for Joukowsky A i r f o i l v/ith Spoiler 65 17 L i f t C oefficient for Joukowsky A i r f o i l with Spoiler 66 V Page 18 L i f t Coefficient for Joukowsky A i r f o i l with Spoiler ...6? 19 L i f t C o e f f i c i e n t for Joukowsky A i r f o i l with Spoiler 68 20 Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler . .70 21 Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. 7 1 22 Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. . • . . « « . * e « s . . > . * e « e • 7*-23 Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. 7 3 2h Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l 25 Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l t l i S p o i l ©i*« • * » * o « © « o < > © • « « • • © • • • ' 7 ^ 26 Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l w i t h Spoxlox*« • • • • • • e o « « * o * * * o o * » • t^tl, 27 Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. » • » » « « . • * . . • > » 7 8 28 Positions of and Pressure Dis t r i b u t i o n Along Separation Streamlines 7 9 29 Pressure Distribution for Basic Clark Y A i r f o i l . . .81 30 L i f t Coefficient for Basic Clark Y A i r f o i l 82 31 Experimental L i f t C oefficient for Clark Y A i r f o i l with Spoiler. 83 vl Page 32 L i f t C oefficient for Clark Y A i r f o i l with Spoiler . .84 33 L i f t C oefficient for Clark Y A i r f o i l with Spoiler .85 34 L i f t Coefficient for Clark Y A i r f o i l with Spoiler 86 35 Experimental L i f t C oefficient for Clark Y A i r f o i l with Spoiler 89 36 L i f t Coefficient for Clark Y A i r f o i l with Spoiler 9P 37 L i f t Coefficient for Clark Y A i r f o i l with Spoiler 91 33 L i f t Coefficient for Clark Y A i r f o i l with Spoiler 9 2 39 Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler 9 5 40 Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler 94 41 Pressure Distribution for Clark Y A i r f o i l with Spoiler 95 42 Pressure Dis t r i b u t i o n for Clark Y A i r f o i l with Spoiler 96 43 Pressure Distribution for Clark Y A i r f o i l with Spoiler. ..97 44 Pressure Dis t r i b u t i o n for Clark Y A i r f o i l with Spoiler 98 Page Pressure Dis t r i b u t i o n for Clark Y A i r f o i l with Spoiler 99 Pressure Distribution for Clark Y A i r f o i l with Spoiler . . .100 v i i i LIST OF TABLES Page I Zero L i f t Angle Comparisons . „ 44 II Pressure Tap Positions on Joukowsky A i r f o i l 53 i x ACKNOWLEDGMENT The author wishes to express his sincere gratitude to Professor G. V. Parkinson for his encouragement and valuable dir e c t i o n throughout t h i s study. It has indeed been a p r i v i l e g e to work with him. Mr. R.Hirschfield and Mr. M. A. Lundberg, summer re-search assistants, designed the wind tunnel models and assisted with the wind tunnel measurements. Their help i s g r a t e f u l l y . acknowledged. The author would also l i k e to thank the Computing Center of The University of B r i t i s h Columbia for the use of their ,-. f a c i l i t i e s . y The research for t h i s thesis was supported i n part by the Defence Research Board of Canada, Grant number 9551-13.• X LIST OF SYMBOLS = Fourier c o e f f i c i e n t s defined i n equation (20) = Fourier c o e f f i c i e n t s defined i n equation (20) C = Chord of a i r f o i l C D = Drag c o e f f i c i e n t C|_ = L i f t c o e f f i c i e n t Cp = Pressure c o e f f i c i e n t = Pressure c o e f f i c i e n t at t r a i l i n g edge Cpc = Pressure c o e f f i c i e n t at spoiler t i p Cj»ff= Incremental pressure c o e f f i c i e n t i n the wake E = Spoiler chordwise location = Complex potential H = Wind tunnel width V\ = Spoiler height L = (-1) 2 }p = Pressure too = Free stream pressure QL= Strength of lower source Q u = Strength of upper source % i _ = Nondimensional strength of lower source % u ~ Nondimensional strength of upper source Rt = Radius of c i r c l e i n the Z 2 plane R 2= Distance between spoiler t i p and center of c i r c l e i n the plane VJ = Free stream velocity i n the Z, plane Wz = Free stream velocity i n the Z^ plane W = Complex velocity X,Y = Coordinates of a i r f o i l i n the Z^ plane #t»Yc= Coordinates of spoiler t i p i n the Z-^  plane j? 0 = Location of center of c i r c l e i n the Z^ plane 2, = Complex variable defining the plane 2 a = Complex variable defining the Z^ plane 2 3= Complex variable defining the Z^ plane Z?4= Complex variable defining the Z^ plane 2 L = Location of lower source i n the Z^ plane 2y= Location of upper source i n the plane <X = Angle of attack o(0 = Zero l i f t angle for basic a i r f o i l <?(go= Zero l i f t angle for a i r f o i l - s p o i l e r combination P = Ci r c u l a t i o n strength 7f = Nondimensional c i r c u l a t i o n strength SL = Angular position of lower source i n the Z^ plane = Angular position of upper source i n the Z^ plane = Defined i n equation (21) ~$ = Complex variable defining the true c i r c l e plane (Fig. 6) *f = Complex variable defining the off c i r c l e plane (Fig. 6) e = Angular variable i n Z^ plane Angle defining spoiler location i n the Z^ plane 6 A = Angle defining the t r a i l i n g edge i n the Z^ plane: ©c = Angle defining the spoiler t i p i n the Z, plane ] J }*>• = Angular variable i n the "5 plane x i i I = R 2 / R l = Free stream density 4^  = Angle defined i n F i g . l a . *f = Angular variable i n the " 5 plane I ' V = Function related to the polar radius i n the "S plane ^ = Constant related to the polar radius i n the plane 1 I INTRODUCTION The current development of V/STOL a i r c r a f t has caused a re-newed interest i n the investigation of the aerodynamic character-i s t i c s of upper surface spoilers, or spanwise fences. These devices are used on wings for r o l l control, i f deflected asymmet-r i c a l l y , or for high drag generation i f deflected symmetrically. Knowledge of their sectional c h a r a c t e r i s t i c s i s fundamental,.-to an understanding of their performance with f i n i t e span on wings. Two dimensional tests of a i r f o i l s with spo i l e r s can be carried out i n wind tunnels, but, as i n a l l wing aerodynamics, a usable theoretical model i s most desirable. Since the upper surface flow downstream of a spoiler i s separated, and since there are presently no theories available to correctly predict base pressure i n such separated flows, c l e a r l y a theoretical model w i l l require at least one empirical parameter. Also, although the transient performance of a i r f o i l s with spoilers after spoiler actuation, and the performance of s p o i l e r s on.air-f o i l s with slotted flaps, are of great interest, i t i s necessary to consider a simpler problem f i r s t , the steady two dimensional flow past a s o l i d a i r f o i l with a fixed s p o i l e r . The most successful of existing theoretical solutions to t h i s problem i s by Woods ( 1 , 2 ) , who uses a l i n e a r perturbation free streamline potential theory to predict the incremental pressure d i s t r i b u t i o n and the l i f t , drag, and incremental pitching moment .2 on an a i r f o i l . w i t h . s p o i l e r in subsonic flow as a function of a i r -f o i l incidence and spoiler height, angle to a i r f o i l surface, chord-wise position, and base pressure. As i s usual with l i n e a r per-turbation theories, i t i s r e s t r i c t e d to thin a i r f o i l s at low i n -cidence with small spoilers. Woods recognized that the a i r f o i l boundary layer-would reduce the effe c t i v e height of a spoiler, and Barnes (3) used the re s u l t s of wind tunnel experiments to devise an empirical modification to Woods' theory for incompres-si b l e flow i n which the effe c t i v e spoiler height i s determined by the boundary layer displacement thickness on the basic a i r f o i l at the spoiler location. Barnes also proposed an empirical equation for predicting spoiler base pressure from the a i r f o i l - s p o i l e r geometry, and demonstrated good agreement with wind tunnel measure-ments on two a i r f o i l s of the predicted l i f t and pitching moment by the modified Woods theory. Barnes' paper gives a useful . l i s t of references to other experimental and theoretical work on spoilers. An additional special but relevant problem of an a i r -f o i l with a ; s p l i t flap and suction has been treated by Mandl Although the present theory also uses conformal mapping of the two dimensional i r r o t a t i o n a l a i r f o i l flow f i e l d , i t i s quite d i f f e r e n t i n approach from that of Woods, and i s not a perturba-tion theory, so that i t offers the advantage that a i r f o i l i n c i -dence, thickness, and camber as well as spoiler height and loc a -tion are unrestricted. However, only normal spoilers on a i r f o i l s are considered. The flow i s two dimensional and incompressible for which Laplace's equation i s applicable. The effect of the a i r f o i l boundary layer i s not considered, although an empirical 3 m o d i f i c a t i o n l i k e t h a t o f Barnes c o u l d be made. I n the f o l l o w i n g s e c t i o n s , the t h e o r y i s d e v e l o p e d , and a p p l i e d t o a Joukowsky a i r f o i l o f 1 1 % t h i c k n e s s and 2..k% camber and t o a lk% t h i c k C l a r k Y a i r f o i l . II THEORY 2.1 Theory for Joukowsky A i r f o i l s In the f i r s t part of the analysis, a thick cambered Joukow-sky a i r f o i l i s used to develop the theory, since the a i r f o i l i s mapped by a simple conformal transformation from a c i r c l e . The fact that Joukowsky a i r f o i l s have a cusped t r a i l i n g edge i s also desirable for the theory, since t h i s permits smooth separation, with a specified velocity, from the t r a i l i n g edge. In the analy-s i s a i r f o i l thickness, camber and incidence as well as spoiler height and position are arbitrary and unrestricted. However, only normal spoilers are considered. 2.1.1 Transformations A thick, cambered Joukowsky a i r f o i l i n the Z^ plane (Fig. la) i s mapped from a c i r c l e of radius R^  and center at ZQ i n the Z 2 plane by the well known Joukowsky transformation: \ I £. = 2, + . . . ( l ) The value of the complex quantity ZQ i s determined from the thickness and the camber of the given a i r f o i l . The magnitude of the radius R^ can be computed from the knowledge of Z Q and geo-metrical considerations. Figure l a . Complex Transform Planes. Figure l b . Complex Transform Planes. 7 The spoiler on the a i r f o i l i s introduced i n the Z^ plane as a r a d i a l straight l i n e segment, which when mapped onto the Z^ plane becomes a normal spoiler with very s l i g h t , but not inappro-priate curvature for p r a c t i c a l spoiler heights. The chordwise location of the spoiler i n the Z^ plane E i s determined by the angular variable 8 0 i n the 7^ plane. This re-l a t i o n i s shown i n F i g . 2. Also the height of the s p o i l e r i n ,the plane i s related to the length of the straight segment i n the Z^ plane as well as the spoiler position E; t h i s i s i l l u s t r a t e d i n F i g . 3 showing the height of the spoiler as a function of J^2L and E. Next, the c i r c l e with spoiler i s mapped onto a s l i t along the r e a l axis i n the Z^ plane (Fig. l b ) : t © 0 g 2 - g 0 s te . . . ( 2 ) *~ 3 - T—— •+• This transformation i s made up of a clockwise rotation through 0O, a translation to s h i f t the center of the c i r c l e to the o r i g i n , a scaling to reduce the radius to unity and f i n a l l y a Joukowsky transformation which maps the complete contour onto the s l i t i n the Z^ plane. Then the s l i t i s mapped onto a c i r c l e of unit radius i n the Z. plane where the uniform flow i s p a r a l l e l to the r e a l axis: 10 •3 -c R where 5 = - ~ . This transformation i s made up of a transla-te tion, scaling, Joukowsky, and rotation. In the Z^ plane the spoiler becomes part of the c i r c l e and the flow separation points A, C are fixed on the perimeter. The location of these separation points i n the plane i s achieved as follows: In the Z 2 plane . C = Z o + RZ e and A > a + I In the Z^ plane -s a n d A = 2 cos <t>6 F i n a l l y , i n the Z^ plane 6 r - G0 -cx ...(4) - i and L / I V T + \ \ 0 A , 8 c - c o s ^ ^ ' ^ ~ f ~ 1 . . . ( 5 ) 11 Here Qc and 0 A are the angular positions of the separation points as defined i n F i g . 4. They can be calculated for a given a i r f o i l -s p oiler combination. The combined transformation derivative v*c"> can be evalu-ated r e a d i l y : d Z 4 r ^ ^ t(e<-eo) 2 U "5 Ti4 ...(6) It can be observed that i?±l has three simple zeros and two simple poles i n the region corresponding to the flow f i e l d and the a i r f o i l boundary: ( t r a i l i n g edge) 1 - a zero at iL^ — +1 2 - a zero at H4 = €• (spoiler tip) 3 - a pole at £ 2 0 + i ? , £ (spoiler base) (on a i r f o i l surface) 4 - a pole at £ 2 _ Z c _ R t e 5 - a zero at - - C Hence points A and C are c r i t i c a l points of the combined trans-formation from Z-. to Z. with simple zeros of at both 1 4 c*24 12 points. Accordingly, angles are doubled at points A and C, and stagnation streamlines at A and C i n the Z^ plane would become tangential separation streamlines at A and C i n the Z-^  plane. Similarly, the pole at the spoiler base w i l l r e s u l t i n the predic-tion of a stagnation point at D i n the Z^ plane. The remaining simple zero and simple pole on the a i r f o i l surface coincide on the boundary, and cancel. 2.1 .2 Mathematical Flow Model The actual flov; about the a i r f o i l separates from the spoi l e r t i p C and the t r a i l i n g edge A, and the r e s u l t i n g wake i s found, experimentally, to be at nearly constant pressure over the back face of the spoiler and the upper surface of the a i r f o i l behind the spoiler. Since there are presently no theories available to correctly predict base pressure i n such separated flows, c l e a r l y a theoretical model w i l l require s p e c i f i c a t i o n of the base pres-sure c o e f f i c i e n t . 1 This si t u a t i o n suggests a free streamline model for the flow outside the wake of the type used by Woods ( 1 ) or by Roshko ( 5 ) . However, the geometrical d i f f i c u l t i e s of the present problem , appear to rule out t h i s approach, and instead the flow exterior to the a i r f o i l and i t s wake boundaries (the streamlines separar ti n g from the spoiler t i p and the t r a i l i n g edge) i s modelled, by adding to the basic flow past.the transform c i r c l e i n the Z^ plane, suitable s i n g u l a r i t i e s inside the region corresponding to the wake to represent i t s effect on the outer flow. There are advantages to an open wake representation, since 13 the bounding streamlines w i l l probably give reasonable approxima-tions to the t r a j e c t o r i e s of vortices formed from the actual bounding shear layers, and these t r a j e c t o r i e s are of int e r e s t be-cause of the effect of wake vortices from wing spoilers on down-stream aerodynamic surfaces. Accordingly, source s i n g u l a r i t i e s were chosen, and i t was found that they had to be located on the body surface, i n the wake region, to s a t i s f y the separation.pres-sure boundary conditions. The flow model i n the Z^ plane (Fig. 4) consists of uniform flow p a r a l l e l to the r e a l axis past a c i r c u l a r cylinder of unit radius with separation points at A (the t r a i l i n g edge) and C (the spoiler t i p ) . Separation i s achieved by adding either one or two sources on the surface of the cylinder betv/een points A and C . Both p o s s i b i l i t i e s have been considered, and they w i l l be referred to as the 1-source and the 2-source models respectively. C i r c u l a -tion about the cylinder i s introduced, for l i f t control, by-adding a vortex at the center of the c i r c l e . When a source i s placed on the surface, an image source must be added on the surface and a sink at the center of the c i r c l e i n order to s a t i s f y the usual, boundary condition on the cylinder. This i s equivalent to placing a double source at the surface and a sink at the center as a l i m i t i n g case for a source outside with the image and the sink inside. For the 1-source model, the strength of the lower source i n the 2-source model i s set equal to zero, consequently the depen-dence of the flow equations on the lower source and i t s position vanishes. The complex potential for the 2-source model i s : Figure 4. Singularities i n Z. - plane. 15 2 4 ZTT TT + ^ - U ( 2 4 - 2 L ) TT i^lBiL In (2 4) ZTT . . . ( 7 ) and the complex ve l o c i t y : 2LTT 2 4 TT 2 4 - 2 o TT 2 4-ZL 2 T 24 ...(8) where i s the velocity of the uniform incident flow in the plane, related to the corresponding ve l o c i t y i n the physical plane as follows: V2 - — I - 5 % 1 2 1 u Variables T , Q0 , Q L , » ZL a r e the c i r c u l a t i o n , source strengths and the source positions respectively. On the surface of the cylinder: 2 4 = e 16 therefore the dimensionless complex vel o c i t y on the cylinder sur-face i s : V i " * i - L ( 9 ) r where zirvfe The t o t a l number of unknowns for the 2-source model i s f i v e 2.1.3 Boundary Conditions The flow given by equation ( 7 ) s a t i s f i e s the boundary condi-tion that the flow i s uniform at i n f i n i t y and the a i r f o i l and spoiler boundary AEDC i s a streamline. It remains to s a t i s f y the condition of separation at points A and C i n the Z^ plane, cor-responding to stagnation points at A and C i n the Z^ plane, cand to specify the value of the pressure c o e f f i c i e n t at these points i n the Z^ plane. < The condition of separation i n the Z. plane i s s a t i s f i e d i f : ( t » » » £u > )» a n d f o r t n e 1-source model i s three ( i > 3u » So )• W ( Z O = o for 0 - 6 C and 6- 6^ . This leads to 17 % Cdt(^-h) t ^ L C o t ( O c - g u ) _4 Sin ©c - 2 = O . . . (10) and % Co\(°±zh) + ^ L Co\(Shzh) - 4*i" - - O .. .(ID The pressure c o e f f i c i e n t at the two separation points i n the physical plane i s specified through the use of Bernoulli's equa-tion which applies to the flow outside the wake, including the separation streamlines, so that or p l e u 2 - u 2 -The values of C p and C p are made equal to the measured pressure c o e f f i c i e n t i n the wake, thus specifying the values of | I u U and by equation (12). This leads to two new boundary C U conditions imposed on the flow f i e l d . In general the complex velocity i n the Z^ plane i s related to the complex velocity i n the Z^ plane through the transforma-tion derivative: 18 W ( 2 t ) = - W C * 4 ) This expression i s an indeterminate form at the spo i l e r t i p and the t r a i l i n g edge, since at these points, both W(Z, ) and are zero. At the spoiler t i p C the c r i t i c a l term i n the expression for J^ lLl- i n equation (6) i s given by , d24 1 (24 ) • Using equation (4) and the fact that on the cylinder i n the Z, plane Z^ = € , the c r i t i c a l term becomes Since i t i s the magnitude of the velocity at separation that re-la t e s to the pressure c o e f f i c i e n t , the variation of ±n ^hi neighbourhood of C can be written 4ii ex. -z s'w ( e - 6 c ) Thus the velocity near the spoiler t i p i n the physical plane i s determined by the proportionality 1 9 u ©c which i s an indeterminate form at 9 = 6 C # Using L'Hospital's rule and l e t t i n g ©—*~©Q The constant of proportionality i s made of the remainder of the transformation derivative evaluated at point C and the r a t i o , u giving the magnitude of the v e l o c i t y of separation from the J spoiler t i p i n the physical plane as u lc L * t l t6c I 4 2. ^ ^ . . . ( 1 3 ) Similarly, at the t r a i l i n g edge A the c r i t i c a l term i n the 20 e x p r e s s i o n f o r i^i. i n e q u a t i o n (6) i s g i v e n bv (Z~ - 1). I n o r d e r to e v a l u a t e the i n d e t e r m i n a t e form, Z2 must be r e l a t e d t o the a n g u l a r v a r i a b l e © i n the Z^ p l a n e . From e q u a t i o n (2) 2 , = 1 £ 3 - t 4-2, where Z~ i s g i v e n by e q u a t i o n (3) f o r p o i n t s on the s l i t as 'ollov/£ z 3 = } u ^ . | coi>(e-ec) + a t -I w r i t t e n as d E 4 ..(14) r phus the v a r i a t i o n of i n the., neighbourhood of A can be ill oc Hi e 4H4 2 16c 2 3 - *- J 4 - 2* + 2o - I where i s d e f i n e d by e q u a t i o n (12+). The v e l o c i t y near the t r a i l i n g edge i n the p h y s i c a l p l a n e i s determined by the propor-t i o n a l i t y WUi) VJ 21 U s i n g ; L ' H o s p i t a l ' s r u l e to e v a l u a t e the i n d e t e r m i n a n c y a t 9 — 6 ^ N 2. • 4 2L ' A g a i n the c o n s t a n t of p r o p o r t i o n a l i t y i s the remainder of the t r a n s f o r m a t i o n d e r i v a t i v e e v a l u a t e d a t p o i n t A and the r a t i o ^ 2 . U Hence, the magnitude of the v e l o c i t y of s e p a r a t i o n from the t r a i l i n g edge i n the p h y s i c a l p l a n e i s g i v e n by: A 2 COS ©A + E q u a t i o n s ( 1 0 ) , ( 1 1 ) , ( 1 3 ) and ( 1 5 ) s a t i s f y the requirements t h a t smooth s e p a r a t i o n o c c u r s a t the s p o i l e r t i p and a t the t r a i ] i n g edge w i t h the p r e s s u r e s p e c i f i e d a t both p o i n t s to be equal t o . t h a t measured e x p e r i m e n t a l l y . These e q u a t i o n s a r e now used tc s o l v e f o r the five'unknowns , ^ , S C > S U , X , so t h a t ano-t h e r c o n d i t i o n i s , needed to s o l v e f o r a l l the unknowns i n the 2-source model. Such a c o n d i t i o n i s i n t r o d u c e d i n s e c t i o n 2 . 3 , . 1 . The 1-source model removes the a m b i g u i t y by d r o p p i n g -what 22 appears to be the least s i g n i f i c a n t boundary condition, the speci-f i c a t i o n of the pressure c o e f f i c i e n t at the t r a i l i n g edge. It i s found that the a i r f o i l pressure distribution,,even quite close to the t r a i l i n g edge on the underside, i s not strongly dependent on the value of C p , which i s therefore l e f t unspecified. In the previous equations + t i s taken to be zero, so %^ and Su are eliminated and equations (10), (11) and (13) are used to solve for 51 , % 0 , y . Equation (15) merely gives the value of Cp , A which i s found to be more positive than the empirical value assumed to apply over surface ABC. Thus, there i s a pressure d i s -continuity at A predicted by the 1-source model. 2.2 .Theory for Arbitrary A i r f o i l s A l o g i c a l extension to the theory would be i t s application to arbitrary thick a i r f o i l s , i f i t i s to have some p r a c t i c a l value for design purposes. This i s achieved by the use of Theodorsen's transformation (6 ), which w i l l map any a i r f o i l onto a c i r c l e . The problem, then, becomes similar to that of a Joukowsky a i r f o i l as viewed i n the Z-^ plane (Fig. l a ) . In general, p r a c t i c a l ; a i r -f o i l s do not have a cusped t r a i l i n g edge. However, since the • theory does require a cusp at the t r a i l i n g edge to s a t i s f y the condition of smooth separation, the t r a i l i n g edge must be a r t i -f i c a l l y modified into a cusp. This modification i s applied to the upper surface of the a i r f o i l a f t of the spoiler. Thus, the .al-tered portion of the a i r f o i l i s completely within the wake, caused by the spoiler, and has no effect on the outer flow f i e l d . It i s suggested that a t h i r d order polynomial be used to replace the 23 upper surface of the a i r f o i l to span the l a s t 10% of the chord. This w i l l permit the location of sp o i l e r s upto the 90% chord station, as measured from the leading edge. Spoilers positioned aft of t h i s are of l i t t l e p r a c t i c a l i n t e r e s t . The four c o e f f i c i -ents of the polynomial are determined by specifying both ordinate and slope of the two end points, to match those of the a i r f o i l lower surface at the t r a i l i n g edge and the a i r f o i l upper surface at the 90% chord station. ;> As a s p e c i f i c application, i t was decided to consider a 14% thick Clark Y a i r f o i l , since force measurements on such an a i r -f o i l f i t t e d with normal spoilers were available for comparison,. The a i r f o i l surface i s defined by a f i n i t e number of points tabu-lated i n Riegels ( 7 ). Both the basic and the modified a i r f o i l s are shown i n Fi g . 5« •- . 2.2.1 Transformations A 14% thick modified Clark Y a i r f o i l i s generated i n the plane so that the chord i s aligned with the r e a l axis. The t r a i l i n g edge i s located at X = +2, and the midpoint between the leading edge and i t s center of curvature at X = - 2 , so as to re-semble the orientation of a Joukowsky a i r f o i l . The Joukowsky :, i transformation w i l l map the a i r f o i l i n the Z-^  plane onto the ~$ plane i n F i g . 6. -: \ 1 s l .(16) C L A R K Y A I R F O I L 14% T H I C K ( M O D I F I E D T R A I L I N G E D G E ) Figure 5. Clark Y A i r f o i l with Spoiler. 26 » The r e s u l t i n g curve i n the plane w i l l be nearly c i r c u l a r i n shape, since most wing sections have a general resemblance to each other and to the Joukowsky a i r f o i l . The coordinates of the contour i n the plane are defined by the r e l a t i o n : S = e and the corresponding points i n the Z^ plane are found by using equation (16) : 2 . - e +e Since € =r C O S M, + t S l V N , Syw. + t S i n ^ A . then i t can be shown that: 2 , - 2 CosUy^) Co^u + 2 i smV^ f/O Sivy* In the plane, the coordinates X, Y of the a i r f o i l surface are known. Hence, i t i s possible to obtain expressions for and i n terms of X and Y as follows: 27 Y 2 Sm r 2 CoS ...(17) Eliminating 'Yv'(y )^ from the above r e l a t i o n s gives: . . . ( 1 8 ) Because Siry/. i s known i n terms of the a i r f o i l coordinates X and Y, the value of f(f-) can be found from equation (17). Next, the contour i n the plane i s mapped onto a c i r c l e of radius g i n the ^ plane (Fig. 6). The transformation re-l a t i n g the to the plane i s the general transformation: ...(19) where the values of the r e a l c o e f f i c i e n t s An and Bn are to be de-termined from the a i r f o i l shape. The coordinates of points on the c i r c l e i n the plane are defined by: 28 R e l a t i n g the two complex v a r i a b l e s ^ and *J f o r p o i n t s on the c o n t o u r s w i l l g i v e the i d e n t i t y : When t h i s i s compared w i t h e q u a t i o n (19), i t becomes o b v i o u s t h a t : oo E x p r e s s i n g i n p o l a r form on the c i r c l e : "J = € ( Cos <f + t Sm ^ ) w i l l y i e l d : (V(f)'V>)+L(r-*) ~ € CAn + *tBn)(co5nf - L S W 29 Equating the r e a l and imaginary parts gives the two Fourier ex-pansions: ^(/A)-Vo = 2. (A wCoSi<f + Bw Sm**) € ...(20) .and 0 0 -wYo £(«f) = ^ _ / U = 2 (A^<>ivMrtu? _ CoS *^>) € ...(21) Here, ' / o i s related to the radius of the c i r c l e i n the ^ plane. Consequently, i n order for the deviation of the near c i r c l e from the true c i r c l e to be a minimum, the value of "V*© i s taken to be Yo = -L j ffr) 4y --(22) o In order to obtain the values of the c o e f f i c i e n t s An and Bn, equations (19)1 (20) and (21) are approximated by a f i n i t e number of terms. Since equation (20) requires an expression of ^ i n terms of if , and since /Y/ i s o r d i n a r i l y known as a function of yu. from the a i r f o i l coordinates, t h i s leads to an i t e r a t i y e process, which converges rapidly to give the values of / 1 f (*tf ) and £(«•?). A f i r s t approximation i s to set € (*f ) = 0 and to solve for "Vo from equation (22), then to evaluate an i n i t i a l set of 30 c o e f f i c i e n t s An and Bn using equation (20), and f i n a l l y to use these c o e f f i c i e n t s i n equation (21) to obtain the second approxi-mation to € ( <C ). This process i s repeated u n t i l no change i n the outcome of € (^C) i s observed. When th i s occurs, the f i n a l values of the c o e f f i c i e n t s An and Bn are known, and can be used i n equation (19) to define the mapping function r e l a t i n g the and ^ planes. Nov/ that mapping an arbitrary a i r f o i l i n the Z^ plane onto a c i r c l e i n the plane has been achieved, the problem becomes similar to that of the Joukowsky a i r f o i l as viewed i n the plane. As before, the spoiler on the a i r f o i l i s introduced i n the *J plane as a r a d i a l straight l i n e segment, which when mapped onto the physical plane becomes a normal spo i l e r with very s l i g h t curvature. The chordwise location of the spoiler on the a i r f o i l (E) i s determined by the angular variable 9o i n the plane. Here, 6o i s the angle between the positive r e a l axis and the r a d i a l segment representing the sp o i l e r . The dependence of E on Go i s shown i n Fig. 7. Also, the height of the spoiler i s shown to be a func-tion of "f and E i n Fi g . 8. ; The transformation r e l a t i n g the and planes becomes: i6o • • • ( 2 3 ) 33 Here, the value of i s taken to be equal to the radius of the c i r c l e i n the plane, The mapping of the s l i t i n the Z^ plane onto the c i r c l e i n the. Z^ plane i s unaltered and i s given by equation (3). Similarly, the location of the separation points A and C i s given by equa-tions (4) and (5). Care must be taken when using equation (4) for the Clark Y a i r f o i l , since o< i s the angle of attack meas-ured from the chordline and not from the lower surface as i s conventional for t h i s class of a i r f o i l s . The combined transformation derivative can be' written as follows: d£*. d24 ~ a? d 1 J dz 3 This can be evaluated r e a d i l y : oo dZ« -s'-i - Z. n ...(24) 3k This expression for ^1 has the same number of simple zeros and simple poles as did the expression i n equation ( 6 ) . Conse-quently, the spoiler t i p and the t r a i l i n g edge are c r i t i c a l points of the mapping and permit the s a t i s f a c t i o n of smooth separa-tion. If the a i r f o i l t r a i l i n g edge had been l e f t unaltered, j - , , ^!. would s t i l l have had a zero at point A, but i t would cl 2.4 not have been a simple zero. The order of the zero would be l e s s than unity, so that the vel o c i t y term would dominate, and a stag-nation point would appear at the t r a i l i n g edge. (When the basic Clark Y a i r f o i l i s mapped onto the plane, the r e s u l t i n g con-tour i s indicated by the dotted l i n e i n Fi g . 6.) r 2.2 .2 Boundary Conditions The flow model for the Clark Y a i r f o i l i s the same as that shown i n Fig. 2+ for the Joukowsky a i r f o i l . Consequently, expres-sions for the complex potential F(Z^), and the complex v e l o c i t y W(Z^) are given by equations (7) and (8) respectively. However, i t w i l l be convenient to rewrite the expression for the complex vel o c i t y as follows: w(&0 _ i_ ^4 H 2 4 - Z o J ^ [ZA-ZL . . . ( 2 5 ) 35 Again, V"2 i s the incident flow i n the Z^ plane and i s related to the uniform flow i n the Zn plane by: The boundary conditions for the a r b i t r a r y a i r f o i l are the same as those for the Joukowsky a i r f o i l , i n that smooth separation must occur at the spoiler t i p and the t r a i l i n g edge with a speci-fi e d value of the pressure c o e f f i c i e n t at both points. The condi-tion of separation i n the Z^ plane at points A and C i s s a t i s f i e d by setting W(Z^) equal to zero at the corresponding points. -, This r e s u l t s i n the two expressions given by equations (10) and (11 ) . In order for the pressure c o e f f i c i e n t to be specified at the separation points, i t i s necessary to obtain expressions for W(Z^) at these points. These were shown to be of an indeterminate form; consequently, i t i s appropriate to consider them separately. At the spoiler t i p C the c r i t i c a l term i n the expression for i n equation (21+) i s given by: c(eo-'x) ) In section 2 .1 .3 i t was shown that the variation of d S 4 i n the neighbourhood of C can be written as: c ^ 4 V * ' 3 6 Using the expression for the complex velocity given by equation ( 9 ) , the v e l o c i t y i n the physical plane i s determined by the pro-p o r t i o n a l i t y : U Z s m ( ^ ) Evaluating t h i s equation at Q = Qc using L 1Hospital's rule gives: Introducing the constant of proportionality gives the magnitude of the velocity of separation from the spoiler t i p i n the physical plane as: X . ( 2 6 ) Similarly, at the t r a i l i n g edge A the c r i t i c a l term i n the expression for d £ 4 i n equation (2.1+) i s given by: so that the variation of written as: d2* i n the neighbourhood of A i s 034 (V-0 Using the expression for the complex v e l o c i t y W(Z^) given by equation (25) , the velocity i n the plane near the t r a i l i n g edge i s determined by the proportionality: u which i s an indeterminate form at "S = + l . Using L 1 Hospital's rule by d i f f e r e n t i a t i n g the numerator and the denominator with respect to the variable , and l e t t i n g w i l l give: wfe»)l U 1 . _L _ 2 E-4 2 u -1 r a* L \ A 38 I n the Z^ p l a n e t h e t r a i l i n g edge c o r r e s p o n d s t o : 2 4 ^ € Hence, the v e l o c i t y i n the p h y s i c a l p l a n e i s g i v e n by: \ U U * 4 ^ 2 . ' T 4 v / I n t r o d u c i n g the c o n s t a n t of p r o p o r t i o n a l i t y g i v e s the magnitude of the v e l o c i t y o f s e p a r a t i o n from the t r a i l i n g edge i n the p h y s i c a l p l a n e a s : ' A Y ( S > ) ' r i i 4 U ^ j .(27) 39 Equations (26) and (27) when used together with equation (12) specify the value of the separation pressures at the s p o i l e r t i p and the t r a i l i n g edge. The magnitude of t h i s pressure i s taken equal to that measured experimentally. Thus, equations (10), (11), (26) and (27) are used to solve for the unknowns ^ y , ^ L , Su , St » 3" • Again the need for another condition i s evident for a unique solution to the 2-source model. This w i l l be discussed i n section 2.3.1. As was the case with the Joukowsky a i r f o i l , the 1-source model has only three unknowns to completely specify the flow f i e l d , and equations (10), (11), and (26) are used to determine these unknowns, namely , $u , )f . Also, the pressure discontinuity at A w i l l be present when using the 1-source model for the Clark Y a i r f o i l . 2.3 , Calculations Since the four equations r e s u l t i n g from the previously^men-tioned boundary conditions are not s u f f i c i e n t to determine the f i v e unknown flow parameters defining the 2-source model, i t i s necessary to impose an additional condition on the flow for t h i s model. 2.3.1 Additional Boundary Condition During the early stages of t h i s work, the lower source posi-tion i n the 2-source model was l e f t unspecified to serve as a free parameter and to permit the determination of the remaining unknowns. It was observed that the l i f t , for any a i r f o i l - s p o i l e r . 40 combination, predicted by the 2-source model was consistently lower than that predicted by the 1-source. It was also found that as the position of the lower source approached point A, corres-ponding to the t r a i l i n g edge i n the Z^ plane, the strength of the source decreased monotonically and became zero at point.A. Thus, the effect of the lower source on the flow f i e l d became vanishingly small as i t s position approached point A. In p a r t i -cular, when &L = 0.95 ©A> the pressure d i s t r i b u t i o n given by the 2-source model was indistinguishable from that predicted by ithe 1-source model except near the t r a i l i n g edge on the underside of the a i r f o i l where the 1-source model w i l l always res u l t i n a pressure discontinuity at the t r a i l i n g edge. >• When comparing the l i f t r e s u l t s from the 1-source model with those obtained experimentally, i t was found that the theore t i c a l values were almost always larger than the measured ones. This suggests a c r i t e r i o n for choosing a value for the lower source position for the 2-source model i n order to provide better agree-ment between theory and experiment. However, to match the value of the l i f t c o e f f i c i e n t for the 2-source model with experiment for every angle of attack, spoiler position and height, would make the theory too empirical and useless for the prediction of the loading d i s t r i b u t i o n before any experiment i s performed. .,• A study of the experimental l i f t as a function of angle of attack for the two a i r f o i l s tested with s p o i l e r s shows that.the zero l i f t angle, for a given a i r f o i l with a normal spoiler, de-pends very l i t t l e on the spoiler position when i t i s varied be-tween the 50% and 90% chord locations. This, together with-.the 41 Ac fact that the 1-source model gave predictions for values of ~ — i n good agreement with experiment, w i l l serve as a c r i t e r i o n for choosing the value of Su i * 1 the 2-source model. Thus, by using the value of predicted by the 1-source model and by specifying the value of the zero l i f t angle for a given a i r -f o i l - s p o i l e r combination, the l i n e a r r e l a t i o n between l i f t and incidence i s known for the 2-source model. Then, the value of Su i s chosen to give the appropriate l i f t s pecified by t h i s r e l a -tion. The two theoretical models d i f f e r only by the presence of an additional source which has a secondary e f f e c t on the flow f i e l d . Thus, to say that both models should have the same value of i s not at a l l unreasonable, p a r t i c u l a r l y when the 1-source model provides good agreement with experiment for values ~3tcT In general when using conventional potential flow theory to of solve for the flow over a given a i r f o i , i t i s observed that thel i f t i s always overestimated by the theory for values of i n c i -dence i n the normal operating range for the given wing section. This i s due to the v o r t i c i t y within the boundary layer. The discrepancy i n l i f t becomes smaller as the zero l i f t angle i s approached, no doubt due to the reduction i n v o r t i c i t y i n the boundary layer, hence the better the agreement i n the ov e r a l l c i r c u l a t i o n about the a i r f o i l . By t h i s observation, i t seems j u s t i f i e d to specify the value of the zero l i f t angle for the 2-source model to match that measured experimentally. Thus, the present theory requires the sp e c i f i c a t i o n of the wake pressure c o e f f i c i e n t for both models, and i n addition, i t requires the knowledge of the zero l i f t angle for the 2-source model. It was observed that varying the pressure c o e f f i c i e n t i n the wake had l i t t l e e f f e c t on the d i s t r i b u t i o n over the rest of the a i r f o i l , but produced a proportional change i n the r e s u l t i n g value of l i f t . I t was also found that the experimentally measured pressure i n the wake, for the two a i r f o i l s tested, was very,nearly the same for equivalent spoiler height and location and a i r f o i l incidence. Thus, i n the absence of experimental values for.the wake pressure for a given a i r f o i l section, i t i s possible to make a reasonable estimate of t h i s value based on data obtained from other a i r f o i l s . i Similarly, i n the absence of experimental values for the zero l i f t angle, needed for the 2-source model, i t i s possible to make a reasonable estimate of Ofyo using a l i n e a r i z e d theory developed by Woods ( 2 ) and l a t e r modified by Barnes ( 3 ). In his report, Barnes gives an expression for the l i f t on any a i r f o i l f i t t e d • with a normal spoiler as follows: To- • • • (28) 43 where Gp^ i s an incremental spoiler base pressure c o e f f i c i e n t in-troduced by Woods, and given by Barnes as an empirical formula applicable to a l l a i r f o i l s as follows: - z .5 i l _ o.v8 c In equation (28) (X0 i s the zero l i f t angle for the a i r f o i l with-out spoiler, and (X c, Y c) are the coordinates of the spoiler t i p i n the Z^ plane. To obtain the zero l i f t angle for the a i r f o i l -spoiler combination, i s made to vanish and 0( becomes (X^^ . Thus the equation reduces to: (U* jiff H ^ t 2 ^ ^ ^ ) T ~ °AS] = - \ \ . . . ( 2 9 ) This i s a l i n e a r equation i n 0ty which can eas i l y be solved for any a i r f o i l - s p o i l e r combination. A l l the incidence variables are measured from the chord l i n e which i s the same as the r e a l axis i n the Z^ plane. The predicted values for C*^ are compared i n Table I with experimental values. It i s clear that t h i s l i n e a r -ized theory gives poor predictions for spoilers mounted at the. ^ 0% chord station, but the agreement improves as the spoiler is,: moved back to the 90% chord location. It i s suggested that the predicted 44 value for for the 70% chord station be used to represent the average of the r e l a t i v e l y constant values measured experimentally. This would give a value for 0ty0 reasonably near the observed quantities, and would serve as an input parameter for the 2r-source model during the i n i t i a l stages of an investigation into the loading c h a r a c t e r i s t i c s of an a i r f o i l - s p o i l e r combination. Joukowsky Clark Y Spoiler Geometry From Measured Chord Line Measured From Lower Surface E/c h/c Theory Expt. Theory ' Expt. 0 . 5 .05 5.94 3 .00 2.88 1.40 0.7 .05 4. 45 2.70 0.89 1.40 0.9 .05 4.29 2.60 0.23 0 .00 0.5 .10 10.16 6.60 7.09 5.20 0.7 .10 8.00 6.10 4.58 4 .70 0.9 .10 7.41 6.10 3.35 3 . 9 0 Table I: Zero L i f t Angle Comparisons 45 2.3-2 Method of Solution Theoretical solutions were obtained for a Joukowsky a i r f o i l of 11% thickness and 2.4% camber both with and without spoiler, and for a 14% thick Clark Y a i r f o i l both with and without spo i l e r . Solutions for the basic Joukowsky a i r f o i l were obtained by the c l a s s i c a l method i n which the Kutta condition i s s a t i s f i e d at the t r a i l i n g edge, determining the c i r c u l a t i o n about the,air-f o i l . The l i f t and pressure d i s t r i b u t i o n were computed for a range of angle of attack. The value of the complex quantity ZQ corresponding to the given camber and thickness i s (-0.09 +30.050). When using the 1-source model to solve for the flow over the Joukowsky a i r f o i l f i t t e d with a normal spoiler, equations (10), (11) and (13) are modified by setting ^-u= 0. It i s obvious that variables ^ u , and Y appear l i n e a r l y i n equations (10) and (11), consequently i t was possible to obtain expressions for %» and x n terms of the remaining unknown So . These expressions are then used i n equation (13) to obtain a lengthy equation u±n So > which i s solved by Newton's i t e r a t i v e method. With the flow parameters ^ 0 , So and known, the a i r f o i l surface' velocity over the contour AEDC i s found from: u ..."(30) where L i s given by equation (6), and the corresponding value of the pressure c o e f f i c i e n t i s found from equation (12). 46 The pressure c o e f f i c i e n t was evaluated for 98 points on the a i r -f o i l surface over contour AEDC, and was taken to be a constant over contour CBA, corresponding to the measured wake pressure. The r e s u l t i n g pressure c o e f f i c i e n t d i s t r i b u t i o n i s integrated numerically using the trapezoidal rule to obtain the value of -; l i f t c o e f f i c i e n t C^. The above procedure was carried out for a range of angle of attack o< for the a i r f o i l f i t t e d with spoilers of 5 and 10% chord height, each at 50, 70 and 90% chordwise locations. Values of for each of the a i r f o i l - s p o i l e r con-figurations, used for solving for the unknowns i n the 2-source: model, are obtained from the r e s u l t i n g values of as a function of (X . Thus the complete solution to the 1-source model i s a pre r requisite to obtaining the values of the flow parameters i n the 2-source model. As a f i r s t step for t h i s solution i t i s necessary to define the l i f t as a function of incidence. This i s achieved by using the value of dc^cltf predicted by the 1-source model for the same configuration, and then specifying the value of 0(QO . The zero l i f t angle i s either obtained experimentally or an-esti-mate of i t s value can be derived from equation (29). Next,; equations (10), (11) and (13) are used to obtain expressions for . > ^L. A N C * 8" i n terms of the remaining two unknowns£ 0 and <Si_ . These are used i n equation (15) to give a single equation i n So and SL. „ To solve for these, a rapidly con-verging process i s established i n which an i n i t i a l choice for £i_ i s made. Then <Su i s obtained by using the secant i t e r a t i v e method. Once an i n i t i a l set of values for the flow parameters i s 47 determined, the a i r f o i l surface v e l o c i t y over the contour AEDC i s found from: u o k , / d ^ . . . ( 3 D where i s given by equation (6), and the corresponding value of the pressure c o e f f i c i e n t i s found from equation (12). The r e s u l t i n g pressure d i s t r i b u t i o n i s integrated giving the l i f t . This value of l i f t i s then compared with the previously specified l i f t curve, and i f i t i s found that the calculated l i f t i s too low, S u i s moved closer to point A i n the Z^ plane, and vice versa. The whole procedure i s repeated u n t i l the resulting, l i f t -agrees with the specified value. This was carried out for .a range of incidence for the a i r f o i l f i t t e d with s p o i l e r s of J and :10% chord height, each at 50, 70 and 90% chordwise locations. In the case of spoilers at the 90% chord station, i t was found that the l i f t curve predicted by the 1-source model was already lower than the experimental curve. In these situations i t i s impossible -to make the 2-source model agree better with measured values, since i t can only be made to produce smaller l i f t values than those pre-dicted by the 1-source. However, i t remains desirable to obtain a solution to the 2-source model, since i t removes the pressure discontinuity at the t r a i l i n g edge. Thus, i t i s suggested that a value for the lower source position be taken as ' S u = 0.95 0^. This v / i l l not a l t e r the l i f t appreciably, nor w i l l i t change the 48 pressure d i s t r i b u t i o n except near the t r a i l i n g edge. To obtain solutions for the basic Clark Y a i r f o i l , i t i s necessary to calculate values for c o e f f i c i e n t s An and Bn defining the mapping from the * S to the planes. This i s achieved by representing the surface of the a i r f o i l by 100 d i s t i n c t points separated appropriately to account for rapid curvature change. These are then used i n equations (1?) and (18) to compute the cor-responding values of yW- and 'VCyM-). Values of c o e f f i c i e n t s An and Bn are then computed from equation (20) by the i t e r a t i v e pro-cess described i n section 2.2.1. These were calculated for-values of n up to 40, since i t was f e l t that t h i s would give s u f f i c i e n t accuracy i n the determination of the mapping function. Thus, the flow about the basic Clark Y a i r f o i l i s determined i n the ~$ plane with the Kutta condition again s a t i s f i e d at the t r a i l i n g edge. This determines the c i r c u l a t i o n about the a i r f o i l and r e s u l t s i n the prediction of a stagnation point at the t r a i l -ing edge due to the fact that i t i s not cusped. The l i f t and pres-sure d i s t r i b u t i o n were computed for a range of incidence. When a spoiler i s f i t t e d on the upper surface of the a i r f o i l i t becomes necessary to modify the upper surface near the t r a i l i n g edge to produce a cusp at A. This i s done by following the" pro-cedure outlined i n section 2.2. Next, a new set of c o e f f i c i e n t s An and Bn must be evaluated, since the modified t r a i l i n g edge w i l l have an effect on the mapping function r e l a t i n g the V and . ^ planes. With the mapping function defined i t becomes possible to solve for the flow over the a i r f o i l - s p o i l e r combination. ; Using the 1-source model, equations (10), (11) and (26) are 49 mo d i f i e d by setting 0. Then, following the same procedure outlined for the Joukowsky a i r f o i l , flow parameters ^ y , . Su and are evaluated and the a i r f o i l surface v e l o c i t y over contour AEDC i s found from: U i i 21 ...(32) where ^ Z | / ( j 2 4 i s given by equation (24). As i n the Joukowsky air-f o i l case, the l i f t s and pressure d i s t r i b u t i o n s were computed for a range of incidence and for spo i l e r s of 5 and 10% chord height, each at 50, 70 and 9°% chordwise positions, again r e s u l t i n g i n predictions for <JC L /c io( for the various configurations. For the 2-source model, the same procedure as o u t l i n e d : f o r the Joukowsky a i r f o i l i s again used to determine the flow para-meters, and the value fordC^/fa predicted by the 1-source model together with s p e c i f i c a t i o n of PQ„ provides the additional con-d i t i o n needed to solve for a l l the unknowns. The ve l o c i t y over the surface of the a i r f o i l i s given by: i u 4 .4»in»-T» + k C ^ ( g £ a ) + W ( g £ ) (33) where given by equation (24). Values of l i f t and pres-sure d i s t r i b u t i o n s were computed for a range of angle of attack 50 and f o r s p o i l e r s of 5 and 10% chord h e i g h t , each at 50, 70 and 90% chordv/ise l o c a t i o n s . 51 III EXPERIMENTS 3•1 Joukowsky A i r f o i l A Joukowsky a i r f o i l of 27 i n . span, 12.08 i n . chord, 11% thickness and 2./+% camber was constructed. Due to the theoreti-c a l cusped t r a i l i n g edge, some construction modifications i n the neighbourhood of the t r a i l i n g edge were needed. Since t h i s a i r -f o i l was to be used with normal spoilers mounted on the upper sur-face i t was possible to modify the upper surface near the t r a i l -ing edge so that the l a s t part of the a i r f o i l had an approximately constant thickness of 1/8 i n . In the presence of spoi l e r s the modified part of the upper surface i s completely within the,.wake and has no effect on the outer flow f i e l d . The o r i g i n a l and > modified p r o f i l e s are shown i n F i g . 9. The a i r f o i l was constructed out of two spanwise sections of wood joined at the center with an aluminum portion containing a t o t a l of 37 pressure taps. Twenty-four of the taps were d i s -tributed on the upper surface and the remainder were on the lower surface. The chordwise locations of these taps are given i n . Table II and the i r positions on the a i r f o i l surface are shown i n Fi g . 9. The a i r f o i l was b u i l t with end plates to allow for the mounting of two spanwise spoilers, one having a height of 5% chord and the second a height of 10% chord. Each of these ; spoilers could be mounted i n 5 d i f f e r e n t chordwise positions: JOUKOWSKY AIRFOIL 11% THICKNESS 2.4%CAMBER Figure 9. Joukowsky A i r f o i l with 10% Spoiler. vn no 53 UPPER SURFACE LOWER SURFACE Chordwlse Chordwise Tap No. Position X Tap No. Position X 1 -2.027 25 -1.960 2 -2.010 26 -1.860 3 -1.960 27 -1.694-4 -1.894 28 -1.527 5 -1.827 29 -1.194 6 -1.740 30 -0.860 7 -1.627 31 -0.427 8 -1.482 32 +0.006 9 -1.280 33 +0.439 10 -1.079 34 +0.873 11 -0.878 • 35 +1.306 12 -0.676 36 +1.639 13 -0.475 37 • +1.839 14 -0.274 15 -0.073 16 +0.129 17 +0.330 18 +0.531 19 +0.733 20 +0.934 21 +1.135 22 +1.337 23 +1.538 24 +1.739 Table I I : Pressure Tap Positions on Joukowsky A i r f o i l 54 50, 60, 70, 80 and 90% chord. The gap between a i r f o i l and spoiler was sealed with tape for each configuration. The a i r f o i l was mounted on a six-component s t r a i n gauge bal-ance system at the l/4-chord position. Measurements for l i f t , drag and pitching moment were recorded for a range of incidence. The.pressure taps were connected to a multitube manometer bank f i l l e d with alcohol. The pressure readings on the a i r f o i l sur-face and on the 10% spoiler were recorded for angles of attack Oi^ , £ * £ 0 + C*^o + 8° for a l l possible s p o i l e r positions. Pressure taps were not b u i l t into the 5% spoiler because of i t s 5 small size. The test Reynolds number was 4.4 x 10 . 3.2 Clark Y A i r f o i l i Measurements for l i f t , drag and pitching moment for a 14% thick Clark Y a i r f o i l were available before the theoretical solu-tions were computed. They were obtained by Mr. M. A. Lundberg, a summer research assistant working under the d i r e c t i o n of Pro-fessor G. V. Parkinson, to investigate the a p p l i c a b i l i t y of:the standard wind tunnel wall corrections to a i r f o i l - s p o i l e r con-figurations. Four a i r f o i l s of 9", 14", 19" and 24" chord were constructed. They were b u i l t with end plates to allow for the mounting of two spanwise spoilers having heights of 5% and 10% chord. The sup-port system for each a i r f o i l was at the mid-chord position. No pressure taps were i n s t a l l e d i n any of these a i r f o i l s . , Again, the six-component s t r a i n gauge balance system was,used to measure l i f t , drag and pitching moment for a l l the a i r f o i l s 55 f i t t e d with a 10% spoiler for a range of incidence and spoiler locations at 50%, 70% and 90% chord. Only the 24 i n . model was used with a 5% spo i l e r . Pressure measurements i n the wake region were obtained by i n s e r t i n g a P i t o t tube behind the spoiler and re-cording the value of pressure at several locations within the 5 wake. The test Reynolds number was 3 x 10 . A l l the measurements for both a i r f o i l s were made i n the low speed wind tunnel of the Mechanical Engineering Department of !The University of B r i t i s h Columbia. This tunnel has a test section of 3 by 2 1/4 f t . , over a length of 8 2/3 f t . , and produces a very uniform flow, with turbulence l e v e l l e s s than 0.1 percent oyer a wind speed range 0 - 150 fps. There exists a certain amount of controversy over the use of established methods to correct for wind tunnel wall interference. The present configuration posed an additional complication i n that separation occurs at the spoiler t i p and at the t r a i l i n g edge with the formation of a broad wake. It was observed that when using the corrections established by Pope and Harper ( 8 ), i n which the wake'blockage term i s taken equal to 1/2 (c/H) C^, the l i f t curves for the d i f f e r e n t sizes of model did not collapse and a tendency to over-correct was evident, p a r t i c u l a r l y for the l a r -ger models. It was further noticed that a value for the wake blockage term equal to 1/4 (c/H) C^, as suggested by Pankhurst and Holder (9), would be more suitable for the present configura-tion and would give a better collapse for the data. Thus, using expressions for the correction of l i f t c o e f f i -cient and angle of attack given by Pope and Harper together,- with 56 the m o d i f i e d wake b l o c k a g e term, the l i f t c u r v e s f o r the Joukow-sky a i r f o i l and t h e 11+ i n . C l a r k Y a i r f o i l were c o r r e c t e d f o r t u n n e l w a l l e f f e c t s . Next, the p r e s s u r e c o e f f i c i e n t d i s t r i b u t i o n was c o r r e c t e d as f o l l o w s : where i s the a n g l e of a t t a c k a t which the d i s t r i b u t i o n was measured. The p r e s s u r e c o e f f i c i e n t i n the wake was a l s o c o r r e c t e d u s i n g e q u a t i o n (34)> r a t h e r than u s i n g an e x p r e s s i o n suggested by M a s k e l l (10) a p p r o p r i a t e f o r s e p a r a t e d f l o w s , s i n c e the two methods of c o r r e c t i o n produced r e s u l t s t h a t a r e v e r y n e a r l y the same. 1.(34) • • • 57 IV RESULTS AND COMPARISONS k'x Joukowsky A i r f o i l To check that the experimental a i r f o i l behaved l i k e a Joukow-sky a i r f o i l , i t was tested without a spoiler, and the r e s u l t i n g Cp-distribution for <X = 6°, uncorrected for tunnel wall effects, i s compared i n Fig . 10 with theoretical solutions. The dashed curve shows the d i s t r i b u t i o n for 0< = 6°. As expected, i t pre-di c t s a higher l i f t than i s actually observed because the theory does not account for the reduction i n c i r c u l a t i o n caused by»the boundary layer v o r t i c i t y . However, i f comparison i s made at. equal l i f t (corresponding to &< = 5«1° for the theoreti c a l calculation, as shown by the s o l i d curve), the agreement i s seen to be excel-lent, and the small bump i n the experimental d i s t r i b u t i o n caused by the a r t i f i c i a l l y thickened t r a i l i n g edge i s evident. The v a r i -ation of l i f t c o e f f i c i e n t with angle of attack i s shown i n F i g . 11, i n which agreement between theory and experiment, here cor-rected for wall effects, improves as the zero l i f t angle is,-, approached, due to the reduction i n boundary layer v o r t i c i t y . Figure 12 shows the effect of a 5% spoiler when mounted at 50, 70 and 90% chord positions. As the spoiler i s moved forward the value of AC-\.l(\o(, i s reduced, implying an increase i n spoi l e r effectiveness at the forward position. However, no appreciable change i n the value of the zero l i f t angle i s observed. Figures 13, 14 and 15 show comparisons of theoretical and experimental -- o E X P E R I M E N T , a = 6 ° T H E O R Y , a = 6 ° q T H E O R Y .' cc = 5.1° -- C H O R D = 4 . 0 2 7 -f\ --— 1 i * - — ° J g T ; — |, j - 2 H S ^ + 2 Figure 10. Pressure D i s t r i b u t i o n for Basic Joukowsky A i r f o i l . 59 1.4 • 1.2 -1.0 -. / \ / ^ / o / / .8 - // / o 1 . 6 - // / o // . 4 -4 / o / 1 1 If • 1° ' / ~0~ E X P E R I M E N T / T H E O R Y 1 1 i i i - 8 - 4 / / " | | 1 1 1— 4 8 1 2 16 2 0 o a 1 . Figure 11. L i f t Coefficient for Basic Joukowsky A i r f o i l . 60 Figure 12. Experimental L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 61 1.4 4-Figure 13. L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 62 Figure 14. L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 63 Figure 15. L i f t C o e f f i c i e n t for Joukowsky A i r f o i l with Sp o i l e r . 64 l i f t c o e f f i c i e n t curves for a 5% s p o i l e r . In each figure the 1-source model i s seen to give excellent prediction for values of c\Ci I&& • However, as the spoiler i s moved forward, the l i f t i s over-estimated by the model. The 2-source model, on the other hand, offers considerable improvement for the 50% spoiler position, and none for the 90% location because the 2-source model can never predict a larger l i f t than that of the 1-source. Thus, i n F i g . 15, the l i f t c o e f f i c i e n t curves for the two theore t i c a l models become indistinguishable. In F i g . 16 the effect of a 10% spoiler located at 50, 70 and 9 0 % chord i s shown. Again, no appreciable change i n the zero l i f t angle i s observed. However, the sp o i l e r effectiveness i s i n -creased when the location i s moved forward, r e s u l t i n g i n a de-crease i n values of <ACu/do< . Figures 17, 18 and 19 show compari-sons of theoretical and experimental l i f t c o e f f i c i e n t curves for a 10% spoiler. In each case, the 1-source model seems to r e s u l t i n good prediction for values of <iCL /di&< > "but, as for the 5% spoiler case, the l i f t i s over-estimated by the model for the forward locations of sp o i l e r s . The 2-source model, i n comparison, offers considerable improvement for the 5°% location, and some for the 70%, but none for the 90% chord position. ;; Most of the experimental l i f t c o e f f i c i e n t curves for a i r f o i l -spoiler configurations have a s l i g h t curvature, which i s due, to the formation of a separated bubble ahead of the s p o i l e r . The extent of t h i s bubble becomes larger and i t s pressure l e s s posi-t i v e as the angle of attack i s increased, thus r e s u l t i n g i n a larger contribution to the l i f t . At the higher values of incidence Figure 16. Experimental L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 66 Figure 17. L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. Figure 18. L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 68 Figure 19. L i f t Coefficient for Joukowsky A i r f o i l with Spoiler. 6 9 i t i s desirable to have the measured l i f t larger than the theore-t i c a l value i n order to produce better agreement i n the pressure d i s t r i b u t i o n over the unseparated part of the a i r f o i l . Figures 20, 21 and 22 show comparisons of theoretical and experimental pressure c o e f f i c i e n t d i s t r i b u t i o n s for d i f f e r e n t chordwise positions of a 5% spoiler with the a i r f o i l at c< = 11° and with C p matched. The experimental values have been corrected for tunnel wall eff e c t s . In each figure the 1-source model i s compared with the experimental d i s t r i b u t i o n and i t i s seen to give reasonable agreement. However, as the spoiler i s moved for-ward, the upper surface suction and the lower surface pressures are over-estimated by the model, and the effect of C p not being A specified becomes more noticeable. In the three figures, i t i s seen that the 2-source model gives much better agreement with the experimental values over the a i r f o i l surface. One inevitable discrepancy between theory and experiment.is evident i n a l l the Cp-distribution figures. The theory predicts a stagnation point at the base of the upstream spoiler surface^ so that C p =1. Actually, the adverse pressure gradient upstream of the spoiler causes boundary layer separation from the a i r f o i l , with reattachment on the spoiler face. The constant pressure ; separation bubble can be c l e a r l y i d e n t i f i e d i n the figures, ,and i t s extent i s p a r t i c u l a r l y well defined i n F i g . 20 by comparison of the experimental variation with the curve for the 2-source model. Figures 23 and 24 show Cp-distributions for the a i r f o i l v/ith a 10% spoiler near the zero l i f t angle. The 1-source model gives - 5 -3 -2 0 + 1 4 C E/ c = . 5 0 , h/c = . 0 5 , a = 11° c = 4 . 0 2 7 \ ' o E X P T . \ 1 - S O U R C E \ 2 - S O U R C E \ \ \ \ \ \ 0 \ ^ > ^ ( 9 O O O © O Q 9 Q j - 2 Q r, ^ Q o o Q O Q 0 "^—'+2 Figure 20. Pressure Distribution for Joukowsky A i r f o i l with Spoiler. Figure 22. Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. Figure 23. Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. Figure 24. Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. 75 good agreement for the 90% spoiler position and can not be im-proved upon by using the 2-source model. For the 50% spoiler l o -cation i n Fig . 23, i t i s seen that the 2-source gives much better agreement than the 1-source and that the extent of the separated bubble i s reduced for the lower incidence. ] Figures 25, 26 and 27 give comparisons of the a i r f o i l at (X = 13° with three positions of the 10% spoil e r . Both the 1-source model and the 2-source model are presented and compared with experiment, and similar comments to those made about Figs. 20, 21 and 22 can be made here. It can be seen that the separation bubbles caused by the 10% spoiler are larger than those f o r ; t h e 5% spoiler. In F i g . 28 the separation streamlines are plotted for' about one chord length downstream for the case of a 5% spoiler at 90% chord with the a i r f o i l at (X = 8°. Both the 1-source and the 2-source models are shown. Cp-distributions along the upper stream-l i n e for each model are also plotted on the figure. It i s seen that the streamlines become almost p a r a l l e l a short distance downstream. The apparent s h i f t i n the streamlines of the two . models i s due to the larger overall c i r c u l a t i o n about the a i r f o i l as predicted by the 1-source model r e s u l t i n g i n the downward s h i f t for the corresponding streamlines. The asymptotic separation of these streamlines i s the quotient of the t o t a l wake source strength and the free stream ve l o c i t y . The Cp-distribution;along the streamlines decays gradually towards zero. .,• •5 --4 -•3 -1 -0 + 1 Figure 25. Pressure Dis t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. - 5 - 3 -1 0 +1 _ c E/c = . 7 0 , h/c = .10 , a = 13° c = 4 . 0 2 7 o E X P T . 1 - S O U R C E 2 - S O U R C E [-© o O O O 1 - 2 " ^ ^ o o o ^ o o ? n — ^ + - , Figure 26. Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. Figure 27. Pressure D i s t r i b u t i o n for Joukowsky A i r f o i l with Spoiler. 2 - S O U R C E Figure 28. Positions of and Pressure D i s t r i b u t i o n along Separation Streamli 80 2+.2 Clark Y A i r f o i l The pressure d i s t r i b u t i o n for the basic Clark Y a i r f o i l at <X = 6° i s shown i n F i g . 29. The effect of the f i n i t e angle at the t r a i l i n g edge becomes evident i n that a stagnation point w i l l always be present at the t r a i l i n g edge. The v a r i a t i o n of l i f t c o e f f i c i e n t with angle of attack i s shown i n F i g . 30. As usual, agreement between theory and experiment i s improved as the zero l i f t angle i s approached due to the reduction i n the boundary layer v o r t i c i t y . .; Figure 31 shows the variation of the measured l i f t c o e f f i -cient' with angle of attack for the a i r f o i l f i t t e d with a 5% .spoiler at three chordwise positions. The effect i s the same as was ob-served before, i n that the forward locations of the spoiler cor-respond to the lower values of dCu/dlfl . For t h i s a i r f o i l there seems to be a noticeable s h i f t i n the value of the zero l i f t angle for the 90% chord location. However, i t i s important to r e c a l l that the data for the 5% spoiler were obtained by using a 22+ i n . chord model, which represents a blockage r a t i o of 0.667, and the a p p l i c a b i l i t y of the wind tunnel wall corrections becomes some-what questionable. Naturally, the values of elc^tf are also affected by t h i s large blockage r a t i o . Figures 32, 33 and 32+ show comparisons of theoretical and experimental l i f t c o e f f i c i e n t 1 curves for a 5% spo i l e r . In each figure the 1-source model.is seen to give quite good predictions for values of dCi_/clo< , except for the 50% location (Fig. 32). However, as the spoiler position i s moved forward the l i f t i s over-estimated by t h i s model, and the 2-source model seems to provide only a s l i g h t improvement at Figure 29. Pressure Distribution for Basic Clark Y A i r f o i l . 82 1.6 + - . 4 + Figure 30. L i f t Coefficient for Basic Clark Y A i r f o i l . 83 84 Figure 32. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. 85 1 . 6 - -Figure 33. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. 86 Figure 34. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. 87 the 50% location due to the disagreement i n the value of d C u / d < X • Better agreement between the 2-source model and experiment i s observed at the 70% location (Fig. 3 3 ) , but no improvement over the 1-source model i s shown at the 90% position (Fig. 3k)» Figure 35 shows the effect of a 10% spoiler located at three chordwise positions, and similar comments to those made e a r l i e r can be made here, except that values of d l C / d ^ are more r e l i a b l e , since the r e s u l t s are from a Ik i n . model. Figures 36, 37 and 38 show comparisons of experimental and theoretical l i f t c o e f f i -cient curves for the 10% spoiler. Again, the excellent predic-tion by the 1-source model for values of clCt/d^ i s evident. ; Also, the 2-source model seems to give progressive improvement over the 1-.source for values of the l i f t as the spoiler i s moved forward. No experimental d i s t r i b u t i o n s of the pressure coefficient, were available for the Clark Y a i r f o i l , hence, i n the following figures, only comparisons between the 1- and the 2-source models are shown. Figures 3k, k® and 2+1 show the pressure d i s t r i b u t i o n s for the a i r f o i l at (X = 10° f i t t e d with a 5% spoiler at three d i f -ferent locations. The 2-source model, because i t gives lower l i f t , has lower suction peaks than the 1-source, also l e s s pres-sure over the lower surface for the 50% and 70% locations. ,For the 90% position the two di s t r i b u t i o n s are i d e n t i c a l except .-near the t r a i l i n g edge, where the 1-source w i l l have a pressure d i s -continuity. Figures 1+2 and 2+3 show Cp-distributions for the a i r f o i l with a 10% spoiler near the zero l i f t angle. The tendency for/ the two models to approach each other as the spoiler i s moved back i s 88 e v i d e n t and i s c o n s i s t e n t w i t h the g e n e r a l t r e n d f o r a l l p r e s s u r e d i s t r i b u t i o n s . F i g u r e s 44} 45 and 46 g i v e comparisons f o r t h e a i r f o i l a t 0* = 12° w i t h t h r e e p o s i t i o n s of the 10% s p o i l e r , and a g a i n com-ments s i m i l a r t o those made f o r the 5% s p o i l e r a r e a p p l i c a b l e h ere. Figure 35. Experimental L i f t Coefficient for Clark Y A i r f o i l with Spoi l e r . 90 Figure 36. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. 9 1 Figure 37. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. 92 Figure 38. L i f t Coefficient for Clark Y A i r f o i l with Spoiler. E / c = . 5 0 , h / c = . 0 5 , cc = 10° c = 4 . 0 4 3 1 - S O U R C E 2 - S O U R C E Figure 39. Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler. E / c = . 7 0 , h / c = . 0 5 , a = 10° c = 4 . 0 4 3 Figure 40. Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler. E / c = . 9 0 , h / c = . 0 5 , cc = 10° c = 4 . 0 4 3 1 - S O U R C E • 2 - S O U R C E - 1 0 Z2-+ 2 + 1 -Figure 41. Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler. E / c = . 5 0 , h / c = .10 , a = 5 . 2 c = 4 . 0 4 3 1 - S O U R C E 2 - S O U R C E +2 Figure 42. Pressure Dist r i b u t i o n for Clark Y A i r f o i l with Spoiler. Figure 43. Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler. E / c = . 5 0 , h / c = .10 , cc = 12 c = 4 . 0 4 3 1 - S O U R C E 2 - S O U R C E - 1 0 +1 Figure 44. Pressure Dist r i b u t i o n for Clark Y A i r f o i l with Spoiler. E/c = . 7 0 , h/c = .10 , <x = 12° c = 4 . 0 4 3 Figure 45. Pressure D i s t r i b u t i o n for Clark Y A i r f o i l with Spoiler. Figure 46. Pressure Dis t r i b u t i o n for Clark Y A i r f o i l with Spoiler. 101 V CONCLUSIONS It has been demonstrated that the 1-source model gives good agreement with experiment for the l i f t and pressure d i s t r i b u t i o n when spoilers are mounted near the t r a i l i n g edge. This model also predicts values for tACu/o\ot that are i n excellent agreement with experiment for a l l possible a i r f o i l - s p o i l e r configurations. For the other locations of spoilers (nearer the mid-chord), the j2-, source model gives much better agreement with experimental l i f t and pressure d i s t r i b u t i o n s . In general, the 2-source model-is; more.desirable since i t removes the pressure discontinuity at the t r a i l i n g edge. The only discrepancy i n the pressure d i s t r i b u t i o n , which i s common to both theoretical models, i s the experimentally evident separation bubble ahead of the spoiler. This i s a boundary layer separation phenomenon that potential flow cannot cope with, and agreement between theory and experiment i n that region i s not . expected. The choice of specifying the zero l i f t angle as an additional condition for the 2-source model i s not an uncommon step towards making potential flow agree more closely with experiment. Pinker-ton (11), working with NACA Four D i g i t a i r f o i l s , suggested that the Kutta condition, determining the c i r c u l a t i o n and consequently the l i f t , be dropped and that instead the c i r c u l a t i o n be deter-mined from the measured l i f t . Naturally t h i s procedure produces 102 i n f i n i t e suction at the t r a i l i n g edge. However, i t does not a l t e r the d i s t r i b u t i o n appreciably over the rest of the a i r f o i l . The s i g n i f i c a n t contribution of the present theory i s the prediction of loading d i s t r i b u t i o n s that are i n good agreement with experiment. Linearized theories, such as those of Woods and Barnes, give good predictions for l i f t , but as expected, not very good pressure d i s t r i b u t i o n s , hence the need for a nonlinear ;theory not r e s t r i c t e d to low incidence and small sp o i l e r s . I t should be possible to obtain better agreement between theoretical and experimental pressure d i s t r i b u t i o n s by finding an empirical r e l a t i o n for the pressure i n the separated bubble as a function of the geometrical parameters for the a i r f o i l - s p o i l e r configuration. Since t h i s pressure i s constant for a given]con-figuration, the present theoretical prediction over the separated part of the a i r f o i l ahead of the spoiler could be replaced by :,the constant value obtained from the empirical r e l a t i o n . 3 103 • REFERENCES 1. Woods, L. C. "The Theory of Subsonic Plane Flow" Cam-bridge University Press, 1961. 2. Woods, L. C. "Theory for A i r f o i l Spoilers" A.R.C. R.&M. 2969, May, 1953. 3. Barnes, C. S. "A Developed Theory of Spoilers on A i r f o i l s " A.R.C. C P . 887, July, 1965. 4. Mandl, P. "Potential Flow about an A i r f o i l with a S p l i t Flap" Canadian Aeronautical Journal, Vol. 6, A p r i l , I960, pp. 123-125. 5. Roshko, A. "A New Hodograph for Free Streamline Theory" NACA TN 3168, 1954. 6. Abbott, I. H., and von Doenhoff, A. E. "Theory of Wing Sections" Dover, 1949, pp. 53-63. i 1 7. Riegels, F. W. " A i r f o i l Sections; Results from Wind Tunnel Investigations, Theoretical Foundations" Translated from the German by D. G. Randall, London, Butterworths, 1961. 8. Pope, A. and Harper, J . J . "Low-Speed Wind Tunnel Testing" John Wiley and Sons, 1966, Chapter 6. 9. Pankhurst, R. C. and Holder, D. W. "Wind-Tunnel Technique" Pitman and Sons Ltd., London, 1952, Chapter 8. 10. Haskell, E. C. "A Theory of the Blockage Ef f e c t s on Bluf f Bodies and Stalled Wings i n a Closed Wind Tunnel" R.&M. No. 3400, November, 1963. ;•. 11. Pinkerton, R. M. "Calculated and Measured Pressure Dis-t r i b u t i o n over the Midspan Section of the NACA 4412 A i r -f o i l " NACA Report No. 563, 1936. 

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