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UBC Theses and Dissertations

Prediction method for spoiler performance Tam Doo, Peter A. 1977

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A PREDICTION METHOD FOR SPOILER PERFORMANCE by PETER A.jjTAM DOO B.Sc, University of Manitoba, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1977 (c) Peter A. Tarn 000 ,1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 2 R t h O r l - n W 1Q77 ABSTRACT The p r e d i c t i o n of the aerodynamic c h a r a c t e r i s t i c s of a f i n i t e rectangular wing with part span s p o i l e r s i s attempted using the l i f t i n g l i n e theory of Prandtl. Required inputs to the theory are se c t i o n a l values of l i f t c o e f f i c i e n t , p i t c h i n g moment c o e f f i c i e n t , zero l i f t angle of attack, and aerodynamic center at selected points along the span. The value of these parameters f o r the spoilered wing sections i s calculated by Brown's l i n e a r i s e d t h i n a i r f o i l theory f o r s p o i l e r s . This theory, i n common with other s e c t i o n a l s p o i l e r theories, requires as input the base pressure c o e f f i c i e n t i n the s p o i l e r wake. The base pressure c o e f f i c i e n t must be determined by experiment, since at the present time i t cannot be predicted t h e o r e t i c a l l y . The e f f e c t of base venting on spoilered section c h a r a c t e r i s t i c s i s examined experimentally. I t i s found that f o r small base vents of around ten percent of s p o i l e r height or l e s s , the vented section c h a r a c t e r i s t i c s are l i t t l e d i f f e r e n t from the unvented. Thus f o r the purposes of preliminary design, the unvented section c h a r a c t e r i s t i c s may be used with l i t t l e l o s s of accuracy, i f the s p o i l e r vent i s about ten percent of s p o i l e r height or l e s s . The r e s u l t s of the f i n i t e wing theory are compared with experiment. Good agreement i s found. The method i s subject to the l i m i t a t i o n s of the l i f t i n g l i n e theory, which l i m i t s i t s a p p l i c a b i l i t y to unswept wings of moderate to high aspect r a t i o s operating at low subsonic speeds. The method i s also subject to the a d d i t i o n a l l i m i t a t i o n s imposed by the sec t i o n a l theories employed. i i i TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 2 THEORY 4 2.1.1 The L i f t i n g Line Theory 4 2.1.2 The Jones Edge Correction Factor 8 2.2 Application to Wings with Spoilers 10 2.3 Brown's Thin A i r f o i l Theory for Spoilers 12 2.3.1 The Acceleration Potential 14 2.3.2 Conformal Transformations . . 15 2.3.3 Boundary Conditions 17 2.3.4 Flow Model 18 2.3.5 Base Vented Spoilers 22 2.4 Experimental Two Dimensional Base Pressures .. 23 2.5 Experimental Finite Span Base Pressures 25 o 2.6 Empirical Relationships for Base Pressures ... 27 2.6.1 C , Averaged across Span 27 pb 2.6.2 Variation of Sectional Properties with Base Pressure 31 2.6.3 Use of Experimental Two Dimensional Spoilered Section Parameters in Finite Wing Theory 33 3 EXPERIMENTS 35 3.1.1 Base Venting Experiments 35 3.2 Fi n i t e Wing Experiments 36 3.3 Base Pressure Measurements 37 iv Chapter Page 3.4 Wind Tunnel Wall Corrections 38 4 RESULTS AND COMPARISONS 4.1 Base Venting Experiments 40 4.2.1 Two Dimensional NACA 0015 A i r f o i l Experiments 40 4.2.2 Rectangular wings of NACA 0015 Section F i t t e d with Part Span Sp o i l e r s 41 4.3.1 Two Dimensional Base Pressure Experiments .... 44 4.3.2 Three Dimensional Base Pressure Experiments .. 44 5 CONCLUSIONS 46 FIGURES 48 REFERENCES 98 APPENDIX 99 V L I S T OF TABLES Table Page I Values of C , f o r two dimensional Clark Y and pb NACA 0015 A i r f o i l s with Normal Unvented Spoilers 24 II Base Pressure C o e f f i c i e n t s f o r Spoilered NACA 0015 & 12.9% Clark Y A i r f o i l s '. 2 g I I I S e c t ional C h a r a c t e r i s t i c s f o r a NACA 0 0 1 5 . A i r f o i l F i t t e d with 9.7% Unvented, Normal Sp o i l e r s 31 i v i L I S T OF FIGURES Figure Page 1 F i n i t e Wing with Spoiler and I t s Vortex Model 48 2 A i r f o i l i n the Phy s i c a l Plane 49 3 Complex Transform Planes 50 4 V a r i a t i o n of m with C , for a NACA 0015 A i r f o i l os pb Section with Normal Unvented Sp o i l e r 51 5 V a r i a t i o n of C & x /c with C , for NACA 0015 macs acs pb A i r f o i l Section with Normal Unvented Sp o i l e r 52 6 V a r i a t i o n of a, with C , f o r NACA 0015 A i r f o i l l o s pb Section with Normal Unvented Sp o i l e r 53 7 Modified Joukowsky A i r f o i l Section of 11% Thickness & 2.4% Camber with Base Vented Normal Sp o i l e r 54 8 NACA 0015 A i r f o i l Section with 9.7% Unvented Normal Spoiler 55 9 12.9% Thick Clark Y A i r f o i l Section with 10% Unvented Normal Sp o i l e r 56 10 Base Pressure Measurement System 57 11 L i f t C o e f f i c i e n t f o r Joukowsky A i r f o i l Section with Base Vented Normal Sp o i l e r s 58 12 L i f t C o e f f i c i e n t f o r Joukowsky A i r f o i l Section with Base Vented Normal S p o i l e r s 59 13 P i t c h i n g Moment C o e f f i c i e n t f o r Joukowsky A i r f o i l Section with Base Vented Normal Sp o i l e r s 60 14 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015 Section 61 v i i Figure Page 15 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015 A i r f o i l Section with Normal Unvented Sp o i l e r 62 16 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r NACA 0015 A i r f o i l Section with Normal Unvented S p o i l e r 63 17 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r , 64 18 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler 65 19 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 66 20 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r 67 21 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 68 22 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r 69 23 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 70 24 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 71 25 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 72 26 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Sp o i l e r 73 v i i i Figure Page 27 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler 74 28 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler 75 29 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spo i l e r , 76 30 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spo i l e r 77 31 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spo i l e r 78 32 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler 79 33 E f f e c t i v e Moment Arm of Incremental L i f t Due to Normal Unvented Spoiler on Rectangular Wing of NACA 0015 Section 80 34 E f f e c t i v e Moment Arm of Incremental L i f t Due to Normal Unvented Sp o i l e r on Rectangular Wing of NACA 0015 Section 81 35 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings with Symmetrically Deployed Sp o i l e r s 82 36 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings with Symmetrically Deployed S p o i l e r s 83 37 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings of NACA 0015 Section 84 38 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings of NACA 0015 Section 85 39 C , D i s t r i b u t i o n f o r NACA 0015 Section with 9.7% pb Normal Unvented Spo i l e r 86 i x Figure Page 40 C D i s t r i b u t i o n f o r a 12.9% Thick Clark Y Section with 10% Normal Unvented Sp o i l e r 87 41 V a r i a t i o n of C , with S p o i l e r P o s i t i o n Along Chord pb fo r Rectangular Wings with S p o i l e r s 88 42 V a r i a t i o n of C , with S p o i l e r Span f o r Rectangular Wings pb with S p o i l e r s - 89 43 V a r i a t i o n of C , with Aspect Ratio f o r Rectangular pb Wings with S p o i l e r s 90 44 V a r i a t i o n of C , with S p o i l e r P o s i t i o n Along Chord pb • - • fo r Rectangular Wings with S p o i l e r s 91 45 V a r i a t i o n of C , with S p o i l e r Span f o r Rectangular pb Wings with S p o i l e r s 92 46 V a r i a t i o n of C ^ with Aspect Ratio f o r Rectangular Wings with S p o i l e r s 93 47 V a r i a t i o n of C ^ with S p o i l e r Span for Rectangular Wing with Normal Unvented S p o i l e r s 94 48 V a r i a t i o n of C , with S p o i l e r Span for Rectangular pb Wings with Normal Unvented Sp o i l e r s 95 49 V a r i a t i o n of C ^ with S p o i l e r Span f o r Rectangular Wings with Normal Unvented S p o i l e r s 96 50 V a r i a t i o n of C_ t / C* u and G/G* with s/c 97 ACKNOWLEDGEMENTS The author wishes to thank Dr. G.V. Parkinson f o r h i s guidance i n the preparation of t h i s t h e s i s . His many suggestions during the course of t h i s research were invaluable and much appreciated. The author also wishes to thank the Mechanical Engineering Machine Shop f or b u i l d i n g the models and the Computing Center at the U n i v e r s i t y of B r i t i s h Columbia f o r the use of t h e i r f a c i l i t i e s . This research was supported i n i t i a l l y by the Defense Research Board under Grant Number 9551-13 and by the National Research Council under Grant Number A586. x i L I S T OF SYMBOLS a x = IJx = ~dz x c o m P o n e n t °* a c c e l e r a t i o n Bx a z = 3z = ~ 9x Z c o m P ° n e n t °f a c c e l e r a t i o n a c c e l e r a t i o n vector A Fourier c o e f f i c i e n t n 2 AR = 2b /S aspect r a t i o of wing b wing semispan b g s p o i l e r span b t coordinate along s p o i l e r span measured with respect to inboard s p o i l e r t i p c wing chord c root semichord of e l l i p t i c disk e r c f l a p chord n s p o i l e r e d s e c t i o n a l drag c o e f f i c i e n t unspoilered s e c t i o n a l l i f t c o e f f i c i e n t C, s p o i l e r e d s e c t i o n a l l i f t c o e f f i c i e n t Is r C unspoilered p i t c h i n g moment c o e f f i c i e n t about o r i g i n mo C m o g s p o i l e r e d p i t c h i n g moment c o e f f i c i e n t about o r i g i n C c unspoilered p i t c h i n g moment c o e f f i c i e n t about the aerodynamic center C spoi l e r e d p i t c h i n g moment c o e f f i c i e n t about the aerodynamic center C h a l f wing l i f t c o e f f i c i e n t L h a l f wing p i t c h i n g moment c o e f f i c i e n t about o r i g i n ^MAC h a l f wing p i t c h i n g moment c o e f f i c i e n t about the aerodynamic center Cp g pressure c o e f f i c i e n t f o r spoilered s ection x i i CL h a l f wing r o l l i n g moment c o e f f i c i e n t R C , base pressure c o e f f i c i e n t i n s p o i l e r wake pb C , base pressure c o e f f i c i e n t i n s p o i l e r wake averaged over pb incidence C , base pressure c o e f f i c i e n t i n s p o i l e r wake averaged over pb incidence and s p o i l e r span =* = C ' Value of C . at the 0.7 chord p o s i t i o n pb pb d height of s p o i l e r base vent E semiperimeter of wing/semispan, Jones Edge Correction Factor F(Z)= (J) + i ^ complex a c c e l e r a t i o n p o t e n t i a l G * G value of G at the 0.7 chord p o s i t i o n h s p o i l e r height k chordwise coordinate of e l l i p t i c disk k g chordwise coordinate of edge of e l l i p t i c disk K = -C , c a v i t a t i o n number pb L' s e c t i o n a l l i f t f o r unspoilered section 1/ s e c t i o n a l l i f t f o r spoi l e r e d s e c t i o n Z c a v i t y length mQ unspoilered dC^/da (/rad.) for a i r f o i l section m s p o i l e r e d dC-/da ( /rad.) f o r a i r f o i l section os ± M Q unspoilered s e c t i o n a l p i t c h i n g moment about o r i g i n M Q S s p o i l e r e d s e c t i o n a l p i t c h i n g moment about o r i g i n M unspoilered s e c t i o n a l p i t c h i n g moment about aerodynamic center ac M s p o i l e r e d s e c t i o n a l p i t c h i n g moment about aerodynamic center 9.C S h a l f wing p i t c h i n g moment about o r i g i n h a l f wing p i t c h i n g moment about aerodynamic center chordwise coordinate of f l a p hinge pressure i n flow f i e l d pressure i n s p o i l e r wake fr e e stream pressure r o l l i n g moment of h a l f wing chordwise coordinate of s p o i l e r base h a l f wing area chordwise coordinate of s p o i l e r t i p f r e e stream v e l o c i t y non-dimensional perturbation v e l o c i t y i n x d i r e c t i o n non-dimensional perturbation v e l o c i t y i n z d i r e c t i o n downwash chordwise coordinate chordwise coordinate f o r aerodynamic center of unspoilered s e c t i o n chordwise coordinate f o r aerodynamic center of spoil e r e d s ection chordwise coordinate of wing aerodynamic center spanwise coordinate spanwise p o s i t i o n at which the induced angle of attack i s evaluated spanwise coordinate of inboard s p o i l e r t i p spanwise coordinate of outboard s p o i l e r t i p coordinate perpendicular to xy plane z' complex transform plane Z = x + i z complex a i r f o i l plane a geometric angle of attack f o r a i r f o i l section a = a + a, absolute angle of attack f o r unspoilered a i r f o i l section a l o a = a + a.. absolute angle of attack for spoilered a i r f o i l s e ction 3.S XO S cx = a + a. e f f e c t i v e angle of attack for a i r f o i l s e ction e a 1 induced angle of attack l o zero l i f t angle of attack for unspoilered section a ^ o g zero l i f t angle of attack f o r spoilered section a s t a l l s t a l l angle f o r wing or a i r f o i l section 6 s p o i l e r e r e c t i o n angle £, v complex transform planes ri f l a p angle 8 angular v a r i a b l e i n £-plane u=cos ^(y/b) span v a r i a b l e f o r f i n i t e wing p o s i t i o n of inner s p o i l e r t i p i n terms of span v a r i a b l e y V<2 p o s i t i o n of outer s p o i l e r t i p i n terms of span v a r i a b l e u p a i r density <|> a c c e l e r a t i o n p o t e n t i a l <J>e v e l o c i t y p o t e n t i a l for e l l i p t i c d i s k ip a c c e l e r a t i o n stream function T=cos \ k / k ) chordwise v a r i a b l e f o r e l l i p t i c d i s k e T c i r c u l a t i o n 1 1. INTRODUCTION S p o i l e r s a r e v e r s a t i l e aerodynamic c o n t r o l surfaces which are used on many modern a i r c r a f t . They may best be described as small f l a p s that have been moved ahead of the t r a i l i n g edge to the upper or lower wing s u r f a c e . S p o i l e r s may be deployed symmetrically to c o n t r o l l i f t and drag, or a s y m m e t r i c a l l y to produce r o l l and yaw. Since s p o i l e r s may be used together w i t h f u l l span f l a p s , t h e i r presence i n V/STOL a i r c r a f t i s becoming i n c r e a s i n g l y common. The behavior of s p o i l e r s on wings i s , however, d i f f i c u l t to p r e d i c t because f l o w separates from the s p o i l e r edges, and a t u r b u l e n t wake i s formed behind the s p o i l e r . The standard p o t e n t i a l f l o w methods of a i r f o i l theory cannot t h e r e f o r e be used. They must be modified to account f o r the presence of the wake. Woods (1) was among the f i r s t to t a c k l e the problem. He developed a l i n e a r i s e d t h i n a i r f o i l theory f o r s p o i l e r e d wing s e c t i o n s . Barnes (2) l a t e r modified the theory to account f o r the presence of the boundary l a y e r on the a i r f o i l . Here at the U n i v e r s i t y of B r i t i s h Columbia, work on s p o i l e r theory was begun by J a n d a l i and Brown i n an e f f o r t to improve on the accuracy of p r e d i c t i o n . J a n d a l i developed an a n a l y t i c t h i c k a i r f o i l theory f o r a i r f o i l s e c t i o n s w i t h normal s p o i l e r s (3). Brown developed a t h i n a i r f o i l theory, and a numerical t h i c k a i r f o i l t heory (4). A l l of the above t h e o r i e s apply f o r s p o i l e r e d a i r f o i l s w i t h wakes which do not r e a t t a c h to the a i r f o i l s u r f a c e . The present work extends the s p o i l e r theory i n t o three dimensions. Since low speed a p p l i c a t i o n s are of p r i n c i p a l i n t e r e s t i n Canada, and s i n c e the s e c t i o n a l t h e o r i e s developed by J a n d a l i and Brown are a p p l i c a b l e 2 to i n c o m p r e s s i b l e f l o w , i t was decided t h a t i t would be appropiate to extend the theory i n t o three dimensions by means of P r a n d t l ' s l i f t i n g l i n e theory ( 5 ) . The l i f t i n g l i n e theory o v e r p r e d i c t s the l i f t f o r sma l l e r aspect r a t i o s . Jones (6) has proposed a m o d i f i c a t i o n to the l i f t i n g l i n e theory which c o r r e c t s the o v e r p r e d i c t i o n and t h i s i s i n c o r p -orated i n t o the theory. The l i n e a r i s e d t h i n a i r f o i l theory of Brown i s used to c a l c u l a t e the s p o i l e r e d s e c t i o n parameters which are r e q u i r e d as in p u t to the l i f t i n g l i n e theory. Other t h e o r i e s , such as those of Woods or J a n d a l i , or Brown's numerical t h i c k a i r f o i l theory, may of course be used. I f the s e c t i o n a l parameters d e r i v e d from two dimensional base pressure i n p u t s to the s p o i l e r t h e o r i e s are used i n the l i f t i n g l i n e theory, e r r o r s w i l l r e s u l t because the three dimensional base pressures d i f f e r from the corresponding two dimensional v a l u e s . Flow around the v e r t i c a l edges o f the f i n i t e span s p o i l e r reduces the negative base pressure and c r e a t e s spanwise g r a d i e n t s of base pressure which are absent from two dimensional f l o w s . Since s p o i l e r e d s e c t i o n c h a r a c t e r i s t i c s are a f u n c t i o n of base pressure c o e f f i c i e n t s , i t i s c l e a r that three dimensional base p r e s s u r e - c o e f f i c i e n t s must be used as input to the s e c t i o n a l t h e o r i e s i n order to o b t a i n s e c t i o n a l parameters appropiate to f i n i t e span f l o w s . A l l of the above mentioned s p o i l e r t h e o r i e s are developed f o r unvented s p o i l e r s . While the p o s s i b i l i t y e x i s t s that some of the t h e o r i e s may be m o d i f i e d to take i n t o account the e f f e c t of base v e n t s , t h i s course of a c t i o n i s not attempted i n the present t h e s i s . Instead the e f f e c t of base v e n t i n g i s examined e x p e r i m e n t a l l y . I t i s found t h a t f o r small base vents of about ten percent s p o i l e r height or l e s s , the vented s p o i l e r behavior i s l i t t l e d i f f e r e n t from the unvented. Thus for the purposes of preliminary design, the unvented s p o i l e r c h a r a c t e r i s t i c s may be used f o r s p o i l e r s with base vents of about ten percent or l e s s . The r e s t r i c t i o n to base vents of about ten percent or l e s s i s not a serious l i m i t a t i o n , since i n p r a c t i c e , most base vents are about t h i s s i z e . 2. THEORY 2.1.1 The L i f t i n g L i n e Theory The l i n e a r i s e d l i f t i n g l i n e theory as formulated by P r a n d t l , i s a p p l i c a b l e to unswept wings of moderate to l a r g e aspect r a t i o o perating at low mach numbers. The wing i s placed i n a r i g h t handed orthogonal coordinate system as shown i n F i g u r e 1. The o r i g i n of the system i s l o c a t e d at the aerodynamic center of the wing r o o t s e c t i o n . The f r e e stream v e l o c i t y i s i n the p o s i t i v e x d i r e c t i o n . The wing i s modelled as a l i f t i n g l i n e of bound v o r t i c e s l o c a t e d on the y a x i s , c o v e r i n g the span of the wing, and a system of t r a i l i n g v o r t i c e s i n the plane of the f r e e stream v e l o c i t y . The t r a i l i n g v o r t i c e s induce a downward v e l o c i t y over the wing, c a l l e d the downwash w^, which a l t e r s the d i r e c t i o n of the onset f l o w , and thus reduces the e f f e c t i v e s e c t i o n a l angle of a t t a c k by an amount known as the induced angle of a t t a c k a^. The s e c t i o n a l l i f t c o e f f i c i e n t i s t h e r e f o r e given by C l = m o ( a a + a i > " TTT-^pU c The s e c t i o n a l l i f t i s s t i l l given by the Kutta-Joukowsky Law, L = pOT (2) S u b s t i t u t i o n of equation (2) i n t o equation (1) g i v e s 5 C = ~ = m ( a + a.) (3) 1 Uc o a 1 In t h i s equation, only the c i r c u l a t i o n i s unknown, s i n c e the induced angle of a t t a c k may be expressed i n terms of the c i r c u l a t i o n by the B i o t - S a v a r t Law. Thus w i 1 f dr/dy J -b Jo Glauert (7) has shown that the c i r c u l a t i o n of a f i n i t e wing may. be expressed i n terms of -the F o u r i e r s e r i e s r = 4bU £ A S i n ny (5) n=l where U i s the f r e e stream v e l o c i t y , b the wing semi-span, and y i s defined i n terms of the spanwise coordinate by y = b Cos y (6) S u b s t i t u t i o n of equations (5) and (4) i n t o (3) leads to the fundamental equation f o r the unknown c o e f f i c i e n t s A R 00 QU E A S i n ny [ Siny + n ] = a Siny (7) , n m c a n=l o 6 where c, m and a are the s e c t i o n a l chord, l i f t curve slope and absolute o a ' * . angle of attack. The parameters c, mQ and may vary along the span, depending on whether or not the wing has section changes, taper and twist. The equation must be s a t i s f i e d f o r a l l points between 0 and TT for y. Solution f o r the unknown A n's i s achieved by considering a f i n i t e number of terms of the Fourier s e r i e s , say m terms. By choosing the number of span v a r i a b l e s y, equal to the number of terms i n the truncated Fourier s e r i e s , a system of m equations i n m unknowns i s ob-tained, and hence a s o l u t i o n f o r the unknown A 's. Since only m terms of n the Fourier s e r i e s are considered, equation (7) i s s a t i s f i e d for only m points along the span. Wieselsberger (8) has shown that i t i s necessary to include the wing t i p s i n c a l c u l a t i o n s f o r flapped wings. Since equation (7) i s degenerate at the t i p s where y i s equal to 0 or TT, L'Hospital's r u l e must be applied to obtain the r e s u l t a (0) = E n 2A a , n n=l oo a (ir) = E n A ( - l ) n + 1 a , n n=l (8) Once the s o l u t i o n f o r the unknown Fourier c o e f f i c i e n t s i s found, the l i f t L, r o l l i n g moment R, and the p i t c h i n g moment about the o r i g i n , may be computed. Since dL = L'dy = pUT(y) dy 7 dM = ( -L* x + M ) dy (9) 0 ac ac J ' dR = L*y dy L,R, and Mg are obtained by i n t e g r a t i n g across the span. The p i t c h i n g moment about the o r i g i n M Q i s r e l a t e d to the p i t c h i n g moment about the aerodynamic center M by M 0 = "LXAC + MAC <10> Since the r e s u l t s are to be compared w i t h r e f l e c t i o n plane experiments, the i n t e g r a t i o n s are made over the semi-span. The r e s u l t s may be expressed i n c o e f f i c i e n t form by means of the formulae C = L L % P u 2 s MAC CMAC = - T - < n > %pU Sc c - R R %pU 2Sb where S i s the planform area of the h a l f wing. When equations (9) are i n t e g r a t e d and s u b s t i t u t e d i n t o equations (11), the f o l l o w i n g r e s u l t s are obtained: 8 LR S A ( 2 i - 1 ) (2i-3)(2i+l) U J ; 3 C Mn = \ f «=2C dy - Cos a MO 2 J mac 2 r TT/2 m x E A,„. 1 N S i n ( 2 i - l ) u Sinu dy (14) ac . n ( 2 i - l ) i = l o In equation (14),. a i s the angle of attack of the wing root s e c t i o n . 2.1.2 The Jones Edge Correction Factor Jones (6) has proposed a c o r r e c t i o n for the l i f t i n g l i n e theory, which i s known to overpredict l i f t for the smaller aspect r a t i o s . In wing section theory the Kutta condition, which determines c i r c u l a t i o n and hence l i f t , depends on the edge v e l o c i t y induced by the r e l a t i v e normal v e l o c i t y of the s e c t i o n . Jones has shown that the v e l o c i t y p o t e n t i a l on the surface of an e l l i p t i c disk, l y i n g i n the xy plane with i t s center at the o r i g i n of the coordinate system, and moving with u n i t v e l o c i t y i n z d i r e c t i o n , i s given by k (|>e = '-f Sin T -1 where T i s the chordwise v a r i a b l e given by T = cos ( k / k g ) , E i s the' semi-perimeter of the disk divided by the semi-span, k i s the chordwise coordinate, and k^ the chordwise coordinate of the edge of the e l l i p t i c 2 2 disk, the value of which i s given by k^ = c g ( 1 - y /b ). Here y i s the spanwise coordinate, b the length of the semi-major axis (semi-span) and c^ the length of the semi-minor axis (root semi-chord), of the e l l i p t i c d i s k . . For an i n f i n i t e disk, E = 1. Thus the r a t i o s of the edge v e l o c i t i e s f o r the i n f i n i t e and f i n i t e e l l i p t i c disks i s 1/E. The f a c t o r E i s c a l l e d the Jones Edge Co r r e c t i o n Factor. I t i s an exact c o r r e c t i o n for e l l i p t i c wings, but i s approximate for other planforms. From t h i s c o r r e c t i o n i t can be seen that the s e c t i o n a l values of l i f t and c i r c u l a t i o n must be reduced by a f a c t o r of 1/E i n three dimensional flows. The various equations i n Section 2.2.1 must be modified accordingly. In p a r t i c u l a r equation (7) must be rewritten as, oo z n=l A Sin ny [ 8bE Siny + n ] = a Siny (7a) n m c o 10 2.2 A p p l i c a t i o n to Wings with Spoilers Experimental and t h e o r e t i c a l i n v e s t i g a t i o n s of two dimensional a i r -f o i l s with s p o i l e r s (2), have shown that the e f f e c t of the s p o i l e r on the s e c t i o n a l c h a r a c t e r i s t i c s i s to a l t e r the l i f t curve slope m , the zero o l i f t angle of attack a ^ Q » the aerodynamic center ^ a Q> and the p i t c h i n g moment about the aerodynamic center, M ( The absolute angle of attack ac i s r e l a t e d to the geometric angle of attack by ( a & = a - a^ 0))» l e t m and a, be the l i f t curve slope and zero l i f t angle of attack of the os l o s • K & s p o i l e r e d sections. Then i n applying equation (7a) to a wing with a part span s p o i l e r , a , the absolute angle of attack of the spoil e r e d section 3.S (a = a - a n ), and m must replace a and m over the spoi l e r e d wing cLS X O S O S 3. O sections. This r e s u l t s i n d i s c o n t i n u i t i e s i n the l i f t curve slope and angle of attack d i s t r i b u t i o n s across the wing. For an i n f i n i t e Fourier s e r i e s , the p o s i t i o n s of the d i s c o n t i n u i t i e s are exactly f i x e d , since a l l values of the span v a r i a b l e y are covered by the s e r i e s . For a f i n i t e Fourier s e r i e s however, the values of the l i f t curve slope and angle of attack must change from one value to another over two adjacent values of . y. This may be considered to be a gradual change i n s e c t i o n a l l i f t curve slope and angle of attack over a f i n i t e range of y, and i s an approximation of the r e a l s i t u a t i o n . The p o s i t i o n s of each d i s c o n t i n u i t y may be made to l i e midway between two adjacent values of y. Once equation (7a) i s solved, the various aerodynamic c o e f f i c i e n t s may be found by applying equations (12) to (14) . For an untwisted rectangular wing of constant section with part span s p o i l e r s , equation (14) may be further s i m p l i f i e d . Let C and x be the p i t c h i n g IUcLCS 3 . C S moment c o e f f i c i e n t about the aerodynamic center, and the aerodynamic center of the sp o i l e r e d sections. C , the p i t c h i n g moment c o e f f i c i e n t mac about the aerodynamic center of the unspoilered sections, i s constant since the wing i s of constant s e c t i o n . Also, x = 0 for a rectangular ac wing over the unspoilered sections. Hence equation (14) reduces to m - y2 x E A / Sin ( 2 i - l ) u Siny dy (15) i = l y 1 c where b i s the s p o i l e r span, y and y„ are the spanwise p o s i t i o n s of the inner and outer s p o i l e r t i p s . .Similarly y^ and y^ are the p o s i t i o n s of the inner and outer s p o i l e r t i p s i n terms of the span v a r i a b l e y. SlAC m a y ^ e obtained from C by applying equation (10). The s p o i l e r e d s e c t i o n parameters m , a, , C , and x may be os l o s macs acs cal c u l a t e d using any of the previously mentioned theories f o r s p o i l e r s . In the present work, Brown's l i n e a r i s e d t h i n a i r f o i l theory f o r s p o i l e r s (2) i s used. Brown has developed the theory to predict l i f t only. Bernier (9) has extended the theory to include the p r e d i c t i o n of the p i t c h i n g moment c o e f f i c i e n t . 2.3 Brown's Thin A i r f o i l Theory for Spoilers The a i r f o i l section of chord c, i s positioned in the physical plane as shown in Figure 2, with i t s leading edge at the origin. The free stream velocity U, i s in the positive x direction, and the a i r f o i l i s inclined at a small angle a to the free stream. The spoiler, inclined at an angle 6 to the chord, i s of height h. Its base i s located at a distance s from the leading edge. The a i r f o i l may also have a flap of chord c^ deflected at an angle T) to the chord. The wake behind the spoiler i s modelled as a constant pressure cavity of f i n i t e length. The pressure i n the cavity may be defined i n terms of the base pressure coefficient, which i s given by where P £ i s the pressure in the spoiler wake, or in terms of the Cavitation Number, K given by The a i r f o i l - c a v i t y combination i s of total length %. In the linearised physical plane as shown i n Figure 3, the wetted a i r f o i l surface and cavity boundary occupies a s l i t on the positive x axis. The f i e l d i n the linearised physical plane i s mapped conformally onto the upper half £-plane external to a unit semicircle centered at the o r i g i n by a s e r i e s of transformations. The wetted a i r f o i l surface i t s e l f i s mapped onto the unit s e m i c i r c l e , while the c a v i t y boundaries occupy the r e a l axis external to the unit s e m i c i r c l e . The flow model adopted i s s i m i l a r to that of Parkin (10), who has extended the complex a c c e l e r a t i o n p o t e n t i a l method of Biot (11) to solve the f o i l - c a v i t y problem. As the theory i s l i n e a r i s e d , the various geometric parameters of the a i r f o i l may be considered separately and superimposed to give the complete s o l u t i o n . Complex a c c e l e r a t i o n p o t e n t i a l s s a t i s f y i n g the boundary conditions imposed by the geometric parameters of the a i r f o i l are found i n the £-plane and superposed to give the complete s o l u t i o n f o r the a i r f o i l with s p o i l e r . The s o l u t i o n i n the Z-plane i s found by matching corresponding points i n the Z and ^-planes. The a i r f o i l c o e f f i c i e n t s may be obtained by applying the Blas i u s Equations The aerodynamic center and the p i t c h i n g moment c o e f f i c i e n t about the aerodynamic center are found using (16) -x C acs mos c macs (17) dC -x mos acs dC 1 c 2.3.1 The A c c e l e r a t i o n P o t e n t i a l Newton's second law f o r an incompressible f l u i d element suggests the existence of an a c c e l e r a t i o n p o t e n t i a l <}>, which has the property Vcf) = a = -V -Biot (9) has shown that the a c c e l e r a t i o n p o t e n t i a l function i s harmonic, so that a conjugate function ip, the a c c e l e r a t i o n stream function e x i s t s as w e l l as the complex a c c e l e r a t i o n p o t e n t i a l F(Z) = <j> + Hp For a small perturbation to the f r e e stream v e l o c i t y U, the Euler and Cauchy-Riemann equations are r e l a t e d to <j> and ip by the l i n e a r f i r s t order d i f f e r e n t i a l equations, + u 5 u = D 3 i 9 t 8x 8x 9 t + U 3x U 9x (18) where u, w, (f> and are non-dimensional but not U, x, z, or t . For steady flows, equations (18) may be integrated to give u = cj> + K/2 (19) w = - \> where K i s the c a v i t a t i o n number. The constants of i n t e g r a t i o n are determined by choosing the constant value of (j) to be equal to zero on the c a v i t y boundaries, and by the conditions at i n f i n i t y . The l i n e a r i s e d pressure c o e f f i c i e n t i s given by C p = -2u = -2(f) - K (20) 2.3.2 Conformal Transformations The a i r f o i l i n the l i n e a r i s e d p h y s i c a l Z-plane i s shown i n Figure 3. The s p o i l e r base and t i p are at x = s and x = t r e s p e c t i v e l y . The f l a p hinge point i s at x = n, the t r a i l i n g edge at x = c, and the c a v i t y termination i s at x = H. The f i r s t transformation maps the ca v i t y termination to i n f i n i t y , and the point at i n f i n i t y to -1. The second transformation h 9 r * V = a ( z' ) where a =( ) maps the e n t i r e z' plane onto the upper h a l f of the v-plane. The a i r f o i l l i e s between -1 _< v _< b on the r e a l a x i s . The upper and lower boundaries of the c a v i t y l i e on the r e a l axis between b <_ v <_ <*>, and -°° _< v <^  -1 r e s p e c t i v e l y . The f i n a l transformation V = ^  ( C + | ) - ^ where b = a ( ^  ) i s a Joukowsky transformation which maps the wetted a i r f o i l surface onto a u n i t s e m i c i r c l e centered at the o r i g i n . The c a v i t y boundaries remain on the r e a l axis, external to the un i t s e m i c i r c l e . By combining the transformations, the equation Z = fco = l a ~ 2 [ ^(b+pq+c"1) - ha-b)]z . 1 + a " 2 [ Mb+l)(C+? _ 1) - ha-b)}2 i s obtained. Major points of i n t e r e s t i n the £-plane are a) the a i r f o i l nose 6 o - C o s _ 1 < r i r > » b) the s p o i l e r base c) the f l a p hinge point d) the point at i n f i n i t y 2.3.3 Boundary Conditions The boundary conditions f o r the problem are: ( i ) cf) = 0 on the c a v i t y boundaries; ( i i ) the Kutta condition, cj) i s continuous at the s p o i l e r t i p and the a i r f o i l t r a i l i n g edge; ( i i i ) the a i r f o i l surface normal boundary condition «--< - S = ( i v ) the boundary cond i t i o n at i n f i n i t y F(Z) = -K/2 ; (v) the body-cavity system to be the equivalent of a s i n g l e closed body. In p o t e n t i a l flow, t h i s equivalent body must have zero drag. Equation (16) gives Im F(Z)dZ = 0 2.3.4 Flow Model Complex a c c e l e r a t i o n p o t e n t i a l functions are found i n the £-plane to s a t i s f y the boundary conditions enumerated i n Section 2.3.3. Separate functions f o r incidence, camber, thickness, s p o i l e r and f l a p , are found and superimposed to give the complete s o l u t i o n f o r the a i r f o i l . The complex ac c e l e r a t i o n p o t e n t i a l functions i n the various planes are i n v a r i a n t at corresponding p o i n t s . The accelerations d i f f e r only by the d e r i v a t i v e s of the mapping functions. Thus The functions given below s a t i s f y the boundary conditions ( i ) to ( i i i ) . The s i g n i f i c a n c e of the various terms i n each function i s explained i n Reference (2). Incidence function dF dF dZ dZ d? + iD o (22) Camber function Thickness function Spoiler function Flap function 19 F c(?) 00 M - i Z -J (23) (C-e i e o)( ?-e- i e°) n=0 ? n oo N E (24) S IT 161 19] L ? e i 6 o - l ce" 1 6 0-! + l n 16, r-e (25) F f (0 = ~ i(82-7T) i(e 2 - i r ) / C-e 16: + L C e i 0 o - l ' ?e _ i e°-l + In -19 5 (26) In these equations D q , M , and Nfl are r e a l constants whose values are given by : IT - - i / dy -3-=- d9 + c ; dx o M n TT "I/ dx Cos n9 d9 5 N TT - I I dy t - j — ( Cos9 - Cos9 ) d0 ; dx o ' ' TT n IT J dx ( Cos9 0 - Cos9 ) Cos n6 d9 n > 1 ; where y £ and y are functions representing the camber and thickness of the a i r f o i l . B q and C q are r e a l constants whose values are determined by boundary condition ( i v ) , Fla ( 5l» Fc ( Ci ) + Ft^ 1)+F 8(C 1)+F f(C 1) - - * , (27) where £ i s the point at i n f i n i t y . The unknown constants are contained i n F ± n ( ? i ) - The r e a l and imaginary parts of equation (27) give two simul-taneous equations, which are solved to give where B = o RlXitlmE - (a-%M )] - ImXiRlE + ^KlmXi RlXiImX 2 - ImXiRlX 2 R l E - B R1X 2- %K c — ; •RlXi (28) Xi = i C.e 1 9"-! i C ±e- i e'-l + 1 *2 = i . < C - f- ) E = _ F c ( 5 ± ) _ F t ( 5 i ) _ F g ( C i ) _ F f ( q ) The remaining unknowns are the c a v i t y length I, and the c a v i t a t i o n number K. The c a v i t a t i o n number cannot be predicted t h e o r e t i c a l l y at the 21 present time. K and £ are r e l a t e d through boundary condition (v). Thus only K i s required as input to the theory. By choosing a contour such that • |z| » Z may be expressed as a Laurent s e r i e s expansion, and the closure condition becomes, Rl [ c o e f f i c i e n t z" 1 ] = 0 . The s o l u t i o n of t h i s equation i n terms of K and % i s equation (29), which i s given i n the appendix. An i t e r a t i v e technique must be used to solve t h i s equation, because 9 o , 0 i , and 62 are complex functions of The s o l u t i o n to the problem may now be completed by determining the pressure, l i f t , and p i t c h i n g moment c o e f f i c i e n t s . The pressure c o e f f i c i e n t i s obtained from equation (20). By adding the r e a l parts of equations (22) to (26), which are the ac c e l e r a t i o n p o t e n t i a l s and s u b s t i t u t i n g into equation (20), the pressure c o e f f i c i e n t i n the £-plane ±s obtained. Points on the a i r f o i l may be r e l a t e d to corresponding points on the c i r c l e by equation (21). Thus la~2[h (b+1) Cos6 - % ( l - b ) ] 2 1 + a " 2 [ ^ (b+l ) C o s 9 - h(l-b)}2 The l i f t and p i t c h i n g moment c o e f f i c i e n t s are obtained from equations (16) and the Laurent s e r i e s expansion of £. The equations f o r pressure, l i f t , and p i t c h i n g moment c o e f f i c i e n t are given i n the appendix. 22 2.3.5 Base Vented S p o i l e r s In some a p p l i c a t i o n s , base vented s p o i l e r s are used. T y p i c a l l y the base vent i s about 10% of s p o i l e r height. Although the p o s s i b i l i t y e x i s t s that Jandali's t h i c k a i r f o i l theory or Brown's numerical t h i c k a i r f o i l theory may be modified to include the e f f e c t s of base venting, t h i s course of a c t i o n i s not attempted i n the present work. Instead the e f f e c t s of base venting are examined experimentally. A two dimensional Joukowsky a i r f o i l of 11% thickness and 2.4% camber i s tested with a s e r i e s of base vented s p o i l e r s of height equal to 10% of chord. The base vents on the s p o i l e r s ranged i n s i z e from 10 to 50% of s p o i l e r height. The r e s u l t s of the base venting experiments are presented i n Figures 11 to 13. These f i g u r e s show that f o r base vents of about 10% of s p o i l e r height or l e s s , the vented s p o i l e r c h a r a c t e r i s t i c s are quite close to the unvented. Thus i t may be concluded that f o r base vents of about 10% of s p o i l e r height or l e s s , the unvented s p o i l e r e d s e c t i o n c h a r a c t e r i s t i c s are close enough to the vented to be used f o r preliminary design purposes. 23 2.4 Experimental Two Dimensional Base Pressures Brown's theory, i n common with the other spoiler theories mentioned previously, requires as input, the base pressure coefficient behind the spoiler. At the present time, the base pressure coefficient cannot be predicted theoretically, so experimentally determined values must be used. Figure 39 shows the results of an experiment designed to find the base pressure coefficient behind a two dimensional a i r f o i l of NACA 0015 section f i t t e d with a normal unvented spoiler of height equal to 9.7% of chord. The spoiler i s f i t t e d to the a i r f o i l at positions ranging from 0.48 to 0.77 chord. Figure 40 presents a similar result for a 12.9% thick Clark Y a i r f o i l with a 10% unvented normal spoiler. The spoiler position along the chord ranged between 0.5 and 0.7 chord. If the experimental values of base pressure coefficient shown in the above mentioned figures are used directly in the theory, then a non-linear l i f t curve, inappropiate to a linear theory is obtained. To over-come this, Brown linearised the base pressure distribution. In the present work, an average value of base pressure coefficient, denoted by ^ s used for the following reasons: (i) Since the base pressure coefficient varies in a highly non-linear manner with angle of attack for most a i r f o i l s , the use of linearised values i s no more appropriate than the use of an averaged value. Predictions using both inputs are shown in Figures 15 and 16. It may be seen that good results may be obtained using either input. ( i i ) In taking three dimensional base pressure measurements behind f i n i t e span spoilers mounted on f i n i t e span wings, the downwash induced by the t r a i l i n g v o r t i c i t y reduces the effective angle of attack, so that a = a + a., where a. i s the induced angle of attack. In general a. e a x i ° i 24 w i l l vary along the span and i s not easy to measure. Thus i t w i l l - b e hard to l i n e a r i s e the three dimensional base pressures because of the d i f f i c u l t y associated with f i n d i n g the e f f e c t i v e angle of attack at which each spoilered section i s operating. The base pressure c o e f f i c i e n t C ^ I s averaged over the incidence range given by 0 _< a _< a s t a - Q » where a i s the angle of attack of the spoilered s e c t i o n , measured with respect to the zero l i f t angle of the unspoilered sec t i o n . The r a t i o n a l e behind t h i s choice of angles i s that i n p r a c t i c e , the s p o i l e r s w i l l only be used when the basic wing i s generating p o s i t i v e l i f t and operating below s t a l l . Values of C , for both the Clark Y and pb NACA 0015 a i r f o i l sections i s given i n Table I . AIRFOIL s/c AIRFOIL si c " 8 p b NACA 0015 0.48 0.552 CLARK Y 0.50 0.624 with 9.7% 0.58 0.559 with 10% 0.60 0.612 s p o i l e r s 0.68 0.562 s p o i l e r s 0.70 0.610 0.77 0.551 (12.9% thick) Table I. Values of C , for Two Dimensional Clark Y and NACA 0015 A i r f o i l s pb with Normal Unvented S p o i l e r s . T y p i c a l comparisons between theory and experiment are shown i n Figures 15 and 16. The p r e d i c t i o n of l i f t i s good. The p r e d i c t i o n of p i t c h i n g moment i s however l e s s accurate, because Brown's t h e o r e t i c a l model pr e d i c t s a s i n g u l a r i t y at the s p o i l e r base. This s i n g u l a r i t y , which i s c h a r a c t e r i s t i c of l i n e a r i s e d t h i n a i r f o i l t h eories, causes a p o s i t i v e increase i n the pi t c h i n g moment -prediction. A more accurate r e s u l t f o r p i t c h i n g moment 24a would be obtained by using thick a i r f o i l theories for spoilered a i r f o i l s , such as those of Jandali and Brown (3,4). In these theories, a stagnation point would replace the s i n g u l a r i t y at the spoiler base. However, even i f thick a i r f o i l theories are used, there would s t i l l be errors i n the p i t c h -ing moment prediction, because i n r e a l flows a separation bubble would be formed i n the region immediately i n front of the s p o i l e r , and the stagnation pressure would not be achieved. 25 2.5 Experimental F i n i t e Span Base Pressures Although Brown's theory i s i n good agreement with experiments, s e c t i o n a l parameters obtained from two dimensional base pressure inputs are inappropiate f o r use with f i n i t e span s p o i l e r s , which have wakes that are s i g n i f i c a n t l y d i f f e r e n t from two dimensional s p o i l e r s . Flow around the v e r t i c a l edges of a f i n i t e span s p o i l e r creates spanwise gradients of base pressure which are absent from two dimensional s p o i l e r flows. Base pressure c o e f f i c i e n t s for f i n i t e span s p o i l e r s may be l a r g e r or smaller than the corresponding two dimensional value, depending on the length to height r a t i o of the s p o i l e r , and on p o s i t i o n along the s p o i l e r span. Since sp o i l e r e d s e c t i o n parameters are dependent on base pressure, i t i s c l e a r that t h e i r values i n three dimensional flows w i l l d i f f e r from the two dimensional case. In t h i s context, i t should also be noted that the use of s p o i l e r e d s e c t i o n parameters derived from two dimensional ex-periments w i l l also be inappropriate. Such experimental values must be modified to account f o r the d i f f e r e n c e i n base pressure between two and three dimensional flows. This i s considered i n Section 2.6. Figures 41 to 43 show the r e s u l t s of t e s t s designed to f i n d the base pressures behind f i n i t e span unvented s p o i l e r s mounted normal to the wing surface.' Rectangular h a l f wings of NACA 0015 section and of equivalent aspect r a t i o s ranging from 3.87 to 7.73 are tested i n the r e f l e c t i o n plane c o n f i g u r a t i o n . S p o i l e r spans of 20, 30, 40, and 50% of wing semispan, and of height equal to 9.7% of chord are mounted on the wings at p o s i t i o n s varying from 0.48 to 0.77 chord. The inboard t i p s of the s p o i l e r s are always f i x e d at midspan. The height and chordwise l o c a t i o n of the s p o i l e r s are the same as f o r the two dimensional t e s t s mentioned i n Section 2.4. Figure 41 shows the e f f e c t of varying the chordwise l o c a t i o n of the s p o i l e r with the aspect r a t i o and s p o i l e r span held constant. Figure 42 shows the v a r i a t i o n of the base pressure c o e f f i c i e n t as a function of s p o i l e r span, with aspect r a t i o and s p o i l e r p o s i t i o n along the chord held constant. Figure 43 shows the v a r i a t i o n of base pressure c o e f f i c i e n t as a function of aspect r a t i o , with s p o i l e r l o c a t i o n along the chord and s p o i l e r percent of span held constant. The f i g u r e s give some idea about the complexity of the v a r i a t i o n of base pressures along the s p o i l e r span when changes are made to wing aspect r a t i o , s p o i l e r span, and s p o i l e r p o s i t i o n along the chord. For f i n i t e span wings with part span s p o i l e r s , the l i f t i n g l i n e equation (7a) i s solved for a f i n i t e number of terms, m by choosing m values of the span v a r i a b l e u, and forming a system of m equations i n m unknowns. Some of the m points w i l l f a l l on the spoilered sections of the wing. The base pressure c o e f f i c i e n t s at these points are used as inputs to the two dimensional theory to obtain the spoilered s e c t i o n parameters a. , m , x , and C which are required as inputs to equation (7a). los os acs macs ^ r ^ Since base pressure v a r i e s across the s p o i l e r span, the spoi l e r e d section c h a r a c t e r i s t i c s w i l l also vary. The s o l u t i o n of equation (7a) together with equations (12) to (15), give the aerodynamic c o e f f i c i e n t s of the spoil e r e d wing. Such t h e o r e t i c a l p r e d i c t i o n s are compared with experiments i n Figures 17 to 24. Agreement between theory and experiment i s seen to be good. 27 2.6 Empirical Relationships f o r Base Pressures The theory as developed to t h i s point requires as input the C ^ d i s t r i b u t i o n across the s p o i l e r span. This information must at present be obtained from wind tunnel t e s t s , since no p r e d i c t i o n methods are a v a i l a b l e . This i s a serious defect, since one of the theory's purposes i s to provide performance p r e d i c t i o n s for a v a r i e t y of wing-spoiler combinations without i n c u r r i n g the cost and time p e n a l t i e s associated with the wind tunnel t e s t i n g of every c o n f i g u r a t i o n . From the t h e o r e t i c a l p r e d i c t i o n s , the designer may s e l e c t the wing-spoiler combination most suited to h i s needs. This advantage i s l o s t i f base pressure d i s t r i b u t i o n s have to be measured experimentally f o r each c o n f i g u r a t i o n before the theory can be a p p l i e d . Furthermore, i t would be a r e l a t i v e l y simple task to take l i f t and moment measurements together with the base pressures. Experimental values of l i f t and moment c o e f f i c i e n t s , more accurate than the t h e o r e t i c a l p r e d i c t i o n s may then be c a l c u l a t e d , thus rendering the theory superfluous. The theory would be l i t t l e more than an i n t e r e s t i n g academic exercise. To be of u t i l i t y , some method must be devised to pre-d i c t the three dimensional base pressures behind the s p o i l e r , e i t h e r from experimental measurements or from theory. In the following sections, an attempt to predict the three dimensional base pressures i s made, based on empirical measurements. The method does not eliminate experimental determinations of base pressure c o e f f i c i e n t s e n t i r e l y , but rather reduces s u b s t a n t i a l l y , the amount of experimentation. 2.6.1 Averaged Across Span Figures 47 to 49 show values of averaged across the s p o i l e r span, which w i l l henceforth be denoted by p l o t t e d as a function of non- • dimensional s p o i l e r span b /h. The quantity b /h may be considered to be s s the s p o i l e r aspect r a t i o . The wing-spoiler combinations are the same as those mentioned i n s e c t i o n 2.5. The Figures show that for each chordwise l o c a t i o n of the s p o i l e r s , the value of C , remains nearly constant with pb respect to s p o i l e r aspect r a t i o , over the measured range 4 <^  b /h < 20. This may appear to be a s u r p r i s i n g r e s u l t , since i t may be expected that as the s p o i l e r aspect r a t i o i s increased, the flow would become increas-i n g l y two dimensional, and that the value of C ^ would approach the s e c t i o n a l value Hoerner (12) presents a s i m i l a r r e s u l t f o r f l a t p l a t e s normal to the flow, which i s a somewhat s i m i l a r flow to that of a s p o i l e r mounted on a wing. By combining the r e s u l t s from several sources, he showed that f o r 1 <^  b/h _< 10, where b/h i s the width to height r a t i o of the p l a t e , the drag c o e f f i c i e n t i s nearly constant. For values of b/h greater than 10, the drag c o e f f i c i e n t r i s e s slowly towards the two dimensional v a l u e . However, the two dimensional drag c o e f f i c i e n t i s not approached u n t i l b/h i s about 50 or more. Thus i t would appear that two dimensionality i n flow i s not approached u n t i l very high aspect r a t i o s are reached. Since the drag of a f l a t p l a t e normal to the flow i s mostly pressure drag, a constant value of drag implies that the average pressure over the f r o n t and rear of the p l a t e , and hence the base pressure, i s a l s o constant. I t may immediately be seen that the use of C ^ as input to the s e c t i o n a l theory, has the advantage that i f one s p o i l e r of aspect r a t i o between 4 and 20 i s tested on a f i n i t e wing, then the value of C ^ so obtained w i l l be v a l i d f o r a l l s p o i l e r s of the same height and angle of 29 d e f l e c t i o n 6, mounted on wings of the same section and at the same chord-wise p o s i t i o n , within the s p o i l e r aspect r a t i o range of 4 _< b g/h < 20. The amount of experimentation i s thus reduced to a si n g l e determination of C ^ f ° r each wing section, chordwise s p o i l e r p o s i t i o n , s p o i l e r height, and s p o i l e r i n c l i n a t i o n . I t should be noted that the s p o i l e r aspect r a t i o range of 4 < b g/h 20 i s quite wide, and w i l l l i k e l y cover a l l s p o i l e r lengths that may be used. A summary of the two and three dimensional base pressure c o e f f i c i e n t measurements i s given i n Table I I . The values of C ^ shown, are averages for a l l s p o i l e r lengths at the given chordwise l o c a t i o n . G i s defined as the r a t i o C_u/C_%_. AIRFOIL s/c -C , pb -C , pb G NACA 0015 with 0.48 0.552 0.495 0.896 9.7% unvented 0.58 0.559 0.473 0.846 normal 0.68 0.562 0.470 0.836 s p o i l e r 0.77 0.551 0.448 0.825 12.9% Clark Y 0.50 0.624 0.547 0.876 with 10% normal 0.60 0.612 0.519 0.848 unvented s p o i l e r 0.70 0.610 0.512 0.839 Table I I . Base Pressure C o e f f i c i e n t s f o r Spoilered NACA 0015 & 12.9% Clark Y A i r f o i l s Although Table I I shows.that the values of C p b are quite d i f f e r e n t f o r = =ft the two a i r f o i l s , a p l o t of C p b / C p b against s/c (Figure 50), where =* = i s the value of C ^ at the 0.7 chord p o s i t i o n reveals that the = v a r i a t i o n of C p b / with s p o i l e r p o s i t i o n along the chord i s almost the same f o r both a i r f o i l s . Also a p l o t of G/G* against s/c (Figure 50) where G* i s the value of G at the 0.7 chord p o s i t i o n , y i e l d s a s i m i l a r r e s u l t . The curves i n Figure 50 show that there i s a small decrease i n C ^ as s p o i l e r p o s i t i o n along the chord i s moved toward the t r a i l i n g edge. This decrease may be approximated by the l i n e a r r e l a t i o n C p b = C* b [ 1.0 - 0.445(s/c -0.7) ] (32) Thus need only be measured at s/c=0.7 f o r a given a i r f o i l s e c t i o n . For any other chordwise l o c a t i o n of the s p o i l e r between 0.5 <s/c _< 0.8 equation (32) may be used to f i n d C j^. I f the two dimensional base pressure c o e f f i c i e n t i s known at the 0.7 chord p o s i t i o n , then the r e l a t i o n -ship, C p b = 0.830 C p b [ 1 . 0 - 0.255 (s/c - 0.7) ] (33) _* _ may be used. C p b i s the value of C p b at the 0.7 chord p o s i t i o n . Equations (32) and (33) are v a l i d f o r both the Clark Y and NACA 0015 a i r -f o i l s , f i t t e d with 10% s p o i l e r s . Use of these equations i s suggested f o r other a i r f o i l s , s p o i l e r heights and i n c l i n a t i o n s , as i t i s u n l i k e l y that the v a r i a t i o n s w i l l be too d i f f e r e n t from the above. In Table I I I , the s e c t i o n a l c h a r a c t e r i s t i c s i n three dimensional-flows of the NACA 0015 sec t i o n f i t t e d with 9.7% unvented normal s p o i l e r s , as predicted by Brown's theory, i s given. The values of C ^ used as input are obtained by using equation (32) together with the value of C , pb from Table I I . -s/c pb m O S l o s C macs x /c acs 0.48 0.497 4.595 0.2116 0.0786 -0.0378 0.58 0.482 5.033 0.2049 0.1161 -0.0252 0.68 0.467 5.448 0.1997 0.1569 -0.0120 0.77 0.451 5.827 0.1960 0.1977 0.0010 Table I I I . Sectional C h a r a c t e r i s t i c s of a NACA 0015 A i r f o i l F i t t e d with 9.7% Unvented, Normal S p o i l e r s 2.6.2 V a r i a t i o n of Sectional Properties with Base Pressure In the previous section, i t i s shown that the use of C , , the base pb pressure c o e f f i c i e n t averaged across span and incidence, as input to the theory, has the advantage of reducing by a la r g e amount, the experimental determination of base pressurures. However the use of such an average w i l l only be v a l i d i f the s e c t i o n a l c h a r a c t e r i s t i c s vary l i n e a r l y with the base pressure. Although the theory i s l i n e a r , i t must not be expected that the v a r i a t i o n of s e c t i o n a l p r o p e r t i e s with base pressure w i l l also be l i n e a r . In the s a t i s f a c t i o n of boundary cond i t i o n (v) i n Section 2.3.3, a non-l i n e a r r e l a t i o n s h i p between K, the c a v i t a t i o n number, (and hence C^) and H, the c a v i t y length i s established through equation (29). The angles 60, 9 i , and 6 2 , which appear i n the complex a c c e l e r a t i o n p o t e n t i a l functions i n Section 2.3.4 are r e l a t e d to JL through the conformal transformations. Hence the s o l u t i o n to the problem i s dependent on I, which v a r i e s i n a non-linear manner with the base pressure c o e f f i c i e n t . Figures 4 to 6 show the v a r i a t i o n of s e c t i o n a l properties with base pressure c o e f f i c i e n t , for a NACA 0015 a i r f o i l s e ction f i t t e d with a normal unvented s p o i l e r of height equal to 9.7% of chord. The s p o i l e r i s mounted at chordwise l o c a t i o n s ranging from 0.48 to 0.77 chord. These fi g u r e s show that the s e n s i t i v i t y of section c h a r a c t e r i s t i c s to changes i n base pressure c o e f f i c i e n t i s decreased as the s p o i l e r p o s i t i o n along the chord i s moved towards the t r a i l i n g edge. Also, the v a r i a t i o n of sect i o n c h a r a c t e r i s t i c s i s only very s l i g h t l y non-linear. Thus the use of C . i s j u s t i f i e d , pb In the p r e d i c t i o n of r o l l i n g moment, an a d d i t i o n a l complication a r i s e s . Since the r o l l i n g moment about the o r i g i n i s given by, b p where y i s the spanwise coordinate, the s e c t i o n a l l i f t near the wing, t i p s w i l l contribute more to the r o l l i n g moment because of the weighting f a c t o r y. For example, i f the C ^ d i s t r i b u t i o n across the span i s such that the spoile r e d section l i f t 1/ increases as y increases, then the p r e d i c t i o n of r o l l i n g moment using C ^ w i l l be low. Conversely, i f the C ^ d i s t r i b u t i o n i s such that 1/ decreases as y increases, then the reverse w i l l , b e true. this effect i s minimised i f the l i f t distribution across the spoilered section i s symmetrical about the midspan of the spoiler. Fortunately this is approximately true, as Figures 41 to 46 show. In these figures the distribution across the spoiler span i s approximately symmetrical. Since the sectional l i f t varies in a nearly linear manner with this means that the sectional l i f t distribution across the span i s also approximately symmetric. A second factor which tends to minimise this effect i s the insensitivity of the l i f t to changes in the base pressure coefficient, 2.6.3 Use of Experimental Two Dimensional Spoilered Section Parameters in Finite Wing Theory The theory as developed to this point is capable of predicting the aerodynamic coefficients of a f i n i t e wing with spoiler, using only C , p b as an empirical input. Experimental spoilered section characteristics, i f available, may of course, also be used. It has already been shown, that the use of sectional characteristics derived from two dimensional tests in three dimensional theory is inappropriate because of d i f f e r -ences in base pressures between two and three dimensional flows. Experi-mental two dimensional section characteristics must be modified to account for the difference i n base pressures before they can be used i n three dimensional theory. In section 2.6.1, i t is shown that i f the two dimensional base pressure averaged over incidence, C , , i s known for a spoiler mounted at pb the 0.7 chord position, then the value of C ^ may be obtained by using equation (33). Theoretical predictions may then be made using C p^ and C , as inputs. The d i f f e r e n c e i n s e c t i o n a l c h a r a c t e r i s t i c s due to the pb d i f f e r e n c e i n base pressure c o e f f i c i e n t s may be c a l c u l a t e d . The d i f f e r -ences may then be deducted from the experimental two dimensional . s e c t i o n a l parameters to obtain values appropriate f o r three dimensional flows. 35 3. EXPERIMENTS The experimental part of t h i s thesis consists of three series of experiments. In the f i r s t , the effect of base venting on spoilered section c h a r a c t e r i s t i c s i s examined. In the second the forces and moments generated by f i n i t e wings with part span spoilers are measured, and aerodynamic c o e f f i c i e n t s calculated. In the t h i r d , the base pressure d i s -t r i b u t i o n behind spoilers mounted on two and three dimensional wings i s measured. » 3.1.1 Base Venting Experiments The purpose of these experiments was to determine the effect of base venting on section c h a r a c t e r i s t i c s . A Joukowsky a i r f o i l of 11% thickness, 2.4% camber and 12.08 inch chord was used. The a i r f o i l was constructed mainly of wood, with an aluminium center section containing 37 pressure taps of which 24 were on the upper surface. Since the Joukowsky p r o f i l e was s t r u c t u r a l l y weak near the cusped t r a i l i n g edge, the upper surface i n t h i s region was modified to give an approximately constant thickness of 1/8 inch. The modified p r o f i l e i s shown i n Figure 7. The a i r f o i l was f i t t e d with end plates on which spoilers could be mounted at the 0.5, 0.6, 0.7, 0.8 and 0.9 chord positions, normal to the a i r f o i l surface. This a i r f o i l was used by Jandali to v e r i f y his s p o i l e r theory, and a f u l l description of i t i s given i n reference (3). A set of 5 spoilers of height equal to 10% of chord, with base vents of 10,20,30,40 and 50 % of spoiler height were made for the a i r f o i l . The tests were conducted i n the small low speed aeronautical wind tunnel i n the Department of Mechanical Engineering at the University of 36 B r i t i s h Columbia. I t has a tes t s e c t i o n of 27 inch height and 36 inch width. The tunnel has good flow uniformity and a turbulence l e v e l of l e s s than 0.1 percent over i t s speed range. The a i r f o i l was mounted v e r t i c a l l y and spanned the t e s t s e c t i o n , with small clearances at the roof and f l o o r . The a i r f o i l was attached to a s i x component pyramidal balance located under the tunnel, at the quarter chord p o s i t i o n . Force and moment measurements were taken with the s p o i l e r s attached at the 0.5, 0.6, 0.7 and 0.8 chord p o s i t i o n s , over a f u l l angle of attack range. Pressure measurements were also taken at some angles of attack using a multi-tube manometer. Test Reynolds number was 4.4 ( 1 0 ) 5 . 3.2 F i n i t e Wing Experiments For the f i n i t e wing experiments, h a l f wing models were used i n order to o btain a good range of aspect r a t i o s , with as large a chord and Reynolds number as p o s s i b l e . Rectangular wings of NACA 0015 section were mounted v e r t i c a l l y at the quarter chord p o s i t i o n , i n the same tunnel-balance system mentioned i n Section 3.1. The wings were machined from s o l i d aluminium i n spanwise sections of 0.5 and 2.0 inches. The chord was 5.17 inches. By combining appropriate numbers of each of the two s i z e s of span-wise sections, h a l f wing models of equivalent f u l l aspect r a t i o s of 3.87, 4.83, 5.80, 6.77, and 7.73 were assembled. Holes were d r i l l e d and tapped on the upper surface of the wing, so that s p o i l e r s of 20, 30, 40 and 50 % of ha l f span, could be mounted at the 0.48, 0.58, 0.68 and 0.77 chord p o s i t i o n s . The unvented s p o i l e r s , of height equal to 9.7% of chord were mounted on the wings so that i n a l l cases, the inboard t i p of the s p o i l e r was positioned at midspan. Force and moment measurements were made f o r a l l p o s s i b l e 37 c o n f i g u r a t i o n s over a f u l l range of angle of a t t a c k . In a d d i t i o n , two dimensional t e s t s were made to o b t a i n the s p o i l e r e d s e c t i o n c h a r a c t e r i s t i c s . A two dimensional wing was made by assembling the 5.17 i n c h chord s e c t i o n s i n t o a wing spanning the t e s t s e c t i o n of the wind tun n e l v e r t i c a l l y , w i t h s m a l l clearances a t the roof and f l o o r . F u l l span 9.7% normal unvented s p o i l e r s were mounted on the wing a t the 0.48, 0.58, 0.68 and 0.77 chord p o s i t i o n s . Force and moment measurements were made. The two dimensional t e s t s were made i n order to o b t a i n comparisons between experiment and the p r e d i c t i o n s of the Brown theory. Test Reynolds number was 3 (10)"*. The NACA 0015 s e c t i o n i s shown i n Fi g u r e 8. 3.3 Base Pressure Measurements. Measurements of the base pressures behind the s p o i l e r s f i t t e d to the two and three dimensional wings t e s t e d i n S e c t i o n 3.2 were taken i n t h i s s e r i e s of experiments. This i n f o r m a t i o n was r e q u i r e d as input to the s e c t i o n a l theory. An a d d i t i o n a l set of base pressures was taken u s i n g 12.9% t h i c k Clark Y wings of 5.9 i n c h chord. The wings were made of wood i n spanwise s e c t i o n s of 3, 6, and 12 inc h e s , which were assembled to g i v e a two dimensional model spanning the t e s t s e c t i o n , and r e c t a n g u l a r h a l f wings w i t h equivalent aspect r a t i o s of 4.07, 6.10, and 7.12. I n a l l t e s t s w i t h the C l a r k Y wing, 10% unvented, normal s p o i l e r s were used. They were taped to the surface of the wings a t the 0.5, 0.6 and 0.7 chord p o s i t i o n s . For the two dimensional t e s t s , the s p o i l e r s were f u l l span. For the f i n i t e wing t e s t s , the s p o i l e r s were of l e n g t h equal to 20, 30, 40, and 50% of the h a l f wing span. They were mounted on the wings so that the in n e r t i p of the s p o i l e r s were always f i x e d a t midspan, as was the case 38 for the NACA 0015 wings. The Clark Y section i s shown i n Figure 9. For the two dimensional t e s t s , a s i n g l e pressure tap, located at midspan, halfway between the s p o i l e r base and the t r a i l i n g edge, was used. For the three dimensional t e s t s , ten taps, equally spaced i n the spanwise d i r e c t i o n , s t a r t i n g at a point 5% of s p o i l e r span away from the inner s p o i l e r t i p and moving towards the outer s p o i l e r t i p was used. The l o c -a t i o n of a l l ten taps i n the chordwise d i r e c t i o n was halfway between the s p o i l e r base and the t r a i l i n g edge. The base pressures were measured using the system shown i n Figure 10. Whenever a set of base pressures was to be measured, the external s t a r t switch was depressed. This acti v a t e d the PDP-11 computer, which i n s t r u c t e d the scanivalve to begin scanning the tunnel dynamic pressure taps, and the pressure taps behind the s p o i l e r . The pressures at each tap were seq u e n t i a l l y transmitted by the scanivalve to the barocel, which converted the pressures i n t o voltages. The analog to d i g i t a l converter then d i g i t a l i s e d the voltages and transmitted them to the computer memory f o r storage. A f t e r a l l the pressure taps had been scanned, the base pressure c o e f f i c i e n t s were computed and printed on a typewriter. Test Reynolds Number was 3 ( 1 0 ) \ 3.4 Wind Tunnel Wall Corrections For the Joukowski a i r f o i l , the wind tunnel w a l l c o r r e c t i o n technique employed was the same as that of J a n d a l i (3), who used the corrections established by Pope and Harper (13). The non-dimensional wake blockage term was however, modified to %(c/H)C,, instead of %(c/H)C,, as J a n d a l i d a found that measurements for a i r f o i l s of varying chord lengths collapsed better using the modified term. For pressure c o e f f i c i e n t s , J a n d a l i used the equation, (34) where C and C, are the true pressure and l i f t c o e f f i c i e n t s at a given P 1 angle of attack, and C and C, are the uncorrected pressure and l i f t pu l u c o e f f i c i e n t s . The data f o r the Clark Y and NACA 0015 a i r f o i l s were not corrected f o r wind tunnel w a l l e f f e c t s because of the small s i z e of the wings i n r e l a t i o n to the tunnel dimensions (c/H < 0.2, S/C < 0.2). The corrections to the various c o e f f i c i e n t s were l e s s than 3% and were therefore ignored. 4. RESULTS AND COMPARISONS 4.1 Base Venting Experiments A sample of the r e s u l t s of experiments using the two dimensional Joukowsky a i r f o i l f i t t e d with 10% normal s p o i l e r s with base vents of various s i z e s i s presented i n Figures 11 to 13. I t may be seen from these f i g u r e s that the e f f e c t of increasing the s i z e of the base vent i s to increase l i f t . For the l a r g e r base vents, t h i s e f f e c t i s s u b s t a n t i a l at the lower angles of attack. At higher incidence, the curves converge towards the unvented r e s u l t . For small base vents of around 10% of s p o i l e r height or l e s s however, the increase i n l i f t over the unvented s p o i l e r i s small at a l l angles of attack. The same r e s u l t i s true f o r the pi t c h i n g moment about the aerodynamic center. Thus f o r base vents of about 10% s p o i l e r height or l e s s , the unvented s p o i l e r c h a r a c t e r i s t i c s are s u f f i c i e n t l y close to the vented to be used f or preliminary design purposes. 4.2.1 Two Dimensional NACA 0015 A i r f o i l Experiments Figure 14 shows the unspoilered NACA 0015 sec t i o n c h a r a c t e r i s t i c s . The l i f t curve shows some n o n - l i n e a r i t y at the higher angles of attack and the l i f t curve slope i s s i g n i f i c a n t l y lower than the t h e o r e t i c a l . These e f f e c t s are due to the low Reynolds number [ 3(10)"*] at which the the t e s t s were conducted. For the t h e o r e t i c a l f i n i t e wing pr e d i c t i o n s therefore, l i n e a r i s e d experimental values of the section parameters mQ, a, , and C are used, l o mac 41 Figures 15 and 16 are.comparisons between experimental and theoretical section characteristics of the NACA 0015 a i r f o i l , with 9.7% unvented normal spoilers mounted at the 0.48 and 0.68 chord positions. Two theoretical curves are shown. The broken lines represent the prediction using a base pressure coefficient linearised with respect to incidence, and the unbroken lines represent the prediction using C , , the base pressure coefficient pb averaged over incidence. The use of these inputs i s discussed in Section : 2.4. Both inputs give good predictions for l i f t . The prediction for pitch-ing moment i s less accurate than for l i f t . The reasons for this are already discussed in Section 2.4 4.2.2 Rectangular Wings of NACA 0015 Section Fitted with Part Span Spoilers Figures 17 to 20 show experimental and theoretical l i f t and pitching moment comparisons for rectangular half wings of equivalent aspect ratio equal to 7.73, fi t t e d with part span spoilers. The unvented 9.7% normal spoilers are f i t t e d to the wings at the 0.48 chord position. Their spans are equal to 20, 30, 40 and 50% of semi-span. The spoilers are mounted so that the inboard tip of the spoilers are always at mid-span. Figures 21 to 24 are the corresponding r o l l i n g moment coefficients for the above mentioned half wings. Since the tests are made with half wing models, the wing r o l l i n g moment i s defined as R J5PU2Sb rather than the more usual definition of 42 R ?5pU2S(2b) as the former definition i s more appropriate to half wing tests. In Figures 17 to 24, two theoretical curves are shown. The broken lines represent the prediction using the base pressure coefficient averaged over incidence but varying across the spoiler span. The solid lines represent the prediction using C ^  , the base pressure coefficient averaged over both incidence and spoiler span. C , is obtained from equation (32), with the value of C , pt) pb coming from Table 2. The use of these two inputs i s discussed in Sections 2.5 and 2.6, where the advantages of using C ^  are discussed. The pred-ictions given by both inputs are seen to be very close. This confirms that the use of C ^  as predicted by equation (32) i s sufficiently accurate for-preliminary design purposes. Figures 25 to 32 are similar to Figures 17 to 24, except that the equivalent aspect ratio i s 3.87, and the spoilers are mounted at the 0.68 chord position. Only one theoretical prediction i s shown, that using C . pb as the input to the sectional theory. In a l l of the above mentioned cases, the prediction of l i f t and r o l l for the f i n i t e rectangular wings i s seen to be good. The prediction of pitching moment i s less accurate. This i s to be expected, since the sectional theory's prediction of pitching moment i s less accurate than for l i f t . Any inaccuracies i n the prediction of sectional characteristics, w i l l of course be carried over into the three dimensional theory. Figure 33 compares the predicted and measured variation with respect to relative spoiler span b g/h, of the effective moment arm of the incremental 43 l i f t caused by s p o i l e r e r e c t i o n . The f i g u r e r e f e r s to wing s p o i l e r combinations which are the same as f o r Figures 17 to 24. The data i s presented i n the form AC D/AC T, plo t t e d against b /b. Figure 34 i s s i m i l a r K. . L S to Figure 33, except that the wing-spoiler combinations are the same as for Figures 25 to 32. The dashed l i n e s i n the two figures represent the v a r i a t i o n that would occur i f the incremental l i f t acted at the midspan of the s p o i l e r . The experimental values are averages over the incidence range of -4° to j u s t below s t a l l . The agreement between theory and experi-ment i s seen to be good. An i n s i g h t into the reason for the inward s h i f t of the e f f e c t i v e moment arm of the incremental l i f t may be obtained by examining Figures 35 and 36, which show the spanwise d i s t r i b u t i o n of the non-dimensionalised c i r c u l a t i o n r/4bU. The curves show c l e a r l y that the e f f e c t of the s p o i l e r i s not confined to the spo i l e r e d portions of the wing. There i s also a l o s s of c i r c u l a t i o n and hence l i f t , over the unspoilered sections of the wings. The l o s s of l i f t over the unspoilered sections of the wings i s l a r g e r inboard of the s p o i l e r . Hence the center of incremental l i f t i s s h i f t e d inwards, towards the wing root. Figure 37 shows the spanwise d i s t r i b u t i o n of the dimensionless ' c i r c u l a t i o n r/4bU f o r rectangular wings of NACA 0015 section and of aspect r a t i o 7.73, f i t t e d with symmetrically and asymmetrically deployed normal, 9.7% unvented s p o i l e r s of span equal to 40% of semi-span. The s p o i l e r s are mounted at the 0.48 chord p o s i t i o n , with the inboard t i p s of the s p o i l e r s positioned midway between the wing root and t i p . The lower curve i s f o r symmetrically deployed s p o i l e r s ( s p o i l e r s up on both h a l f wings). The middle curve i s for asymmetric deployment ( s p o i l e r up on one h a l f wing 44 but retracted on the other). The upper curve i s for the unspoilered wing. Figure 38 i s similar to Figure 37 except that the aspect ratio i s 3.87, and the spoilers are mounted at the 0.68 chord position. The two figures show that the curves for symmetric and asymmetric spoiler deployment are in close agreement. This implies that while the half wing tests s t r i c t l y correspond to cases of symmetric spoiler deployment for complete wings, they may be used to model cases of asymmetric deployment as well. 4.3.1 Two Dimensional Base Pressure Experiments Figure 39 shows the variation of base pressure with incidence for a r two dimensional a i r f o i l of NACA 0015 section f i t t e d with a 9.7% unvented normal spoiler. The position of the spoiler i s varied from 0.48 to 0.77 chord. Figure 40 i s similar to Figure 39, except that the section i s a 12.9% thick Clark Y, the spoilers are 10%, and the position of the spoiler i s varied between 0.5 and 0.7 chord. A comparison of the curves for the NACA section shows that a rearward shift i n spoiler position along the chord results in an earlier peak in the base pressure distribution. However, the average value of the base pressure coefficient C , , as pb defined in Section 2.4, does not appear to vary by a large amount. The same trends are apparent for the 12.9% thick Clark Y section. Although the spoilers for both a i r f o i l s are similar in geometry and height, the base pressure distributions are quite different. However, the values of C ^ a r e n o t t o ° f a r apart. 4.3.2 Three Dimensional Base Pressure Experiments Figures 41 to 43 show the spanwise distribution of C ^  , the base pressure distribution averaged over incidence, for part span spoilers mounted on rectangular wings of NACA 0015 s e c t i o n . The wing-spoiler configurations are the same as i n Section 3.3. In the fi g u r e s b i s the spanwise coordinate, measured with respect to the inner s p o i l e r t i p , and moving outwards, toward the outer s p o i l e r t i p . Figure 41 shows the e f f e c t of varying the chordwise p o s i t i o n of the s p o i l e r with aspect r a t i o and s p o i l e r span held constant. Figure 42 shows the v a r i a t i o n of C ^ as a function of s p o i l e r span, with aspect r a t i o and s p o i l e r p o s i t i o n along the chord held constant. Figure 43 shows the v a r i a t i o n of C , as a func-pb t i o n of aspect r a t i o , with s p o i l e r percent of span and chordwise s p o i l e r p o s i t i o n held constant. Figures 44 to 46 are p l o t s s i m i l a r to Figures 42 to 44 except that the section i s a 12.9% Clark Y. Figures 41 to 46 reveal the complex manner i n which C , v a r i e s pb across the s p o i l e r span, as the various parameters are changed. P l o t s of C p b against non dimensional s p o i l e r span b g/h rev e a l a considerable s i m p l i f i c a t i o n . Figures 47 49 show p l o t s of 5 , against b /h. These fi g u r e s / t h a t at each chordwise l o c a t i o n the value of C ^ remains nearly constant for each a i r f o i l , regardless of aspect r a t i o and s p o i l e r length. The implications of t h i s are discussed i n Section 2.6.1 CONCLUSIONS The use of the modified l i n e a r l i f t i n g l i n e theory i s shown to give good p r e d i c t i o n s of l i f t and r o l l i n g moment for f i n i t e wings f i t t e d with part span s p o i l e r s . The p r e d i c t i o n of p i t c h i n g moment i s not as good as for l i f t . This i s due to the f a c t that the Brown theory for f o r spoilered a i r f o i l sections gives p r e d i c t i o n s of p i t c h i n g moment which are l e s s accurate than for l i f t . A required input to the theory i s the base pressure c o e f f i c i e n t behind the f i n i t e span s p o i l e r s . At the present time, the base pressure c o e f f i c i e n t cannot be predicted t h e o r e t i c a l l y . Experiments conducted i n support of t h i s t h e s i s show that the base pressure c o e f f i c i e n t v a r i e s i n a complex manner with a i r f o i l and s p o i l e r geometry. However i t i s found that the base pressure c o e f f i c i e n t averaged over span and incidence, C , pb on any given a i r f o i l s ection, i s independent of s p o i l e r length, and i t s use as an input to the theory gives good r e s u l t s . A method of p r e d i c t i n g C ^ , which greatly reduces the amount of experimental measurements i s presented. The necessity for a base pressure input based on experimental measurements remains a weakness of the theory. Measurements must s t i l l be made for each s p o i l e r height and i n c l i n a t i o n and a i r f o i l s e c t i o n . In add i t i o n , i f the wing i s flapped, the f l a p angle and s l o t s i z e w i l l a f f e c t the base pressure. Since the experimental part of t h i s thesis deals only with 10% s p o i l e r s mounted on unflapped wings, no attempt can be made to develop e m p i r i c a l l y based formulas f o r the p r e d i c t i o n of base pressure, as changes are made to s p o i l e r height and i n c l i n a t i o n , flap angle and slot size. Further experiments w i l l have to be made before this can be attempted. Experiments on base vented spoilers show that base vents of about 10% of spoiler height or less give sectional characteristics which are l i t t l e different from those of the corresponding unvented spoilers. Thus for the purposes of preliminary design, the unvented spoiler characteristics may be used, provided that the spoilers have base vents of about 10% or less. 48 Z P L A N E F i g u r e 2. A i r f o i l i n the P h y s i c a l Plane Z P L A N E s t ; 1 n c I Z ' P L A N E -1 v P L A N E i a e b C P L A N E /Ae 1*' F i g u r e 3. Complex Transform Planes 10 9 m 0 S 8 3 2 \-S/C = 0-77  S/C = 0-6 8  S/C =0-58 S/C=0-48 h/c=0097 8 = 90° -0-30 -0-35 -0-40 -0-45 -0-50 -0-55 -0-60 -0-65 Figure 4 V a r i a t i o n of m with C , for NACA 0015 A i r f o i l Section & os pb with Normal Unvented Spoiler 52 h/c = 0-097 8=90° 0 0 1 n s/c=077 \j - 0 0 1 s/c=0-68 x a c s c - 0 0 2 '. ^ _ _ s / c = 0 - 5 8 h/c = 0 0 9 7 8=90° - 0 0 3 ~ ^ ~ - ^ s / c = 0 - 4 8 - 0 0 4 - 0 0 5 i i i i i i i -0-30 -0-35 - 0 - 4 0 -0 -45 -0 -50 -0-55 _ -0-60 -0-65 Cpb Figure 5 V a r i a t i o n of C & x Ic with C , for NACA 0015 A i r f o i l macs acs pb Section with Normal Unvented Spoiler F i g u r e 7 Modified Joukowsky A i r f o i l S e c t i o n of 11% Thickness & 2.4% Camber w i t h Base Vented Normal S p o i l e r F i g u r e 8 NACA 0015 A i r f o i l S e c t i o n , w i t h 9.7% Unvented Normal S p o i l e r F i g u r e 9 12.9% Thick C l a r k Y A i r f o i l S e c t i o n w i t h 10% Unvented Normal S p o i l e r J TO DYNAMIC PRESSURE TAPS 3 TO WING PRESSURE TAPS SCANIVALVE PDP-II COMPUTER WITH A/D CONVERTER BAROCEL OUTPUT I (VOLTS) EXT. START SW. F i g u r e 10 Base Pressure Measurement System 58 Figure 12 L i f t Coefficient for Joukowsky A i r f o i l Section with Base Vented Normal Spoilers -8 -4 0 4 8 12 16 20 a Figure 13 P i t c h i n g Moment C o e f f i c i e n t f o r Joukowsky A i r f o i l Section with Base Vented Normal Sp o i l e r s J-.l-O Figure 14. L i f t & P i t c h i n g Moment C o e f f i c i e n t s for a NACA 0015 Section l i n e a r i s a t i o n ; o experimental C. ; A experimental C 62 h/c = 0-097 s/c = 0-48 0-3J--I-2 Figure 15 L i f t & Pit c h i n g Moment C o e f f i c i e n t s f o r NACA 0015 A i r f o i l Section with Normal Unvented S p o i l e r . theory (C , input); theory pb ( l i n e a r i s e d C ^ inp u t ) , o C^ experimental;A C m a c s experimental h/c = 0-097 s/c=0-68 -0-3 J—1-2 Figure 16 L i f t & Pit c h i n g Moment C o e f f i e i e n t s f o r NACA 0015 A i r f o i l Section with Normal Unvented S p o i l e r . theory (C ^  input); theory ( l i n e a r i s e d C , input): o C. experimental; A C experimental pb r 1 r macs 64 0 - 3 T I 2 4 A A 8 A 1 6 a 2 0 AR=773 s/c = 0-48 b s /b=0-2 h/c = 0-097 - 0 - 2 - L - 0 - 8 Figure 17 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f or Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory ( C p b input); — theory (C ^  input) ;o C L experiment;A C^^, experiment 65 AR=7-73 s/c =0-48 bs/b = 0-3 h/c = 0097 -0-2^ --0-8 Figure 18 L i f t & Pitching Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory (C input), ( C p b input);o C L experiment; A C ^ ^ experiment Pb 66 16 a 20 AR=7 73 s/c = 0-48 bs/b= 0-4 h/c = 0 097 0-2 - L- 0-8 Figure 19 L i f t & Pitching Moment Coefficients for Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler. theory (C input), theory (C b input) ;o C L experiment;A pb experiment 67 0-3 T 1-2 AR = 7-73 s/c=0-48 bs/b=0-5 h/c=0097 0-2- 0-8 Figure 20 L i f t & P i t c h i n g Moment C o e f f i c i e n t s for Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory input; theory ( C ^ input) ;o experiment;A C j ^ , experiment 68 Figure 21 R o l l i n g Moment C o e f f i c i e n t f or Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory (Cp b input); — theory (C , input); o experiment 69 0-6 16 20 a AR = 3-87 s/c = 0-68 bs/b=0-2 h /c = 0097 0-6 -0-2-1--0-8 Figure 25 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing NACA 0015 Section with Normal Unvented S p o i l e r . theory; o C^ experiment; A C^^, experiment 73 Figure 26 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f or Rectangular Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory; o C experiment; A C experiment Figure 27 L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory; o C experiment; A C experiment 75 0-3T AR=3-87 s/c = 0-68 bs/b = 0-5 h/c=0097 0-6 - 0-2 -1- - 0-8 Figure 28 ' L i f t & P i t c h i n g Moment C o e f f i c i e n t s f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . Theory; o C experiment; A C experiment 76 0-6 O 0-2 AR=3-87 s/c = 0-68 bs/b=0-2 h/c = 0097 0-3 -L-0-4 Figure 29 R o l l i n g Moment C o e f f i c i e n t f o r Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory; o C experiment; R -r-0-6 0 4--0-2 + -0-3 -L-0-4 AR=3-87 s/c = 0-6 8 bs/b = 0-3 h/c = 0097 Figure 30 Rolling Moment Coefficient for Rectangular Half Wing of NACA 0015 Section with Normal Unvented Spoiler. theory; o C R experiment Figure 31 R o l l i n g Moment C o e f f i c i e n t f or Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory; o experiment Figure 32 R o l l i n g Moment C o e f f i c i e n t for Rectangular Half Wing of NACA 0015 Section with Normal Unvented S p o i l e r . theory; o experiment 0-4 0-3 AR = 7-73 s/c=0-48 h/cs 0-097 0-2 0 Figure 33 0-2 0-3 0 4 b s / b 0 5 E f f e c t i v e Moment Arm of Incremental L i f t Due to Normal Unvented Spo i l e r on Rectangular Wing of NACA 0015 Section. Theory; o experiment; A C T a c t i n g at midspan of s p o i l e r 10 81 0-9 0 - 8 AC R L 0 - 7 0-6 h 0-5 0-4 0-3 h 0 - 2 0-1 \-AR=3-87 s/c=0-68 h/c = 0 097 0 Figure 34 0-2 0-3 0-4 b s /b 0 - 5 E f f e c t i v e Moment Arm of Incremental L i f t Due to Normal Unvented Spoilers on Rectangular Wings of NACA 0015 Section. theory; o experiment; AC^ a c t i n g at midspan of s p o i l e r I 1 1 1 1 I I I I t I 0 01 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 10 y/b ;ure 35 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings w i t h Symmetrically Deployed S p o i l e r s 0 2 8 Figure 37 Spanwise L i f t D i s t r i b u t i o n f o r Rectangular Wings of NACA 0015 Section 0-28 0-24 r 4b U 0-20 016 012 0 0 8 0 0 4 NACA 0 015 AR = 3-87 s/c = 0 -68 b s/b = 0 -4 h/c = 0 0 9 7 8=90° a= I RAD d/h=00 ® ® © UNSPOILERED ASYMMETRICALLY SPOILERED SYMMETRICALLY SPOILERED 0-2 0-3 0-4 0-5 0-6 0-7 0-8 y/b 0-9 Figure 38 Spanwise L i f t Distribution for Rectangular Wing of NACA 0015 Section oo 0-60 h 0-45 h 0-40 \-s/c 0-48 0-58 068 0-77 s y m b o l A O • V NACA 0015 AR= 5-80 bs/b = 0-3 h/c= 0097 d/h=00 8=90° 01 0-2 0-3 0-4 0-5 0-6 0-7 ° - 8 b t / b s ° - 9 10 F i g u r e 41 V a r i a t i o n of C ^  w i t h S p o i l e r P o s i t i o n Along Chord f o r Rectangular Wings w i t h S p o i l e r s AR 3- 87 4- 83 5- 80 6-77 7- 7 3 symbol O • o NACA 0015 s/c=0-68 bs/b=0-3 h/c= 0-097 8 =90° 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 Figure 43 V a r i a t i o n of C p b with Aspect Ratio f or Rectangular Wing with S p o i l e r s IC - 0 - 7 0 - 0 - 6 5 - 0 - 6 0 - 0 - 5 5 - O 5 0 - 0 - 4 5 - 0 - 4 0 s/ c 0-5 0-6 0-7 symbol O • 12-9% CLARK Y AR=6I0 b s / b = 0 - 3 h/c=0097 8=90° 0-2 0-3 0-4 0-5 0 -6 0 7 0 8 . . , 0 9 b t / b s ure 44 V a r i a t i o n of C ^  w i t h S p o i l e r P o s i t i o n Along Chord f o r Rectangular Wings w i t h S p o i l e r s - 0 - 6 5 - 0 - 6 0 - 0 - 5 5 0-50 - 0 - 4 5 0-40 AR 4 0 7 6 1 0 712 symbol O • 12-9% CLARK Y s/c = 0-7 b s/b=0-3 h/c=0-097 8 = 9 0 ° -L _L 0 01 0-2 0-3 0-4 0-5 0-6 0-7 0 8 F i g u r e 46 V a r i a t i o n of C p b w i t h Aspect Ratio f o r Rectangular Wings w i t h S p o i l e r s 0-9 U3 - 0 - 7 - 0 6 - 0 - 5 - 0 4 - 0 - 7 C p b - 0 - 6 - 0 - 5 - 0 4 AR symbol NACA 0015 3- 87 A s/c = 0-48 d/h=00 4- 83 O h/c = 0 0 9 7 8=90° 5- 8 0 • 6- 77 O 8 10 12 14 16 18 20 b s / h NACA 0015 s/c = 0-58 d/h = 0 0 h/c=0097 8= 9 0 ° o o _ l J I I I L \7 0 2 4 6 8 10 12 14 16 18 2 0 b s / h F i g u r e 47 V a r i a t i o n of C fe w i t h S p o i l e r Span f o r Rectangular Wing w i t h Normal Unvented S p o i l e r 4> -0-7 AR symbol NACA 0015 Cpb 3-87 A s/c = 0-68 d/h = 00 4-83 O h/c = 0 097 8=90° -0-6 5- 80 6- 77 • o -0-5 7-73 O' o V ° Q • o v 0 -0-4 i 1 i i i i 1 I i 1 0 2 4 6 8 10 12 14 16 18 20 -0-6 -0-5 -0-4 -0-3 b s / h NACA 0015 s/c = 0-77 h/c = 0 097 d/h=00 8=90° A • A O O • O o 0 8 10 12 14 16 b s /h 18 20 F i g u r e 48 V a r i a t i o n of C , w i t h S p o i l e r Span f o r Rectangular Wings w i t h Normal Unvented S p o i l e r s pb , - 0 - 7 C p b - 0 - 6 - 0 - 5 - 0 - 4 AR symbol 4 0 7 A 6 1 0 • 712 O m 0 r-. * • o 12 9 % CLARK Y s/c = 0-5 h/c=0l 8 10 12 0 • X l 4 b s / h 1 6 96 18 -o-7r C p b - 0 - 6 - 0 - 5 0-4 o 12 -9% CLARK Y s/c =0-6 h/c = 0 l Q _L 8 10 12 1 4 b s / h 1 6 18 - 0 7 S p b - 0 - 6 - 0 - 5 - 0 - 4 12 9 % CLARK Y S/C = 0-7 h/c=0l 8 10 12 l 4 b , / h 1 6 o 18 Figure 49 V a r i a t i o n of C ^ with Spoiler Span for Rectangular Wings with Normal Unvented Sp o i l e r s •i r Cpb =* Cpb 0-9 0-8 0-7 A O A N A C A 0015,h/c=0 097, d/h = 0 , =90° I O CLARK Y, h/c=0l, d/h = 0 , 8=90° 0-4 0-5 0 6 0 7 J 0-8 0-9 10 S/C 1-2 6 G* 10 0 9 0 8 A O A 0-4 0 5 0-6 0-7 _L J 0-8 0 9 10 S/C F i g u r e 50 V a r i a t i o n of C , / C , and G/G* w i t h s/c pb pb REFERENCES Woods, L . C , "The Theory of Subsonic Plane Flow", Cambridge U n i v e r s i t y Press, 1961. Barnes, C.S., "A Developed Theory of Spoilers on A i r f o i l s , A.R.C. R & M, CP 887, J u l y 1965. Ja n d a l i , T., "A P o t e n t i a l Flow Theory for A i r f o i l S p o i l e r s " , Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1971. Brown, G.P., "Steady and Non-Steady P o t e n t i a l Flow Methods for A i r f o i l s with S p o i l e r s " , Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1971. Prandtl, L., "Applications of Modern Hydrodynamics to Aeronautics", NACA Report No. 116, 1921. Jones, R.T., "Correction of the L i f t i n g Line Theory f o r the E f f e c t of the Chord", NACA Technical Note No. 817, 1941. Glauert, H., "Elements of A i r f o i l and Airscrew Theory", Cambridge U n i v e r s i t y Press, 1927. Wieselsberger, C , " Th e o r e t i c a l Investigations of the E f f e c t of the Aile r o n s on the Wing of an Aeroplane", NACA TM 510, 1928. Bernier, R., "Steady & Transient Aerodynamics of Spoilers on A i r f o i l s " , M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1977. Parkin, B.R., "Linearised Theory of Cavity Flows i n Two Dimensions", RAND Report, P-1745, 1959. Bio t , M.A., "Some S i m p l i f i e d Methods i n A i r f o i l Theory", Journal of the Aeronautical Sciences, 9, 1942, pl85. Hoerner, S.F., " F l u i d Dynamic Drag", Published by Author, 1965. Pope, A. & Harper, J . J . , "Low Speed Wind Tunnel Testing", Wiley, 1966 APPENDIX K = 2R1 Jla I i ( l + ^ ) a o R l X i [ ImE - ( a - %M ) ] - I m X i R l E R l X i I m X 2 - ImXiReX2 i 9 o -18, / i 9 n -, \ 2 , — i O n , \ ( a e - 1 ) ( a e -1) . So x 2 R 1 X 2 [ I m E - ( a - % M o ) ] - I m X 2 R l E R l X 2 I m X i - I m X 2 R l X i 6 i S i n 6 0 2 - T T + + n TT TT S i n 6 TT , a -e i — o ±6i - i 6 i a Q - e n TT i 9 2 - i 6 2 a -e a -e o o °° nM ±1 E n = l a n+1 oo I E N 1-n 19, -19, i 9 „ . . - i 6 0 a a e -1 a e -1 o o o n = 0 n _ "n - 1 , i 9 n - i 9 o -. \ a o ( a o e " 1 ) ( a o e _ 1 ) R l ImX2 R l X 2 I m X i - R l X i I m X 2 19, -19 ( a e 1 9 o - l ) 2 ( a e ~ i 6 ° - l ) 2 u. o o -ImX r 1 (1 + i _ ) R l X i I m X 2 - I m X i R l X 2 a Q ( 2 9 ) w h e r e 100 2 h 1-b j i i a [ f J^b 2 i a 1 ] lo 1+b + 1+b L I 1+b 1+b J " 1 J a, = i a 1 1+b 1 + 1-b 2 i a 1+b 1+b C = -2 ps C + o n(e 2-Tr) 0 i S i n 6 • + • TT S i n 8 •z ^ fi+ 4B SinG Cos9o~ Cos6 o 2 S i n 6 TT l n ' s i n ^ | e-e.i | S i n % ( 0 + 0 i ) 2n I n s±nh\0-0 2 SinJ2(0+6 2) 0 0 E N S i n n0 2E M S i n n9 + ° 1 Q n s K n Cos0o - Cos 0 (30) Cls= IT Im| 1Bo*al(1 + V - 1 a o C + o 0 i S i n 6 (0 2 -7T ) n + — TT TT la. 10, - i e , , i 6 0 , N 2 L ( a o e -1) (a e o -± e o - l ) + la. TT , a -e <- o i e a -e o - i 6 i TT 1 i e 2 - i e 2 a -e a -e o o 0 0 nM + i £ a 1 E n = l a n+1 + i J l a . E N I n 1 1 n=0 •, i e 0 l - n e i8o , a a e -1 o o - i e , a e o - i e 0 _ 1 n - l (a e i e ° - l ) ( a e - i 9 o - l ) (31) 101 4TT mo s Im iC £ a„ o 2 2r a l ? o (a r - l ) a o o L - 1 o o •ft, < a o V 1 ) a 2 - 1 + iB I a_ o 2 1 -a o a 2 - 1 i£ 2a E n = l M n n n+1 a. o a o a 2 <=±i> - 1 i£ a„ E N n=0 a -5 a -C o o o o 2 2 n-1 a l _ ( n - 1 ) ( n - 2 ) f l n+1 n a a a 0 o o 2 2! a' a * ( n - l ) a_(a -£ ) a n ( a -£ ) 2 2 o o o o o - n - 1 , x 2 a a (a -£ ) o o o o o - 1 ( a o " ? o ) a 2 a 9 ( a „ - ? ^ ) a " ( a „ " ? r t ) z. o o o o o / >- \ 2 , - \ 2 n-1 n-2 . w - . 2 a 0 ( a -£ ) (a ) a a ( a - £ ) ( a - £ ) 2 o o o o o o o o o o ( a o - ? o ) a 2 - 1 + £ 2a, a l ^ o < a o V 1 ) a 2 -1 (a C -1) cr o a l ^ o ( a o ? o - 1 ) a 2 -1 102 + ^ £ 2 a 9 TT 2 ( a o ^ l > ( ao~ V o 1 Z 2 ( a o - C 1 ) 2 a 2 + 5 TT 2 < a o - 5 2 ) ( a o - ? 2 > 2 ( a o - £ 2 ) 2 a 2 ( a Q - C 2 ) 2 a 2 (32) where a Q and a^ are the same as i n equation (29) and 2i a 3 1-b i a f 2 1+b 4 1+b a2 1+b I 4 r« . , . -.2 i a 1+b f l l a + l ^ b l L1+b 1+bJ "2ia 1-b] .1+b 1+bJ 12 

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