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An experimental study of flow about an airfoil with slotted flap and spoiler using Joukowsky profiles Allan, William D. E. 1988

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AN EXPERIMENTAL STUDY OF FLOW ABOUT AN AIRFOIL WITH SLOTTED FLAP AND SPOILER USING JOUKOWSKY PROFILES by William D.E. ALLAN B.Eng., Royal Military College, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE 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 April, 1988 ©William D.E. ALLAN, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 2i AP£ fifi DE-6(3/81) ABSTRACT An experimental study has been c a r r i e d out on an a i r f o i l with s l o t t e d f l a p and s p o i l e r us ing Joukowsky p r o f i l e s . Pressure d i s t r i b u -t ions were measured as funct ions of angle of a t tack , f l ap d e f l e c t i o n angle , s p o i l e r s i z e and i n c l i n a t i o n . The r e s u l t s are uncorrected for wind tunnel w a l l e f f ec t s but the data base i s a v a i l a b l e to c a r r y out the c o r r e c t i o n s . The r e s u l t s w i l l be used to compare with p r e d i c t i o n s of a t h e o r e t i c a l model, yet to be worked out , which combines work p r e v i o u s l y done by W i l l i a m s , J a n d a l i , Parkinson and Yeung. Thi s theory w i l l invo lve the p o t e n t i a l flow about a two-element 'near'-Joukowsky a i r f o i l system. The secondary a i r f o i l i s a s imulated s l o t t e d f l a p . Various s i ze s p o i l e r s are in troduced to the system at a r b i t r a r y angles of i n c l i n a t i o n using methods proposed by Parkinson and Yeung. The experimental r e s u l t s are q u a l i t a t i v e l y reasonable and some i n t e r e s t i n g e f f ec t s are observed. The behaviour of s p o i l e r s , when used wi th s l o t t e d f laps at var ious d e f l e c t i o n angles , corresponds w e l l with requirements of a i r c r a f t i n approach or landing s i t u a t i o n s . S i m i l a r l y , the use of s l o t t e d f laps alone provides the h igh l i f t at low angle of a t tack which i s b e n e f i c i a l to a i r c r a f t taking o f f . Some recommendations are proposed for fur ther t e s t i n g with t h i s equipment. TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v LIST OF SYMBOLS v i i ACKNOWLEDGEMENT v i i i 1. INTRODUCTION 1 2. THEORY 2.1 Two Circles in Uniform Flow 3 2.2 Two Circles with Arbitrary Circulation 5 2.3 Producing Two 'Near-Joukowsky' Airfoils 9 2.4 The Introduction of a Normal Spoiler 10 2.5 To Obtain Arbitrarily Inclined Spoilers 14 3. APPARATUS 16 4. EXPERIMENTATION 4.1 The Process 22 4.2 Uncertainties 25 4.3 Wind Tunnel Wall Corrections 26 5. OBSERVATIONS AND RESULTS 5.1 General Configuration Phenomena 5.1.1 Trailing Edge of Main Joukowsky Airfoil 28 5.1.2 Blockage Effects by Flap External Airfoil 28 5.1.3 Flap Leading Edge Curvature Effects 31 5.1.4 Coefficients of Lift Versus Angle of Attack 32 5.1.5 'Single-Airfoil' Tendencies 32 5.1.6 Trailing Edge Data Points 35 5.1.7 Flap Upper Surface and Spoiler Back Pressure Correlation 35 5.2 Separation Effects 5.2.1 Unseparated Flow 37 5.2.2 Separation on the Lower Flap Surface 37 5.2.3 Separation on the Trailing Upper Flap Surface 40 5.2.4 Pre-Spoiler Separation Bubble 40 5.2.5 Separation on the Trailing Upper Wing Surface 43 5.2.6 Leading Edge Separation Bubble 43 - i i i -TABLE OF CONTENTS (Continued) Page 5.3 Flap Effects 5.3.1 Flap Effects on Coefficient of L i f t ' 46 5.3.2 Flap Deflection Angle Comparison 46 5.A Spoiler Effects 5.4.1 Curve F i t t ing on Spoiler Back Pressure Coeff ic ients . . 50 5.4.2 L i f t Coefficients with Spoilers Extended 53 5.4.3 Spoiler Effects on Pressure Distribution 53 5.4.4 Effects of Spoiler Size 57 5.5 Effects of Spoiler Use with Slotted Flap 5.5.1 Effects of Flap Deflection Angle with 10% Spoiler . . . 57 5.5.2 Flap Effects on L i f t Coefficient with Spoiler Extended 63 5.5.3 Spoiler Inclination Angle Effects with Flap Deflected 63 5.5.4 Effects of Spoiler Size Changes with Flap Deflected.. 67 6. DISCUSSION AND CONCLUSIONS 70 7. RECOMMENDATIONS 73 REFERENCES 76 APPENDIX A - CALCULATIONS FOR FLAP CONSTRUCTION 77 APPENDIX B - REYNOLDS NUMBER CALCULATIONS 81 APPENDIX C - COEFFICIENT OF LIFT CALCULATIONS 83 APPENDIX D - C L vs o SLOPE CALCULATIONS 85 APPENDIX E - THE JOUKOWSKY TRANSFORMATION 86 APPENDIX F - ERROR ESTIMATES 88 - iv -LIST OF FIGURES Page 3.1(a) Conventional Slotted Flap 17 (b) Two-Element Joukowsky Airfoil 17 3.2 Experimental Section 18 3.3 Experimental 'Optimum Gap and Overlap' 20 A.l Data Acquisition Schematic 23 Pressure Distribution Plots 5.1 no spoiler a=0°, 6=0° 29 5.2 no spoiler a=A°, 6=0° 30 5.A no spoiler a=A°, 6=20° 3A 5.5 10% spoiler a=0°, 6=0° £=90° 36 5.6 no spoiler a=0°, 6=10° 38 5.7 no spoiler a=2°, 6=-10° 39 5.8 no spoiler a=-A°, 6=30° Al 5.9 7% spoiler o=6°, 6=10° E=90° A2 5.10 no spoiler o=12°, 6=0° AA 5.11 no spoiler a=8°, 6=0° A5 5.13 no spoiler a=0°, 6=20° A8 5.1A no spoiler o=A°, varying 6 A9 5.15 7% spoiler a=8°, 6=0° £=90° 51 5.16 10% spoiler a=8°, 6=0° £=90° 52 5.18 no spoiler & 10% spoiler a=A°, 6=20° 55 5.19 no spoiler & 10% spoiler o=8°, 6=20° 56 5.20 7% & 10% spoiler o=A°, 6=20° 58 5.21 7% i 10% spoiler a=8°, 6=20° 59 5.22 10% spoiler o=A°, Varying 6 £=90° 60 5.23 10% spoiler a=8°, Varying 6 £=90° 62 5.25 10% spoiler o=8°, 6=20° varying £. 65 5.27 Varying spoiler size a=A°, 6=20° £=90° .... 68 - v -LIST OF FIGURES (Continued) Page C L vs a P l o t s 573 ncTspoiler 6=0° 33 5.12 no spoiler varying 6 47 5.17 10% spoiler 6=20° 54 5.24 10% spoiler Varying 6 £=90° 64 5.26 10% spoiler 6=20° varying I. 66 5.28 Varying spoiler size 6=20° £=90° . . . . 69 - v i -LIST OF SYMBOLS a radius of main circle used in theory b radius of secondary circle used in theory c centre of secondary circle used in theory C chord ... sum of main airfoil and flap chords C n 2-D coefficient of drag Cj chord of flap C L 2-D coefficient of l i f t C£ corrected 2-D coefficient of l i f t Cmc/4 coefficient of moment at the quarter-chord position Cp coefficient of pressure C corrected coefficient of pressure P r C average C pav 6 p C w chord of wing g gravitational acceleration h manometer fluid height H span of aerofoil Q dynamic pressure Re Reynolds number U free stream velocity approaching airf o i l V free stream velocity in wind tunnel a angle of attack or incidence o flap deflection angle, down is positive c solid blockage factor in wall corrections = Ao T general circulation r specific circulation about main airfoil I* specific circulation about flap airfoil A ai r f o i l shape factor used in wall corrections o density of air 'a o density of water "w o TTV48 ( C / H ) j parameter from wall correction theory v kinematic viscosity of air a £ spoiler inclination angle - v i i -ACKNOWLEDGEMENT This research work has been confined to a very short time span due to unforeseen circumstances. In view of t h i s , e n t h u s i a s t i c and r e l i a b l e support was thrown behind me by the s t a f f and fe l low graduate students at the U n i v e r s i t y of B r i t i s h Columbia. Dave Camp from the Mechanical Engineering Workshops was respons ible for a success fu l model produced i n record time despite concurrent design m o d i f i c a t i o n s . Capt . (RCAF Ret'd) Gerry Desroches was inva luab le i n data c o l l e c -t i o n ; he dedicated h imse l f , i n h i s time away from a job at Transport Canada, and through h i s own i n i t i a t i v e , to t h i s p r o j e c t . The enthusiasm of the s t a f f i n the Mechanical Engineering Department does not go without deep a p p r e c i a t i o n . The work environment was as p o s i t i v e l y a f f ec ted by t h i s , as i t was n e g a t i v e l y , by t ime. The Natura l Science and Engineering Research Counc i l of Canada provided f u l l funding for me to complete t h i s work between September 1986 and September 1987. I regret that I s h a l l not be present to provide my fe l low graduate students i n Mechanical Engineer ing the same informal and f r i e n d l y support which I rece ived from them. I am i n t h e i r debt. Most worthy note i s extended to Dr. G . V . Parkinson, my advisor and guiding hand i n t h i s endeavour. His cool and unwavering support , i n the face of a l l the ex traord inary circumstances by which t h i s thes i s was completed would have won him renown as a leader i n my more f a m i l i a r m i l i t a r y wor ld . I t has been a great p r i v i l e g e to work with him. - v i i i -1 CHAPTER 1 INTRODUCTION The aerodynamics of various wing sections and their control surfaces has played an important role in the Mechanical Engineering Department at the University of British Columbia for some years now. Theoretical studies have been carried out on spoilers and various types of flaps. Experimental work has always been necessary as well, to verify the theories proposed. This particular study has concentrated on the use of spoilers with slotted flaps. Spoilers are effective fences raised from the upper surface of the wing section. They reduce circulation around the aerofoil which contributes to a great reduction of l i f t and an augmentation of drag. One could see a use for these effects as air brakes or in the approach and landing of aircraft. As shown in Fig. 3.1(a) they are of varying size and when closed, usually reach back to the trailing edge of the wing. They can be opened to any angle and are commonly used up to 60° deflection. Spoilers are frequently used in conjunction with flaps of some type. In this report, the effects of their use with slotted flaps is investigated. Slotted flaps are individual airfoils which, when retrac-ted, form to the trailing edge of the wing section itself, and when extended allow airflow between the new "trailing edge" of the wing and the leading edge of the flap (see Fig. 3.1(a)). The flaps are used in the take-off and landing of aircraft. They effectively alter the camber of the airfoil to create a higher l i f t wing. This results, of course, in measurably higher drag as well. This plays an important role in 2 reducing landing and take-off speeds. This is desirable for improved safety and in reducing the runway length required to take off or land. As well, control is retained, despite the increased ascent or descent angles of the aircraft. Both theoretical and experimental work has been completed on spoilers used alone, as well as in conjunction with split flaps (refs. 6,16). Studies have also been completed on simple and slotted flaps. To further this work, the theory is expected to be worked out for two-element aerofoils with spoilers. This would make use of the Williams methods (ref. 15) or producing an external airfoil-flap, virtually a slotted flap behind a wing. The work by Parkinson and Yeung (ref. 10) on potential flow about spoilers will be added to the aforementioned two-element systems to complete the theory. Verification for this theory will be possible with the experiments conducted in this study, when the data are corrected for wind tunnel wall effects. CHAPTER 2 THEORY 2.1 Two C i r c l e s i n Uniform Flow To obta in c i r c l e s i n uniform flow, the Milne-Thomson technique from Ref . 8 i s used. I t e f f e c t i v e l y uses image doublets t o p lace a c i r c l e o f r a d i u s b i n the v i c i n i t y o f a c i r c l e o f rad ius a . A c i r c l e o f rad ius a i n uniform flow o f u n i t v e l o c i t y i s shown i n F i g . 2 . 1 , plane £ . F igure 2.1 (2.1) FAO = C e i a + f - e ^ a o t To s imply add a new c i r c l e o f rad ius b , the term (2.2) i s added. (2.2) . . . + , 2 i a b e C - c T h i s term however a l t e r s the shape of the f i r s t c i r c l e of rad ius a . To mainta in i t s o r i g i n a l shape a counter-balanc ing doublet must be p laced i n s i d e the f i r s t c i r c l e . T h i s i s the Milne-Thomson Technique from Ref . 8 (2.3) .2 2 - i a b a e 2 2 C « - * - > Figure 2.3 The f i r s t counter-balancing doublet i s shown i n (3) . Note i t s opposite s i g n to (2) and i t has a 2 2 s c a l i n g f a c t o r b / c . Since b<c i t s va lue i s much l e s s than 1. S i m i l a r l y , the doublet i n (2.1) a l t e r s the shape o f the second c i r c l e o f rad ius b . A new image must be set up at C « c - — according to the Milne-Thomson technique , to r e g a i n the second c i r c l e ' s o r i g i n a l shape, as shown i n F i g . 2 .4 . F igure 2.4 -ao (2.4) «-<c - f - » 2 2 The new term (2.4) has , a / c as the s c a l i n g f a c t o r , aga in l e s s than one. The s c a l i n g f a c t o r s are converging as i s expected, not on ly geometr i ca l l y , but i n magnitude as w e l l . As one progresses i n t h i s manner, the image doublets d imin i sh i n s trength and converge on po in t s f © b y r V 2 Figure 2.5 5 inside the two circles near the outer bound. The next two terms are as follows: (2.5) . . . + ^ ^ r - - a 2 2 2 2 c , b 2 a % (c - —) (£ r ) c (c - b /c) (shown in Fig. 2.5) (2.6) v2 2 b a 1 / a 2.2 (c - - ) • b -ia e (t - (c - r)) c -(not shown to avoid clutter) Each time the scaling factors are increasingly less than unity, promising convergence. The velocity field of this situation is dF (2.7) W0(C) - ^ • 2.2 Two Circles with Arbitrary Circulation The Kutta condition entails a finite velocity or a stagnation point at the trailing edges of airfoils. To be in keeping with reality one avoids infinite velocity at the cusp of a Joukowsky trailing edge. To achieve this with the Joukowsky transformations, circulations must be introduced onto the two circles produced in part 1 of this theory section. Once again an infinite convergent series is encountered. Two c i r c u l a t i o n s are required for a two-element Joukowsky a i r f o i l system. C a l l Tl c i r c u l a t i o n about the main c i r c l e that i s to be made i n t o the main a i r f o i l . Let be the c i r c u l a t i o n about the smal ler c i r c l e which w i l l become the f l a p . The problem w i l l be reduced to two complete i n f i n i t e convergent s e r i e s . As be fore , the Milne-Thomson C i r c l e Theorem from Ref. 8 i s used, but once again the r e s u l t i n g image c i r c u l a t i o n s must be accounted f o r . For s i m p l i c i t y , l e t I = 2TT, SO I = 2n i s u n i t y . Now to p lace c i r c u l a t i o n about the large c i r c l e ; (2.8) FAO = iln(C) + iln(C-c) - iln(C-(c - bVc)) . . . where the l a s t two terms represent the image v o r t i c e s i n s i d e the small c i r c l e caused by the presence of the vortex at the centre of the large c i r c l e , shown i n F i g . 2 .7 . F igure 2.6 © Figure 2.7 7 Once again an i n f i n i t e s e r i e s emerges, wi th terms 2 . 9 and 2 . 1 0 represent ing the f i r s t image p a i r i n s i d e the large c i r c l e , as i n F i g . 2 . 8 2 ( 2 . 9 ) . . . - i l n ( C - f~) 2 ( 2 . 1 0 ) . . . + i l n ( C - 2 8 2 ) c - b / c The v e l o c i t y f i e l d i s dF. ( O ( 2 . 1 1 ) W j t f ) -One must note that the s trength does not d imin i sh i n t h i s i n f i n i t e s e r i e s . The opposite p a i r s of v o r t i c e s , however, do converge on t h e i r complementary inverse p o i n t s . The r e s u l t s are n u l l vortex doublets which have contr ibuted nothing to t o t a l complex p o t e n t i a l . S i m i l a r l y f o r c i r c u l a t i o n about the smal l c i r c l e : ( 2 . 1 2 ) F a (C) - i lnC + i ln (C-c ) - i l n t t - a V c ) - i l n ( C - ( c - b V c ) ) + i l n ( C - ( c 2__)) a c c + ... t © Figure 2.8 8 and s i m i l a r l y the v e l o c i t y f i e l d i s found by derivation (2.13) W,(C) = dF, To combine the cir c u l a t i o n s derived above with the c i r c l e s created i n part one of t h i s section, simple superposition i s required: r r (2.14) W(r.) = W0(C) + 2i W i ( C ) + 2i W * ( C ) The magnitudes of Tj and T3 are determined, as stated above, from the Kutta condition of f i n i t e v e l o c i t i e s at the t r a i l i n g edges of wing and flap sections, which correspond to zero v e l o c i t i e s at the respective points i n the c i r c l e plane, Tj and T 2, as shown i n Fig. 2.9. © Boundary Conditions r hv 1. (2.15) 2. W(0 = 0 at Tj W(C) = 0 at I , With these boundary conditions i n (2.14), two unique equations result for two unknowns Tl and T3. Figure 2.9 Figure 2.10 2.3 Producing Two 'Near-Joukowsky' Airfoils To obtain two Joukowsky airfoils from this existing state of two circles with circulation in proximity, simply apply some translations, rotations, and the Joukowsky Transformation (Appendix E). The thickness and camber of the resulting airfoils is dependent on the translations and rotations carried out. The gap between them is especially dependent on the circles' original distance apart (c-b)-a. To conserve the Kutta conditions, the axes about which the Joukowsky Transformations are performed must pass through the stated points T1 and T 2 as described above, and shown in Fig. 2.11. The Joukowsky Transformation is performed here to produce a Joukowsky main wing and a 'near circle'. This slight alteration in the shape of the smaller circle is due to the small effects the Joukowsky Transformation does have at that relatively large distance from the origin. New axes Figure 2.11 Figure 2.12 are used to create the flap airfoil at a different angle of incidence to the main a i r f o i l , as shown in Fig. 2.12. Now, the resulting two-element J^" Joukowsky airfoil setup is made up actually of two 'near Joukowsky' airfoils, as shown in Fig. 2.13. The main airfo i l is not a true Joukowsky profile because i t has been slightly modified by the second use of the Joukowsky Transformation. The flap's shape is slightly askew because i t originated from a 'near circle' not a perfect circle. It would be a perfect Joukowsky airfoil had the first Joukowsky Transformation not been carried out. To get the velocity field in this final configuration of two 'near Joukowsky' airfoils, one must simply manipulate the known flow field from (2.16), Figure 2.13 (2.16) v * ; dz2/dC dZj/dZj • dZj/dz • dz dC 2.4 The Introduction of the Normal Spoiler To obtain a Joukowsky airfoil with a normal spoiler, one must simply apply the Joukowsky Transformation to a circle with a radial fence. Details for this brief overview can be found in Ref. 7. The Joukowsky Transformation i s of the form (2.17) Z = t + 1/t The fence gets s l ight ly curved in the transformation. This is not undesirable because r e a l i s t i c a l l y , a spoiler retracts to form a section of the upper surface of the wing. This surface should be curved as wel l . Points T and E in F ig . 2.15 indicate the separation of flow. This is in keeping with a real situation when the working f l u i d , usually a i r , is unable to achieve in f in i te velocity around the cusp of the t r a i l i n g edge and the spoi ler . The resulting inf in i te negative pressure gradient induces the separation. One must work backwards from the c i r c l e with radial fence shown in F i g . 2.14 to obtain the flow f i e ld for the a i r f o i l with normal spoiler. Figure 2.14 Figure 2.15 Figure 2.16 I f one t r a n s l a t e s the c i r c l e to be concentr ic wi th the o r i g i n and rota tes i t to p lace the r a d i a l fence along the h o r i z o n t a l a x i s , F i g . 2.16, a Joukowsky Transformation can reduce the shape to a f l a t p l a t e , F i g . 2 .17. The v e r t i c a l ax i s however does not b i s e c t the p l a t e . By t r a n s l a t i n g the o r i g i n to the centre of the p l a t e , F i g . 2 .18, and performing an inverse Joukowsky Transformat ion , a c i r c l e wi th separat ion p o i n t s can be crea ted . Another r o t a t i o n v i l l i n d i c a t e p a r a l l e l uniform flow wi th separat ion at T and E ( F i g . 2 .20) . C i r c u l a t i o n remains unchanged by any of these transformations so i t can be introduced at t h i s p o i n t . The wake source model as proposed by Parkinson and J a n d a l i , (Ref. 9) can be u t i l i z e d to create the s tagnat ion po in t s at T and E . They are c r i t i c a l po in t s and hence angles w i l l be doubled i n the Joukowsky Transformat ion. T h i s w i l l produce the requ ired t a n g e n t i a l separat ion at the s p o i l e r and t r a i l i n g edge. 1 © 1 2 Figure 2.17 © T T Figure 2.18 F igure 2.20 13 The flow in Fig. 2.20 can be produced by placing two double sources of arbitrary strengths on the bound of the circle and corresponding image sinks at the centre. The double sources are required because half of the strength is expended in the interior of the circle, hence having no effect on external flow. (2.18) F(C) = V(C + 1/C) + — In (C - e l 6 l ) + — ln (C - e"i62) IT TT V Q 2 ir /TT 2TT Equation 2.18 describes the potential flow field for Fig. 2.20. The unknowns are , Q2> b^, b^ and T. Four boundary conditions are known using stagnation points and assumed constant back pressure. This assumption is that separation occurs at an empirical back pressure coefficient, C . Using Bernoulli's equation the complex normal pb velocity of separation can be determined: W(C) =0 at T W(C) =0 at E -ISMi - V — at T U P b |W(z)I = Vl-C at E U P b where U is the free stream velocity approaching the a i r f o i l . One more boundary condition is required to solve for a l l five unknowns. This problem can be overcome by assuming a location for one of the sources as done by Jandali and Parkinson in Ref. 7. Parkinson and Yeung proposed a suitable fifth boundary condition in Ref. 10. rw = v ^ d z = w(C) dC = 0 Subscr ipt 'w' i n d i c a t e s wake. Th i s i s based on the fact that the flow i n the wake reg ion makes no c o n t r i b u t i o n to the a i r f o i l c i r c u l a t i o n , which i s only dependent on the flow upstream of the s p o i l e r t i p and the t r a i l i n g edge. I t must be noted that t h i s ana lys i s can extend from the t h e o r e t i c a l shape of Joukowsky a i r f o i l s to the ac tua l shapes of contemporary a i r f o i l s . T h i s i s done by the i n t r o d u c t i o n of a method proposed by Theodorsen i n Ref. 13. T h i s process b a s i c a l l y reforms 'near* c i r c l e s i n t o r e a l c i r c l e s aga in . 2.5 To Obtain A r b i t r a r i l y I n c l i n e d  S p o i l e r s From a Joukowsky a i r f o i l wi th an T i n c l i n e d s p o i l e r , F i g . 2.21, where that i n c l i n a t i o n i s l e ss than 90 degrees, one roust degenerate the a i r f o i l back to a c i r c l e us ing an Figure 2.21 inverse Joukowsky trans format ion . The s p o i l e r w i l l then become a non-r a d i a l fence, F i g . 2 .22. By t r a n s l a t i n g t h i s form one can be i n a p o s i t i o n to use the Karman-T r e f f t z mapping (2.19) s = iR s i n I cot(w/2) Figure 2.22 This reduces Fig. 2.23 to a bounded flow with a semi-infinite plate parallel to the flow as shown in Fig. 2.24. Parkinson and Yeung developed this method thoroughly in Ref. 10. The inside of Fig. 2.24 maps to the outside of the circle. Now a Schwarz-Christoffel Transformation will map the inter-ior of Fig. 2.24 into the upper half complex plane. The parameters in Equation 2.20 are shown in Fig. 2.24. (2.20) u = - | (2-n) + ih - f [n ln (£ + 1) 2 n + (2-n) ln - 1)] 2-n To complete the set of mappings a translation, a scaling, a bilinear transformation and a rotation will map the upper half plane in Fig. 2.25 to the outside of a unit circle. 4 ' 0 ( Y y Figure 2.23 4 E " ( n-2 ) TT "2" T l h i 2 Figure 2.24 0 ~7—7—7—7—r © y y -?—/ J S / Figure 2.25 16 CHAPTER 3  APPARATUS Slotted flaps, as shown in Fig. 3.1(a), normally are nested in the trailing edge of the main wing section. This study's major approxima-tion is the use of a two-element airfoil to simulate a slotted flap. Joukowsky airfoils are used for simpler mathematical modelling. Hence the experimental setup has a small Joukowsky airfoil tucked in behind a larger Joukowsky wing section (Fig. 3.1(b)). Because pressure distribution data was the ultimate goal for experiments in this study, a maximum number of pressure taps was desir-able. This entailed the largest possible flap to be built, s t i l l staying within the reasonable bounds set by flaps in use today. The largest realistic chord of a slotted flap is 30% of the wing total chord. It should be noted that the wing chord is the sum of the chords of the flap and the main a i r f o i l . So in a two-element wing section, the sum of the chords of the large and small airfoils makes up the chord of the system. Since the existing Joukowsky wing section has a 12 inch chord, this would set the chord of the flap at 5.14 inches. As 30% of the ful l chord, the value then of the wing chord is 17.14 inches (see Fig. 3.2). By referring to data from the profiles of commonly used NACA wing sections, available in Ref. 1, i t was found that the maximum common, and realistic thickness of the section at the 70% chord position was 4.5%, although some wing sections had up to 5.4 thickness at the 70% chord point. The 4.5% is to say that a flap which makes up the last 30% of a wing section has a thickness ratio of 15%. So what was required retracted f l a p Figure 3.1(a) Conventional Slotted Flap with Spoiler. f l a p retracted Figure 3.1(b) Two-Element Joukowsky Airfoil. 18 12.08 in 0.7 C 5 y in 0.3 C Figure 3.2 Experimental Section. 1 9 for this study's flap construction was a Joukowsky wing section with a chord of 5.14 inches and a thickness ratio of 15%. Appendix A has a l l calculations used to determine the section required as well as a profile sketch of the flap used in experimenta-tion. Flap position in relation to the wing section was determined by referring to many NACA reports on experiments with slotted flaps and external airfoil flaps. Many different configurations have been used in past experimentation. These were termed "best gap size" or "best overlap". This "gap" referred to the space between the trailing edge of the wing section and the leading edge of the flap section. The "overlap" was the amount by which the leading edge of the flap was positioned upstream, and below the trailing edge of the wing. Although many possibilities were identified, only one was realistically feasible with the thickness of the flap in this study. This configuration was found in Ref. 3. It called for an optimum gap of 2.0% C w at 0° flap deflection, and 2.3% C w at 30° flap deflection. Its best overlap was identified to be 1% Cw, where C w is the chord of the main a i r f o i l . This layout is shown in Fig. 3.3. Graphical methods using a transparency of the proposed flap deflection and a layout of the trailing edge of the main Joukowsky airfoil determined the pivot point for the flap that gave the desired overlap and the optimum gap points. This also left the gap at not less than 2.0% C w at any flap deflection angle. This was also cited in reference 3 as best. The flap itself was constructed on the numerically controlled milling machine. The data points as shown in Appendix A were input to a 20 p i v o t _JY_ 2.0% Cw gap f o r 6=0 2.3% Cw gap f o r 6=30° 6=0 F i g u r e 3.3 E x p e r i m e n t a l Optimum Gap and O v e r l a p . 6=30° 21 built-in curve fitting routine. The milling machine cut out two inch thick aluminum profiles of the required section. These were a l l stacked to form the 27" chord for use in the wind tunnel. The centre section was drilled with nineteen pressure taps and the tubes were run out down the core of the flap span. Endplates were constructed to maintain experimental flap positions as well as to house the spoilers to be tested. Spoilers existed for the Joukowsky wing section. Given the new definition of wing chord however, these took on new "sizes". What was originally a 10% spoiler for the wing section, became a 7% spoiler for this new two-element arrangement. A new spoiler was built 1.7 inches in width to serve as the 10% spoiler used in this study . The spoilers had variable deflection angle capability as well. The 30°, 45°, 60° or 90° inclinations were set by the endplate construction. 22 CHAPTER A  EXPERIMENTATION A.1 The Process The experiments in this study were a l l conducted in the 27 inch closed c ircui t wind tunnel in the UBC Aerodynamics Laboratory. The dynamic pressure chosen was A6.2 mm of water. This corresponded to a Reynolds number of 7.0 x 105 based on the f u l l section chord of 17.IA inches. Calculations are found in Appendix B. The temperature of the tets varied from 75°F to as high as 90°F. This would s l ight ly alter the viscosity used in Reynolds calculations to the order of A%. This study however, is essentially Reynolds Number independent. The only separation incurred in these tests is behind the spoiler and this is independent of the nature of the boundary layer. Similar tests conducted at these endpoint temperatures did not show any perceptible variations in coefficients of pressure. Most tests recorded tempera-tures of 85°F . The pressure taps from the wing were plugged into a scanivalve. This instrument was set up in series with an identical scanivalve which linked in the flap pressure taps. A d ig i ta l readout provided voltage measurements from a pressure transducer. Figure A . l i l lustrates the data acquisition setup. Tufts were attached to the wing and flap 6 inches below the roof of the tunnel. This is out of the boundary layer along the top of the tunnel and 7.5 inches from the pressure taps. This distance should ensure minimal interferences to the pressure data to be taken. The role of the tufts was flow visual izat ion. Fu l l and part ia l separation was easily seen with them. They effectively showed flow through the gap and provided interesting insight into the interaction between the spoiler and the slotted flap. 23 Pressure Transducer Signal Conditioner and Amplifier D i g i t a l Voltmeter Figure A.l Data Acquisition Schematic. 24 The existing Joukowsky airfoil with 12 inch chord had its mount at the quarter-chord position. This allowed moment measurements to be taken for normal testing with no flaps. This was done with the help of the six-degree-of-freedom balance on which the airfoil was mounted. Due to the extended chord length of the wing arrangement and various inter-ferences, this balance was of no use to these experiments beyond its ability to steadily support the setup. The new 30% flap, fitted in behind the wing, placed the mount at about the 18% chord position: far upstream of any theoretical aerodynamic center. The new chord length introduced a need to advance the balance upstream so as to allow the pressure tubes from the flap to descend through the turntable hole in the floor of the tunnel. Early in the testing i t was observed that the added chord, as well as the higher l i f t abilities of the section at high flap deflections was sufficient to bend the whole setup as much as several inches at the top in the " l i f t " direction. To rectify this, a pin was sunken through the top of the wind tunnel to the upper pivot point of the system. This solved the lateral bending of the setup, but the torque created by the system continued to be a problem as will be explained later in this section. It was deemed very important to have an estimate or close experi-mental approximation of the pressures at the trailing edges of the wing and the flap. It was particularly important at the flap because its thickness became too small for the trailing 1.5 inches of the flap to have pressure taps drilled. For accurate plotting of the pressure data, this trailing edge pressure reading was absolutely necessary. To carry out this task small pressure tubes were tightly taped to the trailing edges o f the wing and f l a p . T h e i r openings were about one i n c h below the d r i l l e d p r e s s u r e t a p s f o r the wing and f l a p t o m i n i m i z e adv e r s e e f f e c t s by t h e tube. As w e l l , i t was d e s i r a b l e t o have them c l o s e t o t h e t u n n e l c e n t r e t o a v o i d w a l l e f f e c t s . A c c o r d i n g t o t h e o r y , t h e p r e s s u r e b e h i n d a s p o i l e r s h o u l d be c o n s t a n t . To m o n i t o r t h i s , when s p o i l e r s were used i n the e x p e r i m e n t s , two p r e s s u r e t a p s were a t t a c h e d i n t h e s e p a r a t e d f l o w r e g i o n s b e h i n d the s p o i l e r . A. 2 U n c e r t a i n t i e s Some e x p e r i m e n t a l u n c e r t a i n t i e s must be mentioned t o a i d f u r t h e r r e s e a r c h e r s i n comparing t h e i r t h e o r y t o t h e d a t a and r e s u l t s c o n t a i n e d i n t h i s work. Appendix F l i s t s some q u a n t i t a t i v e e r r o r s . As was p r e v i o u s l y mentioned, the f l a p was added t o an e x i s t i n g Joukowsky a i r f o i l . The mount f o r the a i r f o i l was a t i t s q u a r t e r - c h o r d p o s i t i o n . T h i s a l o n e would be adequate f o r s t u d y w i t h t h e b a l a n c e a t t h e Aerodynamics Lab a t U.B.C. U n f o r t u n a t e l y , t h e f l a p c r e a t e d an extended e f f e c t i v e c h o r d . T h i s p l a c e d the mount a t about an 18 p e r c e n t c h o r d p o s i t i o n , l e a v i n g the s e t u p prone to l a r g e moments. At the h i g h f r e e s tream v e l o c i t y a t which the model was t e s t e d , t h e s e moments tende d t o t w i s t t h e model i n a streamwise d i r e c t i o n a t h i g h a n g l e s o f a t t a c k and f l a p d e f l e c t i o n . There was o b s e r v e d t o be as h i g h as one degree i n a n g l e of a t t a c k e r r o r . T h i s t o r q u e was not v e r y e v i d e n t between -A and +6 degrees a n g l e o f a t t a c k . S i n c e t h e f l a p was s e c u r e d t o t h e main a i r f o i l by s o l i d e n d p l a t e s t h e r e were no c o n c e r n s about i t s d e f l e c t i o n a n g l e b e i n g a l t e r e d by a i r f l o w . 26 Pressure measurements were the main emphasis of experiments in this study. The tiny pressure taps on the model were connected to the scanivalves through small plastic tubes. Although the area is clean and the air relatively dry, i t is not out of the question that foreign objects may have entered the system. This could cause blockages of the tubes. Total blockage is easily detected. Partial blockage is much more difficult to discover. It may appear in a number of forms. Perhaps the most common is a sluggishness in voltage readout from port to port in the scanivalve. The partial blockages would force the researcher to allow more time for the readouts to settle. So effect-ively, the small tubes form dampers in the pressure data acquisition system. 4.3 Wind Tunnel Wall Corrections These data are not corrected for wall effects. In Appendix C, the trapezoidal method of determining experimental values for C^ is shown. This method could be extended to determine Cmc/4 for any given configuration of flap, spoiler and wing. As formula (4.1) shows, both » t C.and C ,, are needed to determine true C. and hence C_. As shown in L mcM L P Ref. 11 3C (4.1) CJ = C. (1 - 2e - o) - f- (C. - 4 C ) r-^ L L ZTT L m ., OCX c/4 where e = Ao; A = airfoil shape factor; o = TT*/A8 ( C / H ) J , To get C P : 1 L p W . 27 The agreed scope of this work did not allow for data reduction to be carried out towards the determination of corrected pressure distribu-tions. All the data have been collected and when the theory for these experiments is pursued, the corrected values C£ and Cp must be used to compare experiment with theoretical predictions. 28 CHAPTER 5  OBSERVATIONS AND RESULTS 5.1 General Conf igurat ion Phenomena 5.1.1 T r a i l i n g Edge of Main Joukowsky A i r f o i l The main Joukowsky a i r f o i l was b u i l t i n 1969 and used by J a n d a l i i n h i s work on s p o i l e r s , Ref . 6. I t was b u i l t of mahogany by the t e c h -n i c i a n s i n the Mechanical Engineer ing Shops. Since Joukowsky a i r f o i l s have cusp t r a i l i n g edges, i t i s p l a i n that t h i s presents cons truc t ion d i f f i c u l t y . To overcome t h i s problem, the upper surface of the a i r f o i l i n the t r a i l i n g i n c h o f chord was r a i s e d . T h i s e f f e c t i v e l y thickened the t r a i l i n g edge f o r s trength and s t i f f n e s s . Unfortunate ly , t h i s a l s o a l t e r e d the c h a r a c t e r i s t i c s of the a i r f o i l from what theory may p r e d i c t . Although the d i screpanc ies are smal l they can be seen i n F i g . 5 .1 . Note the 'bump' at about 65 percent chord on the wing upper surface l i n e . The e f f e c t s o f t h i s are not detr imenta l to the r e s u l t s s ince they are smal l and l o c a l i z e d . 5 .1 .2 Blockage E f f e c t s by F l a p E x t e r n a l A e r o f o i l The t rend o f the c o e f f i c i e n t of pressure along the lower surface of the main f o i l i s to r a p i d l y approach zero a f t e r being u n i t y at the s t a g -n a t i o n p o i n t . T h e o r e t i c a l l y , and e a s i l y shown by experiment ( J a n d a l i , r e f . 6 ) , i t does not reach zero but rather approaches and increases s l i g h t l y again as the t r a i l i n g edge of the a i r f o i l i s approached. Because the f l a p a i r f o i l i s mounted behind and below the main a e r o f o i l i n t h i s s tudy, i t creates a blockage e f f e c t . The conf igura t ion i s shown i n F i g . 3.2 (pg. 18) . One would expect a dece l era t ion i n flow upstream - 5 - 4 H - 3 - 2 CL O - H • WINGUPPER O WINGLOWER A FLAPUPPER_ O FLAPLOWER r 6 0 0 2 0 4 0 PERCENT OF TOTAL CHORD F I G U R E 5.1 P R E S S U R E D I S T R I B U T I O N : T 8 0 100 NO SPOILER oc= 0 deg 6= 0 deg -5-i • WINGUPPER - 4 -- 3 -O WINGLOWER A FLAPUPPER O FLAPLOWER Q_ O - 1 -2 0 4 0 6 0 —r-8 0 100 CO O PERCENT OF TOTAL CHORD m S P 0 I L t R FIGURE 5.2 PRESSURE DISTRIBUTION: a = 4 deg <5= o deg 31 of the flap because of its thickness. This would result in the increase in Cp shown in Fig. 5.2. A great reduction in velocity contributes to the Cp increase to about 0.6 at the trailing edge lower surface of the main ai r f o i l . Although the airfoils of the flap and wing are dissimilar in cross-section as well as size, this dramatic increase in Cp is not observed on the flap. Figure 5.2 shows the flap in an undeflected state. This does not show this Cp increase well. When i t is deflected 20° one can more easily see that no similar effects are experienced by the flap. Figure 5.13 on page 48 better indicates the almost imperceptible increase in Cp along the underside of the flap. This supports the supposition that the effects observed on the underside of the wing in Fig. 5.2 are caused by blockage by the leading edge of the flap. 5.1.3 Flap Leading Edge Curvature Effects As can be seen in Fig. 3.3 on page 20, the Joukowsky flap airfoil section is relatively thick with a visible camber. The large curvature found on the leading edge, and particularly on the lower surface at the leading edge causes some effects worth noting. When the flap is in an undeflected condition, a high negative Cp is found in the area of concern. The pressure coefficient on the lower side is even more negative than that of the upper surface. Pehaps the restriction caused by the small gap above the flap also contributes to these effects. The cause of the high negative pressure coefficients around the lower surface leading edge is the increased airflow velocity in that region. This is only found in the undeflected flap configuration however, no matter what the angle of attack is. As Fig. 5.6 on page 38 shows, at a 32 ten-degree deflection, the effects of the curvature have already been overcome and lower flap surface Cp's are once again higher than those of the upper flap surface. Figure 5.14 on page 49 also shows the absence of these effects at flap deflections of 20 and 40 degrees. 5.1.4 Coefficients of Lift Versus Angle of Attack The C^ vs a curves for the experiments in this research complied very well with what theory might predict. Fig. 5.3 shows that the curve is virtually linear with a dCL/da of about 0.1. See Appendix D for these calculations. According to Houghton and Carruthers, Ref. 5, the theoretical value is 2n/rad which corresponds with 0.1097 per degree. Also in Fig. 5.3 one must note the small upward deflection of the curve towards the lower end of the o scale. This could be caused by the beginnings of negative stall at these low angles of attack. 5.1.5 "Single-Airfoil" Tendencies This two-element Joukowsky airfoil system appears to take on single-a i r f o i l tendencies. As Fig. 5.4 shows, the pressure distribution on the upper surface of the main airfo i l ceases to descend as theory predicts i t should, were i t a single a i r f o i l . Instead i t turns upwards as the flap is approached. This is a form of upstream negative pressure recovery. "Upstream" i s , of course, with reference to the flap. In Fig. 5.4 there is a 20 degree deflection of the flap. This results in a high negative pressure coefficient at the leading edge of the flap. The trailing edge of the wing, because of its close proximity to the flap, tends to this high negative pressure coefficient. A more obvious comparison of this effect is shown on Fig. 5.14, page 49. At a higher 3n - 5 C L o - H H 0 P •o O' • W1NGUPPER O WINGLOWER A [LAPUPPER O FLAPLOWER •©••••O O © o 0 2 0 4 0 6 0 8 0 PERCENT OF TOTAL CHORD FIGURE 5.4 PRESSURE DISTRIBUTION: 100 a= 4 NO SPOILER deg 6= 20 deg 35 flap deflection than Fig. 5.4 shows, this negative increase in Cp is very dramatic. The upper surface of the full wing section appears to effectively "ignore" the gap. ; 5.1.6 Trailing Edge Data Points In the 'Experimentation' section of this report, i t was mentioned that external pressure tap lines were secured to the trailing edges of the wing and flap airfoils. The data recovered from these positions leave the analyst with doubts as to their accuracy. Figure 5.4 shows a point for the trailing edge of the wing in good comparison with those ' positions slightly upstream on the upper surface. According to theory, a stagnation point exists on the trailing edge of an airfoil in airflow. This is in keeping with the Kutta condition. This is not achieved experimentally for the wing in Fig. 5.4 but usually trailing edge data may be expected to represent more of a balance between upper and lower surface pressure data. The trailing edge of the flap in Fig. 5.4, however, has a Cp of about zero. This appears to complement the more exact data upstream of this point. One might suspect that the pressure tap in the first case was more sensitive to the upper surface flow than the lower. This is easily possible as the taps are held in position along the knife-edged trailing edge, by tape. Slight shifts to the upper or lower surfaces of the wing are not unlikely. 5.1.7 Flap Upper Surface and Spoiler Back Pressure Correlation Figure 5.5 shows an interesting effect along the top surface of the flap. The almost constant Cp along this surface corresponds very closely to the back pressure measured behind the 10% normal spoiler. -5 CL O -4 -3H - 2 - H • WINGUPPER O WINGLOWER A FLAPUPPER_ O FLAPLOWER PERCENT OF TOTAL CHORD FIGURE 5 . 5 PRESSURE DISTRIBUTION: Z°\ z%™ 7' 1 0 0 90 de 0 de 37 This happened despite the unstalled condition of the flap. Light flow separation was noted in the last 1.5 inches of the upper surface of the flap. Once again this supports the hypothesis presented in section 5.1.5 that the two-element wing expresses characteristics of a simple one element a i r f o i l . Were a spoiler used on a simple ai r f o i l , flow would remain separated through to the trailing edge on the upper surface. The empirical constant back pressure assumption used in Ref. 7, holds validity because of this. This remarkable effect is consistent throughout tests in this work where spoilers were used, irrespective of flap deflection, and despite an unstalled flap condition. 5.2 Separation Effects 5.2.1 Unseparated Flow In the experiments carried out in this report, very few were entirely free of separation effects, however light. Fig. 5.6 is one that does show a configuration where separation is absent. Succeeding sections and figures will briefly describe some of the experiments where separation was noticeable. The tufts described in Section 3 were the means by which the separation and its degree were visually noted. It is also important to note that the tufts did not indicate any three-dimensional effects in any of the tests in this study. 5.2.2 Separation on the Lower Flap Surface Attention is drawn, when studying Fig. 5.7, to the unusual Cp distribution around the slotted flap. The high negative Cp on the lower flap surface hsa been discussed in section 5.1.3 but the example used in that discussion was far from as adverse as this case. The flap was negatively or upwardly deflected. This is not a practical configuration - 5 - i -4 A - 3 -2H Q_ O • W I N G U P P E R O W I N G L O W E R A F L A P U P P E R _ O F L A P L O W E R 6 0 8 0 100 P E R C E N T O F T O T A L C H O R D F I G U R E 5 .6 P R E S S U R E D I S T R I B U T I O N : a_ 0 " ° S P 0 , L/ R i n , ex— v a e g o= 10 d e g 00 - 5 n -4.H - 3 • WINGUPPER O WINGLOWER A FLAPUPPER_ O FLAPLOWER - 2 CL O - 1 A 0 2 0 4 0 PERCENT OP TOTAL CHORD F I G U R E 5.7 P R E S S U R E D I S T R I B U T I O N : A = 2 I 6 0 80 N O S P O I L E R d e g 6 = 100 - 1 0 d e g 40 in rea l i ty , but for study purposes proved interesting. The underside of that high camber flap aerofoil was experiencing separation but was not completely s ta l led . The tufts were pulled away from the surface but were not drawn anywhere but downstream. In stal led conditions, they would have been usteadily f luttering upstream. 5.2.3 Separation on Trai l ing Upper Flap Surface In the configuration corresponding to F ig . 5.8, moderate separation in the rear several inches of the flap surface was observed. The high flap deflection of 30 degrees in concert with a negative angle of attack of the wing are responsible for th is . The gap before the flap i s in the shadow of the negative angle of attack on the wing. Less airflow can make i t s way through to the upper flap surface to help with boundary layer control and delay separation. The upper flap airflow is more affected by the free stream and deflects from the surface that is effectively at a 25° angle of attack. In this study, i t was shown that the flap can remain unstalled at 40 degree deflections up to 8 degrees angle of attack when spoilers are extended. This form of separation was less l ike ly at positive angles of attack even without spoilers due to the greater exposure of the gap to airflow. 5.2.4 Pre-Spoiler Separation Bubble Yeung in his thesis, Ref. 16, cites some of Jandali's unpublished work on a separation bubble upwind of a spoiler. This can be easily explained because the flow w i l l undoubtedly be slowed down by the v i r tua l fence erected in i t s path. The slowing of the f lu id results in increased C p , followed by local boundary-layer separation and a -5 •4H - 3 J • WINGUPPER O WINGLOWER A FLAPUPPER •m a— mm mm • • • • • • • • • • • • i s O FLAPLOWER Q_ C J o 2 0 4 0 6 0 PERCENT OF TOTAL CHORD FIGURE 5.8 PRESSURE DISTRIBUTION: 8 0 100 cx= — NO SPOILER 4 deg 6= 30 deg • WINGUPPER O WINGLOWER A FLAPUPPER O FLAPLOWER .© •o-• © • •A 7 "T • , , - r - — , , , , 2 0 4 0 6 0 8 0 100 PERCENT OF TOTAL CHORD 5.9 PRESSURE DISTRIBUTION: H°Y%e 7g % ^ ? 0 ° d e 43 constant-pressure bubble. This phenomenon is observed on F ig . 5.9. A 7% spoiler exists in this test and, as one can see, the wing upper surface pressure coefficients actually swing to the positive sense upwind of the spoiler despite a 6 degree angle of attack. Although the tufts did not show this effect, had smoke or some other means of flow visual izat ion been tr ied as Jandali d id , i t would have shown this bubble. 5.2.5 Separation on the Tra i l ing Upper Wing Surface Although the curves drawn for the upper wing on F i g . 5.10 do not" show irregulari ty , flow visual izat ion detected l ight separation on the rear upper surface of the wing. This can easily be explained by the high angle of attack of the system. A highly adverse pressure gradient set up in the upper surface of this wing w i l l seek boundary layer separ-ation. The highly exposed gap between wing and flap at this elevated angle of attack unquestionably serves as boundary layer control, delaying the s ta l l ing of this wing and undeflected flap arrangement. 5.2.6 Leading Edge Separation Bubble In many of the experiments at high angles of attack, regardless of spoiler existence, a constant ripple in pressure distribution was experienced on the upper surface of the wing within the f i r s t 5% total chord. This i s probably caused by a small separation bubble incurred at such a high angle of attack. Reattachment is very quick. Figure 5.10 shows this ripple very c learly . Figure 5.11 is a similar flap deflec-t ion with no spoiler and s l ight ly reduced angle of attack. In this case, the ripple i s absent yet the coefficient of l i f t is only about .2 0 2 0 —r-4 0 - r — 6 0 i 8 0 100 PERCENT OF TOTAL CHORD FIGURE 5.10 PRESSURE DISTRIBUTION: NO SPOILER a= 12 deg 6= 0 deg ~ r — 2 0 ~1~ 6 0 0 2 0 4 0 PERCENT OF TOTAL CHORD FIGURE 5.11 PRESSURE DISTRIBUTION: 8 0 100 4S Cn cx= 8 NO SPOILER deg (5= 0 deg 46 lower. (This corresponds to a dC /da of about 0.05:half the theoretical value). At a = 12° as in Fig. 5.10, C L = 1.47; at o = 8 ° as in Fig. 5.11, = 1.25. Examination of the two figures shows that the flap pressure distributions are almost identical for the two cases. 5.3 Flap Effects 5.3.1 Flap Effects on Coefficient of Lift As Fig. 5.12 shows, the effects of flap deflection are substantial on the lifting capabilities of a wing section. The slopes of the C^  vs a curves are unaltered as would be expected. The various configurations stalled at lower angles of attack as flap deflection angle was increased. The important zero l i f t angle found near -5° angle of attack with no flap, was pushed substantially further negatively with the addition of a flap deflection. Fig. 5.13 shows the great lifting effects of the flap. It is used as a high l i f t device and the reasons are evident when the flap pressure distribution is seen in Fig. 5.13. Another phenomenon observed on Fig. 5.12 is the greater effect on l i f t found between 6 = 0 ° and 6 = 20°, than exists between fi = 20° and 6 = 40°. This will be noted again in the following section. 5.3.2 Flap Deflection Angle Comparison The pressure distributions in Fig. 5.14 give good perspective on the effects of flap deflection angle. Note the great effects on flap pres-sure distribution between 6=0° and 6=20°, and again, but to a lesser degree, between 6=20° and 5=40°. The main airfo i l l i f t increases measurably. In the 6=20° and 6=40° conditions, separation was experienced on the last two inches of the upper flap surface. No separation was incurred in the - 5 - i -4 - 3 • WINGUPPER O WINGLOWER A FLAPUPPER O FLAPLOWER CL O 0 o " • i • G -H —r— 2 0 — r -6 0 0 4 0 PERCENT OF TOTAL CHORD FIGURE 5.13 PRESSURE DISTRIBUTION: a = 0 8 0 100 00 NO SPOILER deg 6= 20 deg 0 2 0 4 0 6 0 PERCENT OF TOTAL CHORD FIGURE 5.14 PRESSURE DISTRIBUTION: NO SPOILER AND CONSTANT a , VARYING S 8 0 100 VO a = 4 d e g 50 6 = 0 ° case. The greatest changes once again appear between 6 = 0 ° and 6 = 20° as was noted in section 5.3.1 and on Fig. 5.12. It is" also interesting to note the pressure trend of the trailing edge of the main airfoil as the flap is approached. As was noted in Section 5.1.5, a single-airfoil behaviour becomes apparent. This could also be due to trailing edge measurement inaccuracy. Since the gap is only 0.28 inches at its widest point, and the pressure tap is almost a third of that, the data point at the trailing edge of the wing could be greatly affected by the leading edge of the flap. 5.A Spoiler Effects 5.4.1 Curve Fitting on Spoiler Back Pressure Coefficients As the data points symbolized in Fig. 5.15 show, spoiler back pressure was measured to be constant. The curve drawn, however, does not indicate such a condition. This is due to the spline curve fitting routine used in plotting these graphs. Some variation in spoiler back pressure measurement was observed when using the 10% spoiler in testing. This is easily seen by the symbols on Fig. 5.16 using upper surface data. When the 10% spoiler was fitted to the end plates, i t was not as snug to the main airfoil as the 7% spoiler was. One will notice four spoiler back pressure data points, excluding the trailing edge point. The forward two of these points were taps 23 and 24 on the main airf o i l . The rearward two points were pressure taps taped to the back of the spoiler as explained in section 3.1. The airflow seeping under the spoiler over pressure taps 23 and 24 on the wing can account for their consistency in being more negative pressure coefficients than the more reliable data obtained on the spoilers' downstream surfaces. - 5-1 QL -3H • WINGUPPER O WINGLOWER A ELAPUPPER am mm mm mm mm mm mm • • O FLAPLOWER A ' ^ - A A , 2 0 4 0 6 0 PERCENT OF TOTAL CHORD 8 0 FIGURE 5.15 PRESSURE DISTRIBUTION: ll ona £\ 7g  % 100 90 de 0 de • 5 - n - 4 -3H • WINGUPPER O WINGLOWER A FLAPUPPER O FLAPLOWER C L a - 2 H 0 —r— 2 0 4 0 6 0 8 0 PERCENT OF TOTAL CHORD FIGURE 5.16 PRESSURE DISTRIBUTION: i ^ V g 0 5 5 100 90 deg 0 deg 53 5.4.2 Lift Coefficients vith Spoilers Extended As would be expected, the zero l i f t angles calculated for wing configurations with spoilers are higher than those without spoilers. Fig. 5.17 shows this angle to be about -3 degrees. Fig. 5.12 on page 47 showed a value of -5° for a similar configuration without spoiler. To be noted as well, the theoretical dC /da value is obtained L experimentally once again: 0.1 even with spoiler extended. 5.4.3 Spoiler Effects on Pressure Distribution Figure 5.18 shows the monumental reduction in lifting capability of the main wing section with the extension of a spoiler. C^ for no spoiler is 1.88. When a 7% spoiler is extended, i t is reduced to 0.82. This control surface's usefulness for the approach and landing of an aircraft is understood when one sees the effects of such a small spoiler in a typical flying configuration of angle of attack and flap deflection. Figure 5.19 shows the effects of a spoiler at a higher angle of attack. An interesting point to note is the small change in flap pressure distribution despite the addition of a spoiler. The pressure distribution about the wing was collapsed with the intrusion of a spoiler, yet the flap is not as significantly affected. The slot is the main reason for this effect. The influx of air from the lower surface serves as boundary layer control as well as maintaining flap effective-ness regardless of what is happening upstream on the upper wing surface. Were the trailing edge points of the flap more certain, one might be able to see them better correspond as well. -1 10 - 8 - 4 - 2 0 2 4 8 10 12 FIGURE 5.17 C L v s a CONSTANT SPOILER AND 6 a S P O I L E R : 1 0 % £ = 9 0 d e g 6= 2 0 d e g CL O SPOILER • NO-SPOILER O 10%&90deg m ^ I i r _ • PERCENT OF TOTAL CHORD FIGURE 5.18 PRESSURE DISTRIBUTION: «= 4 d e g CONSTANT a A N D <5, VARYING SPOILER 6= 2 0 d e g Ln - 5 SPOILER i_ | . , 1 1 . 1 1 1 ' 1 0 2 0 4 0 6 0 8 0 100 • PERCENT OF TOTAL CHORD FIGURE 5.19 PRESSURE DISTRIBUTION: a= 8 deg 6= 20 deg CONSTANT aAND (5, VARYING SPOILER 5 7 A note on the plotting of these curves, only every fourth data point was drawn on the curves to promote clarity. 5 . 4 . A Effects of Spoiler Size Although some change in pressure distribution and hence lifting capability was observed between 7 % and 1 0 % spoilers, i t was nowhere near as evident as the change between no spoiler and a 7 % spoiler. This is not surprising. As Fig. 5 . 2 0 shows, the flap pressure distribution is virtually unchanged with the 3 % spoiler increase. The wing however, experiences a more substantial reduction in l i f t . The for the 7 % spoiler in this configuration is 0 . 8 2 . The same quantity for a 1 0 % spoiler is 0 . 5 6 . Most of the l i f t lost is a result of the spoiler size increase effect on the main wing's lifting capabilities. Another plot is presented in Figure 5 . 2 1 which compares spoiler size effects at a higher angle of attack. Once again the same results are obtained and l i f t is lost from the main airfoil whereas the flap remains unchanged. 5 . 5 Effects of Spoiler Use with Slotted Flap 5 . 5 . 1 Effects of Flap Deflection Angle With 1 0 % Spoiler Figure 5 . 2 2 shows some interesting effects of increased flap deflection when a spoiler is extended. Of worthy note is the increase in spoiler back pressure corresponding to increased flap angle. As Section 5 . 1 . 7 describes, this is reflected in the flap upper surface pressure coefficients as well. When the spoiler is extended with no flap, one can see that the positive pressure coefficient distribution shown in Fig. 5 . 2 on page 3 0 with no spoiler, is collapsed in Fig. 5 . 2 2 , - 5 - 4 H - 3 SPOILER • 7%&90deg O 10%&90deg - 2 C L O 0 2 0 4 0 6 0 8 0 , PERCENT OF TOTAL CHORD FIGURE 5.20 PRESSURE DISTRIBUTION: a= 4 deg CONSTANT a A N D (5, VARYING SPOILER 100 Cn Oo 6= 20 deg - 5 i • PERCENT OF TOTAL CHORD FIGURE 5.21 PRESSURE DISTRIBUTION: a= CONSTANT a A N D 6, VARYING SPOILER - 5 -4H -3H -2H QL O A • 0_ 0 20 A 40 0 20 4 0 6 0 8 0 100 .. PERCENT OF TOTAL CHORD FIGURE 5.22 PRESSURE DISTRIBUTION: S P 0 I L E R : 1 0 % *= 9 0 d e 9 CONSTANT SPOILER AND a, VARYING 6 ON o a= 4 deg 61 to the point of producing a negative l i f t coefficient: C L = -0.025. Although this recovers to a positive value with increased flap deflection, i t is most certainly the result of l i f t created almost entirely by the f lap. Fig.5.23 shows some interesting effects upstream of the spoiler. As mentioned in Section 5 . 2 . 4 , the recirculation zone upstream of the spoiler has a detrimental effect on the main wing l i f t capabi l i t ies . The reduced negative pressure coefficients observed between the 30 and 60 percent chord positions are due to this zone of reduced velocity. Obvious dips are experienced by a l l three curves at the point immediately upstream of the spoiler. Port 22 on the main a i r f o i l i s very close to the base of the 10% spoiler and hence velocity is close to zero. The poor curve f i t t ing immediately behind the spoiler i s explained in Section 5.4.3. The tests described in F ig . 5.23 are very illuminating when considering the effects of spoiler on flap performance. Where the l i f t is drast ica l ly reduced on the main a i r f o i l , even at high angles of attack the flap s t i l l produces high l i f t and i s maintained free of s t a l l effects. In fact, the spoiler extension actually allowed increased angle of attack without s ta l l ing the f lap. With no spoiler and 40 degrees flap deflection, f u l l s ta l l ing occurred after 4 degrees angle of attack. The curve shown on Fig . 5.23 is the same flap deflection, with spoiler this time but at 8 degrees angle of attack. F u l l s ta l l ing occurred beyond that point. This effect is due to the spoilers restr ict ing flow over the top surface of the wing (reducing circulation), hence reducing the s t i l l adverse pressure gradient, and delaying separation unt i l a greater angle of attack is reached. g_ , ' 1 1 1 1 1 1 1 ' I 0 2 0 4 0 6 0 " 8 0 100 PERCENT OF TOTAL CHORD FIGURE 5.23 PRESSURE DISTRIBUTION: S P 0 , L E R : 1 0 % 9 0 d e g CONSTANT SPOILER AND c c , VARYING 6 a = 8 d e g 63 5.5.2 Flap Effects on Lift Coefficient vith Spoiler Extended As one might expect, the l i f t v i l l s t i l l increase vith flap deflec-tion. The magnitude, hovever, is substantially lover than vith no spoiler extension. Fig. 5.12 on page 47 shovs phenomenal vertical translation of C^ vs a curves vith no spoiler extension, as flap deflection angle is increased. This trend is also seen on Fig. 5.24 but by no means to the same extent. Once again more change occurs in the 20° increase from 6 = 0 ° than in the same increase from 6 = 20°. Another interesting note concerns the zero l i f t angles of the three curves on Fig. 5.24. When the spoiler is extended vithout flap extension, the zero l i f t angle is actually positive at about 4°. This reduces to -3° and -6.5° vith 20° and 40° flap deflections respectively. This is especially useful in landing aircraft. A positive l i f t at negative angle of attack allovs steeper glide angle and slover approach speed because drag varies vith l i f t . One further point obvious on Fig. 5.24 is the consistency in dC^/da around 0.1 as predicted by theory and seen before in this vork. 5.5.3 Spoiler Inclination Angle Effects vith Flap Deflected Once again the slotted flap immunity to spoiler deflection is observed in Fig. 5.25. The l i f t curves of the main ving at 8° angle of attack are substantially collapsed. It is easily seen again that the effects of a control surface are more substantial vhen i t is initially deployed. Here a greater l i f t reduction happens betveen £ • 30° and I = 60° than betveen £ = 60° and £ = 90°. This latter point is more readily seen in the C L vs a curves on Fig. 5.26. Very l i t t l e happens betveen £ = 60° and £ = 90°. In 3-i 2H C f 1-c5 • 0_ O 20 A 40 A , A ' .A" a FIGURE 5.24 C L v s a 10% SPOILER AND VARYING 6 .A .O - 5 - 1 C L O 100 .. P E R C E N T O F TOTAL C H O R D C D n i l r D i n q r FIGURE 5.25 PRESSURE DISTRIBUTION: a = 8SPd°eLgER: T° 20 deg CONSTANT a AND 6 AND SPOILER SIZE, VARYING f O l FIGURE 5.26 C L v s a CONSTANT 6 WITH 10% SPOILER, VARYING £ ON 2 0 d e g 67 fact their zero l i f t angle only varies by about half a degree. More substantial effects are noted between I = 30° and I = 60° with" a 3.5° decrease in zero l i f t angle. Again one can see that the slope of the curves in Fig. 5.26 is constant at about 0.1 per degree. The pre-spoiler dip mentioned in Section 5.5.1 is again evident on the curves in Fig. 5.25. 5.5.A Effects of Spoiler Size Changes with Flap Deflection Figure 5.27 shows some further evidence of slotted flap immunity to spoiler l i f t reduction. Virtually no change is noted in pressure distribution around the flap. This is further explained in Sections 5.4.3 and 5.5.1. The boundary layer control from the gap creates reattachment of the flow at the flap after i t was separated at the spoiler tip and the trailing edge of the wing. The phenomenon of i n i t i a l control surface effectiveness is once again noted as l i t t l e change occcurs between 7% and 10% spoilers. Substantial circulation reduction by the 7% spoiler on the undisturbed configuration is apparent as the pressure distribution collapses, apparent on Figure 5.27. This point is reiterated with more evidence presented on the C^  vs a curves of Fig. 5.28. Once again the curve slopes are unchanged from previous observation, and the major l i f t reduction occurs initially: between no spoiler, and a 7% spoiler. Q. O - H SPOILER • NONE PERCENT OF TOTAL CHORD FIGURE 5.27 PRESSURE DISTRIBUTION: ^ E R d e g CONSTANT a AND c5 AND ^.VARYING SPOILER SIZE 1 0 0 9 0 d e g 2 0 d e g oo SPOILER • NONE O 7% FIGURE 5.28 C Lvs a CONSTANT 5 AND £ , VARYING SPOILER SIZE 70 CHAPTER 6  DISCUSSION AND CONCLUSIONS Encouraging results were obtained from pressure measurement experiments on a Joukowsky airfoil with Joukowsky slotted flap and spoiler. The results are qualitative only and are not to be compared quantitatively with theory until wind tunnel wall correction factors are applied. All data are presented or available in a data bank to complete the required corrections. The experimental setup was effectively a two-element Joukowsky airfoi l arrangement with variable secondary airfoil deflection. The secondary airf o i l was modelling a slotted flap. Several spoilers of varying size and inclination were used in the testing. Theory, first proposed by Williams (Ref. 15) for the production of a two-element Joukowsky aerofoil was combined with the work by Parkinson and Jandali (ref. 7), on separated flow and Parkinson and Yeung, on potential flow about airfoils with spoilers (ref. 10). The end product of this theory was a system incorporating two 'near-Joukowsky' airfoils with a spoiler, of arbitrary inclination. Despite the resulting 'near-Joukowsky' airfoils, the assumption was made that they were not too different from true Joukowsky airfoils and experimental results would coincide within error margins. In his paper, Halsey, Ref. 4, shows that the variations on shapes situated away from the origin, under conformal transformations, are small, and can be reduced further by repeated mappings. The use of the Theodorsen method explained in ref. 13 can extend this work to the use of slotted flaps and spoilers on real airfoils. 71 The Joukowsky airfoils are used in this analysis for two reasons. First, an existing Joukowsky model could be put to use and more importantly, much simpler theory would be required to compare with the experiments in this report. As one studies the overview of theory provided in Section 2, one can acquire an idea of the complexity of the theory already required to combine the three works cited earlier in this section. Once the comparisons are made however, Theodorsen mappings will make this work more applicable to real airfoil control system design. Of great encouragement in this report are the qualitative results obtained for the behaviour of slotted flaps and spoilers. The effects they had on l i f t coefficients are complementary to these control surface used in real situations. The high l i f t characteristics of slotted flaps without spoilers demonstrate their effectiveness on takeoff. The ability of the flap to remain unstalled despite large deflection supported the role of slotted flaps over normal or split flaps in aircraft today. Higher l i f t at lower angles of attack will also allow aircraft to take off at lower velocities, reducing runway requirement and hence increasing the effective range of uses of an aircraft. When used in concert with spoilers, the balance of great l i f t reduction and retained control, i.e. unseparated flow over the model, shed light on the configuration's role in approach, air-braking, and landing of aircraft. The much-reduced zero l i f t angles of attack would allow steep, controlled approach paths. As well, high l i f t normally signifies high drag; so slower approaches and landings are possible. These effects were a l l noted experimentally, and, considering the safety 72 which accompanies steep glide paths and slow approach velocity, the roles of spoilers and slotted flaps are supported. Interesting phenomena were noted on the in i t i a l uses of flaps and spoilers. The effects of these controls were more evident when they were first introduced into the flow. For example i t was observed, at various points in this research, that slotted flaps made a more significant difference between 0° and 20° deflection than they did after 20°, to AO0 deflection. Similarly, the effect of increasing spoiler inclination beyond 60° was not remarkable. This supports the requirement of aircraft for maximum effectiveness of controls with minimum added linkage and hence minimum contribution to overall weight of the aircraft. There is l i t t l e need to extend flaps beyond AO0 and no need to incline spoilers beyond 60°. The added deflection makes l i t t l e or no difference to flight characteristics. This latter point also is important in the future theoretical study of this configuration. Since the theory to produce inclined spoilers involves many complex mappings, then perhaps approximations can be made from the less complicated theory to produce normal spoilers. Finally, the research found in this report is useful in providing a better idea to aerodynamicists, of the effects of control surfaces. Spoilers and slotted flaps are not new to the real flying world. Analytical theory can do nothing but improve their effectiveness through further understanding. This experimental work brings real airfoil design and modification closer to the arcane world of conformal mapping and theoretical airfoils. 73 CHAPTER 7  RECOMMENDATIONS This work has shed light on some promising future developments for aerodynamics at the University of British Columbia. Some work related to this research is required and some equipment modifications are recommended. It is understood that the time constraints on this thesis leave much to be done to complete the study of the flow about a Joukowsky airfoi l with slotted flap and spoiler. It is imperative that wind tunnel correction factors be applied to the data bank created in this experimental study. As well, the theories proposed in Section 2 must be concluded to determine velocity potentials for the two-element Joukowsky airfoil on which the experiments were carried out. This will be no mean feat when one considers the multitude of spoiler and slotted flap combinations obtainable and used today in real flight situations. Perhaps some testing in the 'smart' wind tunnel at UBC would provide some insight towards the magnitude of and accuracy of wind tunnel corrections to be applied. It was discovered that the scanivalves used in the testing are prone to blockage and require more frequent maintenance attention. Small leaks are common to such a complex apparatus and can be a severe impediment to an experiment. It is recommended that the instruments be compared to manometer readings before major testing situations. A computerized data acquisition system for the existing scanivalve arrangement would save enormous amounts of time and contribute greatly 74 to the accuracy and reliability of the researcher's work. The monotony and slow process of manual data recording on the existing scanivalve equipment leaves a margin of error because of the human liabilities of fatigue and impatience. Some recommendations to further the accuracy and expand the capability of existing equipment are proposed as follows. To accurately measure the cusp trailing edge pressures on existing Joukowsky airfoils, perhaps a small groove can be cut down the metal surface of the section containing pressure taps. This could be sealed from the surface yet open at the delicate trailing edge. The groove could be accessed from the inside of the section, just as a l l the other taps are. The cover must be thin and must not pose alterations to the cross-section of the a i r f o i l . Problems were also encountered with twisting of the airfoil due to the moments caused by the 43% increase in effective chord of the section: the flap addition. Further studies on this setup should include a means of securing the airfoils from such torque. A mounting device which places the mount at a new effective quarter-chord position is proposed to reduce the magnitude of the moments on the base. The time constraints on this project insisted on a quick production of a 27-inch-span, 5.14-inch-chord Joukowsky airfoil meeting designated shape requirements. This was done in aluminum on the numerically controlled milling machine. The exactness and quality of the result was a testament to the capabilities of its production methods. It is recommended that, i f costs allow, this method be used to produce more test models in the future. The durability and toughness of aluminum are 75 great improvements and the cusp requirement of a Joukowsky trailing edge has been fulfilled. Perhaps an area of further study could be to delve into the use of slats at the leading edge of a wing section. A less common control surface than flaps, the slat is also used in high l i f t low speed situations like take-off and landing of aircraft. The small airfoil extension of the leading edge helps in boundary layer control, and basically is the slotted flap's leading edge counterpart. Control surface theory for airfoils is far behind the requirements of today's complex aircraft industry. Further work in this area would prove fascinating and would be of great economic and engineering importance to contemporary flight advances. 76 R E F E R E N C E S 1. Abbott, I.H. and von Doenhoff, A.E. "Theory of Wing Sections". Dover, 19A9. 2. Brown, G.P. "Steady and Nonsteady Potential Flow Methods", Ph.D. Thesis, University of British Columbia, 1971. 3. Foster, D.N., Irwin, H.P.A.H., Williams, B.R. "The Two Dimensional Flow Around A Slotted Flap", RAE Reports and Memoranda No. 3681, September 1970. 4. Halsey, N.D. "Potential Flow Analysis of Multielement Airfoils Using Con formal Mapping", AIAA Journal, Vol. 17, pp. 1281-8, December 1979. 5. Houghton, E.L. and Carruthers, N.B. "Aerodynamics for Engineering  Students", 3rd ed., London, 1982. 6 . Jandali, T. "A Potential Flow Theory for Airfoil Spoilers", Ph.D. Thesis, University of British Columbia, 1970. 7. Jandali, T. and Parkinson, G.V. "A Potential Flow Theory for Airfoil Spoilers", CASI Transactions, Vol. 3, No. 1, March 1970. 8. Milne-Thomson, L.M. "Theoretical Hydrodynamics", Macmillan &. Co., Ltd., 1955. 9. Parkinson, G.V. and Jandali, T. "A Wake Source Model for Bluff Body Potential Flow", JFM, Vol. AO, No. 3, pp. 577-59A, Feb. 1970. 10. Parkinson, G.V. and Yeung, W.W. "A Wake Source Model for Airfoils with Separated Flow", JFM, Vol. 179, pp. Al-57, May 1986. 11. Pope, A. and Harper, J.J. "Low-Speed Wind Tunnel Testing", John Wiley and Sons, 1966. 12. Shames, I.H. Mechanics of Fluids, 2nd ed., McGraw-Hill Inc., New York, 1982. 13. Theodorsen, T. "Theory of Wing Sections of Arbitrary Shape", NACA Rep. No. All, 1931. IA. Watt, G.D. "Multi-Element Thin Airfoil Theory", Ph.D. Thesis, University of British Columbia, 198A. 15. Williams, B.R. "An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils", RAE Technical Report 71197, September 1971. 16. Yeung, W.W. "A Mathematical Model for Airfoils with Spoilers or Split Flaps", M.A.Sc. Thesis, University of British Columbia, 1985. 77 APPENDIX A CALCULATIONS FOR JOUKOWSKY FLAP CONSTRUCTION As Appendix E w i l l elaborate, the following geometric configuration i s the starting point to create a Joukowsky a i r f o i l . Fig. A-1. When the Joukowsky Transformation i s applied an a i r f o i l results: Fig. A-2 The t/c ratio and camber of the resulting a i r f o i l are dependent on the values of p and e as defined in Fig. A-1. The requirements of the flap to be buil t , as described in Section 3 of this report, include a t/c 78 r a t i o of 0.15, a chord of 5.14 inches and a reasonable camber was requ ired to be i n keeping with r e a l i s t i c s l o t t e d f l a p s . Graphica l means were used to combine these requirements to obta in the a i r f o i l shown i n F i g . A - 3 . The Car te s ian coordinates of the c a l c u l a t e d a i r f o i l were s imply sca led to the proper chord length . Values of p and e are given below to create the set of coordinates l i s t e d i n Table A . l . p = 0.2343 e = 1 3 0 ° = 2.269 radians TABLE A-1 COORDINATES FOR A JOUKOWSKY SLOTTED FLAP 5.087392 5 .063474 5.034182 4 .999739 4 .960370 4 .758093 4 . 4 9 4 4 4 0 4 .182193 3 .832910 3 .457114 3 .064577 2 .664547 2 .265892 1.877157 1.506531 1.161777 0 .850131 0 .578185 0 .351796 0 .176002 0 .054989 - 0 . 0 0 7 9 1 - 0 . 0 1 0 1 7 0 .049991 0 .173749 0 .361692 0 .613808 0 .929239 1.305677 1.738295 2.218204 2 .730750 3 .254400 3 .761330 4 .220495 4 .602802 4 .886739 4 .940532 4 .987562 5 .027609 •5.060695 5 .125012 0 .237742 0 .246118 0 .256559 0 .269019 0 .283432 0 .358719 0 .456257 0 .566638 0 .679930 0 .786745 0 .878932 0 .949957 0 .995101 0 .011519 0 .998225 0 .956005 0 .887294 0 .796008 0 .687344 0 .567541 0 .443602 0 .322974 0 .213175 0 .121358 0 .053793 0 .015266 0 .008373 0 .032764 0 .084444 0 .155357 0 .233657 0 .305112 0 .355917 0 .376513 0 .365094 0 .328954 0 .282556 0 .271376 0 .260896 0 .251363 0 .243003 0 .225052 80 Figure A-3 Flap Joukowsky Section. 81 APPENDIX B  REYNOLDS NUMBER CALCULATIONS The experiments were carried out at a Reynolds Number of 700,000. The results however are theoretically independent of Reynolds number at this order. Knowns: g = 32.2 f t / s 2 p f l = .002378 slug/ft 3 p •= 1.936 slug/ft 3 C = 12 in w C f = 5.14 in so C = C + C, - 17.14 in w I -i, 2 air temperature of 80°F gives v * 1.8 x 10 f t / s : Ref. 12. Re = VC/v vRe/C 1.8xl0~*»7xl05 17.14 in • ^2 f t / i n = 89 ft/s To determine the corresponding manometer height: 1 „ 2 u = 2 pa V = p f g h h - I ^ y i 2 P f g 82 1 .002378 89 2 X 1.936 X 32.2 ft x 12 in/ft h = 1.817 in x 25.4 mm/in h •= 46.2 mm 83 APPENDIX C  LIFT COEFFICIENT CALCULATIONS A trapezoidal scheme was used to calculate coefficients of l i f t for both the main airfoil and the flap. Fig. C-l. Any Ax increments used must be corrected for a and/or 6 depending whether i t is found on the wing or the flap. Since i t was contributed to by positive Cp regions on the lower surfaces and negative C p regions on the upper surfaces, the quantities were added or subtracted as applicable. 84 CL "  1 iCtav>i * Ax. * cos a - I ( C p a v ) . * Ax. * cos o lower wing surface upper wing surface + 2 ( Cpav }i * * c o s ^ a + 6 ) " * ^vavh * Ax. * cos(a+6) lower flap surface upper flap surface When spoilers were used the following adjustment to x increments were made x l x s x j Ak Ax. « x -x. rather than x.-x. 1 s 1 j i Ax.. = \ ~ x s rather than a^-x^ 85 APPENDIX D C L vs a CALCULATIONS This approximation of a C^  vs a curve will demonstrate the dC^/da calculation. a) Determine a when C^ = +1. b) Determine zero l i f t angle, i.e. a when C^ = 0. c) Determine resulting Aa, i.e. (a)-(b). Calculate dCL/da = 1/Aa here dCL/da =0.1 86 APPENDIX E THE JOUKOWSKY TRANSFORMATION The Joukowsky Transformation is I = z + l/z The results of such a transformation on several figures are shown Fig. E - l. A unit circle is flattened to a flat plate. Fig. E-2. A larger circle becomes an ellipse. Fig. E-3. A vertically translated circle becomes a circular arc ai r f o i l . 8 7 Fig. E-4. A horizontally translated circle becomes a symmetrical a i r f o i l . 1 0 © Fig. E-5. A horizontally and vertically translated circle becomes a cambered Joukowsky a i r f o i l . This last transformation is the focus of this paper's attention. The degree to which the airfoil is cambered depends on the magnitude of e. The thickness of the resulting airfoil is dependent on the value of p. Note that the trailing edges of any Joukowsky a i r f o i l , Figs. E-3, E-A, and E-5, are cusped. 88 APPENDIX F ERROR ESTIMATES Quantity Error % Error Remarks Flow Temperature T + 10°F - Affects kinematic viscosity Kinematic Viscosity ± 5x10"6 f t V s 2.8 Affects Reynolds Number Reynolds Number Re ± 3x10* 4.0 Affects Flow Pattern Flow Velocity V ± 3 ft/s 3.4 Affects Dynamic Pressure Dynamic Pressure Q ± 0.6 psf 6.7 Manometer Height h + 1 mm 2.0 Voltmeter Readout ± .03 V 1.8 

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