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Multi-element thin airfoil theory Watt, George Donald 1984

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Multi-Element Thin Airfoil Theory by George Donald Watt B.A.Sc, The University of British Columbia, 1977 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of D O C T O R O F 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 December 1984 © George Donald Watt, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mechanical Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date December 10, 1984 DE-6 (.3/81) i i Abstract A linearized, two-dimensional, potential flow analysis of multi-element airfoil configurations is attempted. Arguments are presented that suggest the accuracy of the linearized theory should be as good as or better than that of the well-known one-element thin airfoil theory — despite the large mean line curvature common to multi-element configurations. The results obtained in this thesis tend to support this expectation. Good success is achieved in developing a general two-element tandem airfoil linearized theory. The effects of incidence, leading or trailing edge flap deflection, and camber on the overall and localized lifts and moments are summarized in integrals which are the tandem airfoil versions of the historical Munk integrals. Analytical solutions for the forces on tandem N A C A airfoils are then obtained from these integrals. The expressions for the forces reduce, when the two airfoil elements come together, to the familiar one-element thin airfoil theory formulas for an airfoil with a simple flap. All the forces are calculated on a small hand-held computer and the results are compared with exact potential flow theory. The overall force results for the tandem airfoil incidence problem are partic-ularly simple and, in fact, this simplicity allows the solution of the incidence problem for an arbitrary number of in-line airfoils. Although thickness has no effect on tandem thin airfoil theory forces, it does affect pressures. The general thickness distribution analysis for tandem airfoils, an exceed-ingly simple analysis involving only elementary functions, is presented. A similar type of analysis enables the design of tandem airfoil camber lines. Examples are given. Modelling the effects of overlap of a staggered two-element airfoil configu-ration is a much more complicated analysis than that of tandem airfoil elements. Never-theless, substantial success is achieved. This includes the development of the equivalent Munk integrals that give the overall and localized forces on airfoil elements of arbitrary chord length, arbitrarily positioned relative to one another (providing the chord lines are approximately parallel to each other and to the flow at infinity — the linearization require-ment). Although analytical solutions are obtained for the overall forces of the incidence and flap deflection problems, they are in terms of parameters which can only be obtained through a trial and error solution of the staggered parallel slit conformal mapping. ui Contents Abstract ii Tables v Figures vi Symbols vii Acknowledgement x 1 Classical Thin Airfoil Theory 1 1.1 Introduction 1 1.2 The potential flow - real flow correlation 5 1.3 The linearizing procedure 12 1.4 One-element thin airfoil theory functions and forces . . . 18 2 Tandem Symmetrical Airfoils 24 2.1 The mappings 24 2.2 The flow functions 29 2.3 The forces 32 2.4 N in-line uncambered airfoils 37 3 General Tandem Thin Airfoil Theory 39 3.1 The thickness distribution analysis 41 3.2 The camber line design problem 47 3.3 Camber line analysis: general functions 52 3.4 Camber line analysis: special functions 60 3.5 Thickness distribution design and other problems . . . . 64 3.6 Munk's integrals extended 65 4 Tandem Airfoil Forces an Easy Way 66 4.1 The overall forces 66 4.2 The localized forces 72 4.3 Segmented boundaries: the segment selectors 76 4.4 Tandem N A C A airfoils, an example 84 5 Staggered Two-Element Thin Airfoil Theory 100 5.1 The staggered parallel slit mapping 100 5.2 The overall forces I l l 5.3 The localized forces 119 6 Conclusions and Recommendations 123 References 126 Contents iv APPENDICES A Exact-Numerical Potential Flow Computations 127 A.l The flow field formulation . 127 A.2 The forces 129 A. 3 Computational accuracy 134 B Elliptic Functions . 141 B. l General 141 B.2 Elliptic function series representations in powers of q* . . 145 B. 3 Elliptic function series representations in powers of q16 147 C The Selector Function Expansion Coefficients 153 C l £{z) and F{z) 153 C. 2 A(u) and 154 C.3 ${OJ) and T(w) . 156 D Computer Program: The Tandem Airfoil Forces 159 V Tables 1 Potential flow predictions of the forces on the tandem airfoils of Figure 8 50 2 Comparisons of staggered slit mapping parameter values at the tandem and biplane slit extremes 109 3 Comparisons of the exact and present numerical predictions of the lift on Williams' configuration A slotted flap arrangement 134 vi Figures 1 Lift characteristics of a NACA 23012 airfoil with a 20% plain flap . . 7 2 N arbitrary airfoils 13 3 Joukowski transformation for the one-element problem 19 4 The tandem airfoil mappings 25 5 N in-line uncambered airfoils 37 6 Solution planes for the thickness and camber problems 41 7 Thin airfoil theory predictions of the pressure distributions over tandem 15% thick ellipses 44 8 Potential flow predictions of tandem airfoils designed by thin airfoil theory to give uniform chordwise load distributions at a = 0° . 51 9 The elementary mappings for the second camber function 55 Figures 10 to 17: The aerodynamic characteristics of tandem NACA 23012 airfoils. 10 Lift curve slope characteristics 89 11 Flap deflection effects on lift 90 12 Camber line lift characteristics 91 13 The aerodynamic centers 92 14 Flap deflection effects on moment 93 15 Camber line moment characteristics 94 16 Ct. vs. a for different 7/ 97 17 C m . - vs. a for different ij 98 18 The staggered parallel slit mappings 101 19 Doublet flow in the A-plane corresponding to zero incident uniform flow in the 2-plane 102 20 The control point Zj in the middle of the ; t h segment 131 Figures 21 to 25: The accuracy of the exact-numerical calculations. NACA 23012 airfoil elements in a slotted flap configuration at a = 0°. 21 Overall lift and drag 136 22 Localized lift 137 23 Localized drag 138 24 Overall leading edge moment 139 25 Localized leading edge moment 140 vii Symbols an p o l y n o m i a l b o u n d a r y c o n d i t i o n coef f ic ients . An F o u r i e r ser ies coef f ic ients fo r th i ckness flow f u n c t i o n . A{OJ) a i r f o i l se lec to r f u n c t i o n . bn p o l y n o m i a l b o u n d a r y c o n d i t i o n coef f ic ients . Bn F o u r i e r ser ies coef f ic ients fo r c a m b e r flow f u n c t i o n . c c h o r d l e n g t h of ove ra l l c o n f i g u r a t i o n : (c^ H h cN). Cj c h o r d l e n g t h of a i r f o i l ; . Ci = -r—z— ove ra l l l i f t coef f ic ient fo r a i r f o i l c o n f i g u r a t i o n . Ci. = - j — ^ — l o c a l i z e d l i f t coef f ic ient fo r a i r fo i l j. Cm* ~ i 1^° ove ra l l m o m e n t coef f ic ient ( abou t the o r i g i n ) . M Cfju'n — i — ^ 7 l o c a l i z e d m o m e n t coef f ic ient f o r a i r f o i l ( abou t t he l e a d i n g edge of i'tt2o<7 a i r f o i l ; ) . Cmac , Cmj- ove ra l l a n d l o c a l i z e d a e r o d y n a m i c cen te r p i t c h i n g m o m e n t coef f ic ients ; E q s . 4.72 a n d 4 .73 . , A C m t he shi f t i n a coef f ic ient 's va lue at a = rj = 0 ° d u e t o t h e presence of a c a m b e r l i ne . Cp = n—^p- p ressure coef f ic ient . C t Cj c l osed con tou rs i n t he c o m p l e x p h y s i c a l p l ane a b o u t the o v e r a l l c o n -figuration a n d a i r f o i l j. dn p o l y n o m i a l b o u n d a r y c o n d i t i o n coef f ic ients . Dn F o u r i e r ser ies coef f ic ients fo r the 2 n d c a m b e r f u n c t i o n . en p o l y n o m i a l b o u n d a r y c o n d i t i o n coef f ic ients . E, E' c o m p l e t e e l l i p t i c i n teg ra ls of the 2 n d k i n d . £ e r r o r i n se lec to r f u n c t i o n e x p a n s i o n coef f ic ients . / l i n e a r i z e d t a n d e m a i r f o i l p h y s i c a l p l a n e geomet ry , F i g . 4 . F(z) flow f u n c t i o n for s o l v i n g t he t a n d e m a i r fo i l a - p r o b l e m . 7{(JJ) flap se lec tor f u n c t i o n . Symbols h H U*) k, k' K, K' m M N q, q' R 3 <• *j i-plane T(«) w(z) = u — tu w'{z) = u' — iv' Xae> XJac Xj> Xjj y(x), y'(x) z = x + iy P 6 viii constants, Eq. 2.37. the gap in a staggered two-element airfoil configuration, Fig. 18. constant, Eq. 5.16. selector functions for the localized airfoil force solutions, Eq. 4.68. modulus and complementary modulus of elliptic functions, complete elliptic integrals of the 1st kind, length of segment Fig. 20. the boundary condition selector. number of airfoil elements in a configuration (Appendix A only). number of airfoil elements in a configuration (except in Appendix A). number of segments in the configuration's boundary and in the bound-ary of the kth airfoil (Appendix A only). theta function parameter, Appendix B. radius to outer annulus circle representing the flap, Fig. 4. flap chord to total chord ratio: (1—f)/{s + l — /). linearized tandem airfoil physical plane geometry, Fig. 4. airfoil segment selector. thickness to chord ratios. "tandemized" staggered slits, Chapter 5, Fig. 18. flap segment selector, complex velocity. nondimensionalized complex disturbance velocity, section 1.3. aerodynamic centers, Eqs. 4.72 and 4.73. leading and trailing edge x-coordinates of airfoil j. airfoil boundaries, section 1.3. linearized complex physical plane. angle of attack of airfoil configuration. orientation of doublet in the rectangle, Fig. 19. vorticity (Appendix A only). staggered slit mapping parameter, Eq. 5.21. "special* camber flow functions. thin airfoil theory solution planes. Symbols ix rj flap deflection angle. 6 angular variable in -^planes. i > i , t ? 2 , t?3 , #4 the four theta functions, Appendix B. A = fi + iv the rectangle solution plane, Figs. 4 and 18. p fluid density. S 3 , E 4 flow function constants. r theta function parameter,Appendix B. <j> velocity potential. V> stream function. ipk value of stream function on &th airfoil element, Appendix A. u = x + *7 modified rectangle solution plane, Fig. 4. u' = x' + *V modified rectangle solution plane, Fig. 4. fl = <f> + i0 complex potential. X Acknowledgement I would like to thank Dr. G.V. Parkinson for the opportunity he has given me here at UBC and for the enthusiasm and support he has shown over the years. A major factor in the successful solution of a complicated problem is the problem's initial formulation: Dr. Parkinson's insightful suggestion to linearize the doubly connected flow about a slotted flap onto tandem slits provided this graduate student with a problem with extremely simple solutions just waiting to be discovered. The research was carried out under a grant from the Natural Sciences and Engineering Research Council of Canada. Additional support was recieved from the Uni-versity Graduate Fellowship and H.R. MacMillan Family Fellowship funds. My thanks to my wife, Ying, who translated this entire manuscript into English. Also, my thanks to Mr. Feng Chen who donated a large part of his summer holidays towards the TgXing of this thesis. T^jX is a computerized typesetting process (sponsored by the American Mathematical Society) that is particularly suited to mathematically oriented dissertations such as this. My thanks to Dr. Larry Roberts for his work in getting TjjjX on-line in time for the processing of this thesis and for his help in achieving that end. CHAPTER 1 Classical Thin Airfoil Theory 1.1 I N T R O D U C T I O N In recent years there has been a great deal of interest in short take-off and landing aircraft. Their success is, to a large extent, attributable to the low-speed, high lift airfoil configura-tions that have been developed over the years. These multi-element configurations often employ multiple slotted flaps and a leading edge slotted slat to achieve large mean line curvature, and thereby high lift, while minimizing boundary layer growth and separation effects which lead to stall. Naturally, the development of these airfoils has been closely followed by the development of theories capable of predicting the aerodynamic forces asso-ciated with them. In keeping with what seems to be a general trend in engineering today, these theories are almost all numerical. They exploit to the fullest the power of big, but fast, large memory, mainframe computers. , There are several numerical theories available that can accurately predict the potential (i.e. inviscid) flows about multi-element airfoils, and in conjunction with bound-ary layer theory they can also give good predictions of the real flow lifts and moments. These potential flow theories are basically of two types. First, there are the more analyti-cally oriented theories which use numerical methods to conformally map arbitrary airfoil shapes to circles, the flow about which is solved as accurately as required by superpos-ing doublets and vortices on the flow in the manner of Williams (1971). The forerunner 1 Classical Thin Airfoil Theory / 1.1 2 of this approach was a method by Garrick (1936) who worked out a two-element con-figuration analogue to Theodoreen's (1931) analytical theory for arbitrary single-element airfoils. A modern example is seen in a paper by Halsey (1979). Secondly, there are the surface singularity panel methods, such as Hess and Smith (1967), which represent airfoil surfaces as a series of discrete line segments over which source strength and vorticity are appropriately distributed. A particularly efficient version of this approach which uses only vorticity, employs the stream function as fundamental variable, and uses a trailing point Kutta condition is given by Kennedy (1977). Of the two approaches to the numerical potential flow problem, the surface singularity methods, especially Kennedy's method, are the simplest and are probably as fast as the others. It is a version of Kennedy's method, presented in Appendix A, which is used in this thesis to calculate exact-numericalf potential flow theory predictions of flows. The success of the above numerical approaches depends on access to a large computer and, without expensive parameter studies, they give solutions to just one prob-lem— trends are not easily established. The object of this thesis is to attempt an analyti-cal description of these multiply connected flow fields, with the aim of obtaining a better qualitative understanding of the problem and simplified but realistic aerodynamic force predictions that can be calculated on today's inexpensive, readily available programmable hand calculators or small microcomputers. Such attempts have been made in the past, especially during the era of the slide rule dependent engineer who had no choice but to develop theories that were as ana-lytical as possible. However, these earlier multi-element airfoil theories were not motivated by small gap slotted flap type configurations but rather by the large gap biplane configu-rations prevalent at the time. In order to get simple solutions, either approximations were made which are incompatible with a small gap analysis (Munk, 1922b; Millikan, 1930; Glauert, 1926) or some geometric considerations, such as camber and different airfoil ele-ment chord lengths, were ignored (Glauert, 1926). Those that were general (Garrick, 1936) f By which is meant a numerical approximation to exact. Classical Thin Airfoil Theory / 1.1 3 are complicated enough that they do not compare favourably with the modern numerical methods. The main reason for this complexity is that the flow field around multi-element airfoils is multiply connected. A one-element airfoil flow is singly connected^ so that, at worst, it is described in terms of the singly periodic trigonometric functions. On the other hand, a two-element airfoil flow, such as one about a slotted flap or leading edge slotted slat airfoil, is doubly connected and a general description of it would require use of doubly periodic elliptic functions which, numerically, are much more difficult to eval-uate. As will be seen, however, elliptic functions have properties that allow them to be manipulated in much the same way that trigonometric functions are. In particular, the theory of the theta functions (Whittaker and Watson, 1927) used in conjunction with a programmable hand calculator allows rapid numerical evaluation of most elliptic functions. The desire to obtain a simple, useful, and general multi-element analytical theory is also motivated to a large extent by the success of the well-known single-element linearized airfoil theory, or thin airfoil theory. It originated in the early part of this cen-tury when Max M . Munk (1922a) first used a linearized boundary condition approach on the ideal flow.about a simple airfoil. He was able to describe the airfoil's overall lift and moment in terms of simple integrals. Subsequent contributions by Birnbaum (1923) and Glauert (1926) resulted in a much improved general formulation of the problem. Others applied the theory to special situations such as a simple attached flap, and calculated local-ized quantities such as flap lift or flap hinge moments. Much of this work is summarized by Abbott and von Doenhoff (1959). Stewart (1942) gives a particularly efficient formulation of the general problem using complex analysis and solving for a complex disturbance veloc-ity. He shows how it is just as easy in thin airfoil theory to specify a pressure distribution and work out a camber line or thickness profile as it is to do the converse. His methods are similar to the ones used in this thesis. | The definition of connectivity used is given in Woods (1961): a region is singly connected if it has one contour separating it from its complement domain, doubly connected if it has two, etc. Classical Thin Airfoil Theory / 1.1 4 One-element thin airfoil theory lift and moment predictions are typically simple analytical expressions which agree quite well with experiment if boundary layer growth and separation effects are minimal. In fact, this agreement is usually better than that of exact potential flow theory which tends to overestimate loadings. Pressure dis-tribution predictions, while generally less accurate and often containing singularities at points where the linearization process breaks down (such as leading and trailing edges), do show good overall qualitative agreement with experiment and are also given by simple expressions. The reason for the theory's simplicity is that application of the linearized boundary condition results in the geometric amplitude parameters for airfoil angle of at-tack, camber, and thickness appearing linearly in the flow solutions. These three geometric effects, which completely determine the airfoil profile and its orientation in the flow, can then be shown to contribute independently to the aerodynamic characteristics of the air-foil. The effects are solved for in separate problems, the solutions of which are linearly superposed as required. One-element thin airfoil theory then is an efficient analytical tool that has been and continues to be a very useful aid to both the student and practising aerodynam-icist. Its simplicity is one very good reason for using it as a starting point for developing an analytical multi-element thin airfoil theory. However, for this one would also want to anticipate that a multi-element extension of the theory would give predictions as realistic as those of the one-element theory. At first this seems unlikely since multi-element con-figurations generally operate with large mean line curvature, as in Figure 2 for example (page 13), so that the linearization requirement, that deviations of the boundary from some average straight line through the airfoil be small, is violated. Undoubtedly, neglect-ing the second order effects of large angles of attack or flap deflections would eventually be a limiting restriction on the use of a multi-element theory. However, as will be seen in the next section, these effects are not the limiting constraint on one-element thin airfoil theory and, in some cases, may not even be in a multi-element extension of it. Classical Thin Airfoil Theory / 1.2 5 This thesis then is devoted to the extension of classical thin airfoil theory into the multi-element regime. Only partial success is achieved but this includes a general tandem two-element thin airfoil theory, extensions of its overall lift and moment results to an arbitrary number of in-line airfoils, and some two-element staggered airfoil results. Also, a method is developed that greatly facilitates the calculation of multi-element thin airfoil theory lifts and moments. Based on Munk's integrals (section 4 of this chapter), in principle the method is applicable to any multi-element configuration. Comparisons are made between the tandem wing thin airfoil theory and exact-numerical potential flow theory. Variations of the aerodynamic forces with slot size and with flap chord size are examined. All of the aerodynamic forces for the tandem wing thin airfoil theory are calculated on a small, portable, inexpensive microcomputer. Some of these results are as simple as those from one-element thin airfoil theory. 1.2 T H E P O T E N T I A L F L O W - R E A L F L O W C O R R E L A T I O N As an example of the relationship between one-element thin airfoil theory, exact poten-tial flow theory, and the real flow consider Figure 1. Here, the nondimensionalized lift characteristics of a NACA 23012 airfoil with a 20% plain flap are shown for various angles of attack and flap deflections. The experimentally measured points in the figure are from Abbott and Greenberg (1939). The exact-numerical potential flow calculations were made using the theory described in Appendix A. Airfoil geometry was obtained from Abbott and von Doenhoff (1959). The predictions of one-element thin airfoil theory shown in the figure are described in section 1.4. There are several characteristics worth noting in Figure 1, as follows. a) The relationship between thin airfoil theory and exact potential theory. The mathematical relationship between the exact and linearized theories is more for-mally presented in the next section of this chapter. They are both two-dimensional, Classical Thin Airfoil Theory / 1.2 6 incompressible, potential-flow theories. Essentially, the difference is that whereas the exact theory applies the tangential velocity boundary condition exactly on the airfoil boundary, the linearized theory applies it instead on some average straight line through the airfoil parallel to the flow at infinity. This is the same as assum-ing that deviations of the airfoil boundary from the average line are small, so that second order terms in a and r}t for example, can be neglected. The errors that this assumption gives rise to are seen in Figure 1. This N A C A airfoil is 12% thick (maximum thickness relative to chord length) and it is this thickness which is primarily responsible for the difference between the exact and linearized theories at low values of a and T}. If the thickness were reduced to, say, 1% then the exact theory would collapse down virtually on top of the linearized theory (camber only being about 2% for this airfoil). For high values of a and rj, this collapse would not be as complete. Note that the angle of zero lift predictions, a^, are virtually identical for the two theories, so that differences between the theories are due to different lift curve slopes, dCi/da. Abbott and von Doenhoff (1959, section 3.5) show how the second order effect of thickness on the lift curve slope for a Joukowski airfoil is given by the factor ( l + A£), where (t/c) is the thickness to chord ratio and A w 0.77. For typical airfoil section thicknesses of 12% to 18% then, one would expect that the exact predictions would be greater than the linearized ones by approximately 9% to 14%, which is consistent with Figure 1. Thus, there is a consistent relationship between one-element thin airfoil the-ory and exact potential flow theory which is, of course, completely independent of real flow effects such as boundary layer growth and separation. Note too, from Figure 1, that second order effects of o and rj are minimal for the ranges considered since the variations of the exact theory's predictions with both a and r\ are almost linear. For larger values of a and rj, these effects eventually result in the exact theory being overpredicted by the linearized theory. FIGURE 1 Lift characteristics of a NACA 23012 airfoil with a 20% plain Bap. Classical Thin Airfoil Theory / 1.2 8 b) The correlation between potential flow theory and experiment. Abbott and von Doenhbff (1959) give experimental lift and moment results for many dif-ferent airfoil section profiles that the United States National Advisory Committee on Aeronautics (NACA) has studied over the years. These results show that, for thicknesses of 12% to 18% (and with no flap deflections), experimental values of lift curve slopes (measured in the minimum drag regime of the profile — i.e. well be-low the stall) are all scattered about the thin airfoil theory prediction with most of them within 5% of that prediction and 95% of them within 10% of it. This accuracy, when combined with less accurate predictions of , results in thin airfoil theory overpredicting lifts at the point of maximum lift to drag (typically 2° < a < 6°) by 5% to 15% — although some poorly designed profiles can be overpredicted by as much as 20% to 30%. It is very rare indeed to see thin airfoil theory underpre-dict real flow results (those profiles that have their lift curve slopes underpredicted invariably have the magnitudes of their zero lift angles overpredicted) so that one can conclude that one-element thin airfoil theory predictions are usually closer to real flow results than exact potential flow predictions are. This is, of course, purely coincidental. It comes from the errors introduced by the linearization of the exact potential flow partially compensating for the necessary boundary layer corrections to this flow. It is these boundary layer corrections ,among other things, that airfoil profile designers try to minimize. By so doing they can delay the onset of stall, minimize drag, and maximize the lift curve slope, all of which also happen to improve the correlation between thin airfoil theory predictions and the real flow results. In practice, however, one must use caution when relying on this correlation since, as is obvious from Figure 1, it breaks down without warning as angle of attack increases. This breakdown begins gradually but is completed abruptly as the airfoil stalls (dC(/da becomes negative). The stall is caused by flow separation from the Classical Thin Airfoil Theory / 1.2 9 airfoil's upper surface well up from the trailing edge. An airfoil that develops high lift gets that lift primarily from a large negative pressure peak on its upper surface just behind the leading edge. The flow over this surface, then, experiences large adverse pressure gradients which promote boundary layer growth and eventually flow separation. The separation begins at the trailing edge and, as the lift increases, so too does the adverse pressure gradient, the separation point moves upstream, the drag increases dramatically while the lift curve slope decreases, and eventually the airfoil stalls. Stall can be gradual, which would be preferable, or it can be sudden as in the case of the 23012 profile. These boundary layer related flow characteristics can be controlled to some extent by choosing a profile with a thickness distribution and camber line that min-imizes upper surface adverse pressure gradients, at least over the primary operating range of a. In other words it is possible to use inviscid theory, in conjunction with a designer's knowledge of how inviscid flow characteristics affect boundary layer growth, to minimize degradation of the real flow. Here one-element thin airfoil theory can and, in fact, has played an important role. The NACA 6-series air-foils, an improved design that followed the 230 profiles (Abbott and von Doen-hoff, 1959), are constructed from camber lines calculated by thin airfoil theory from designer specified uniform load type pressure distributions. These camber lines are simple analytical expressions, c) The effect of flap deflection on the potential flow - real flow correlation. Figure 1 shows how this correlation breaks down as the plain flap is deflected through 10 and 20 degrees. For deflections greater than about 10 or 15 degrees the experimental results of Abbott and Greenberg (1939) also show that the drag increases dramatically, due to flow separation from the flap, as the lift slowly in-creases up to a maximum AC7 (C/ at a = 0°) of 1.5 at i] = 60°. Abbott and von Doenhoff (1959) conclude that one-element thin airfoil theory a . . .permits cal-culation of the angle of zero lift, the pitching moment coefficient, and the chordwise Classical Thin Airfoil Theory / 1.2 10 load distribution with reasonable accuracy . . . for most commonly used wing sections . . . for flap deflections not over 10 or 15 degrees." But even then, thin airfoil theory predictions of quantities such as flap lift and hinge moment are "relatively poor" since they are localized results from a part of the flow experiencing pronounced boundary layer separation. Other one-element configuration high lift devices, such as split flaps and lead-ing edge flaps, have also been used but, while they do increase the lift, none of them can match the high lift to drag ratios obtained by the multi-element configurations. Thus, it appears that one-element thin airfoil theory correlates well with real flow results unless angle of attack or mean line curvature are large enough that bound-ary layer effects appreciably affect the overall flow. The linearized theory's neglect of the second order geometric effects of these large deviations is not the reason for the break-down of the potential flow - real flow correlation. One would expect then that an airfoil configuration that dramatically reduced boundary layer effects on the overall flow would also increase both the degree of this correlation and the range of deviations over which it would apply. One of the reasons the flow separates so easily from the attached flap of Figure 1 is that, due to the adverse pressure gradient over the back of the main part of the airfoil, boundary layer growth and momentum loss in that boundary layer are already appreciable when the flow reaches the flap. For even small flap deflections the further adverse pressure gradient over the flap (resulting from the pressure peak at the deflected flap leading edge) easily causes the flow to separate. However, if the flap were detached from the airfoil enough that the relatively clean flow from below the airfoil came through the slot and swept the wake coming off the back of the airfoil away from the flap, so that boundary layer growth over the flap had to begin anew, then separation from the flap should not be as imminent. This is what appears to happen. Classical Thin Airfoil Theory / 1.2 11 Foster, Irwin, and Williams (1970) made an extensive investigation of the flow around an airfoil with a slotted flap. They measured the lift and drag on the two airfoil elements and compared the lift with exact-numerical potential flow predictions. They also investigated the size and interactions of the boundary layers developed over the airfoil and flap. The nose of their flap was tucked under the trailing edge of their airfoil, as is usually done in order to drop the flap below the wake of the airfoil, and the effect of variation of overlap (the distance the flap leading edge is ahead of the airfoil trailing edge) on the aerodynamic forces was examined. The effect was minimal. Large effects were found, however, for variation in gap (the vertical distance between the lower surface of the airfoil and the upper surface of the flap). It was found that for very small gaps and relatively large flap deflections (TJ = 30° for their 31% flap), the wake from the airfoil was merged with the boundary layer over the flap, the drag was high, and the exact-numerical inviscid lift greatly overpredicted the experimental lift. Although the wing had not necessarily stalled, the flap was experiencing a great deal of separation. As the gap was increased, the airfoil wake and flap boundary layer began to separate from each other and eventually there was a region of flow between them that attained the free stream energy head of the flow. As this happened, flow separation from the flap was reduced, drag was reduced, and lift increased. After the wake and boundary layer were effectively separated by the small region of inviscid flow, the experimental lift variation once again correlated well with the inviscid flow predictions. The report concluded that, while both boundary layer and inviscid effects were important in obtaining the optimum flap position (which for this configuration was about a 2% gap), once in that position the airfoil wake and flap boundary layer would be separated and the overall real flow would be strongly dependent on the inviscid solution for the flow. In conclusion, exact potential flow theory and, even more so, thin airfoil the-ory give realistic predictions of the real flow forces on a one-element airfoil if boundary layer Classical Thin Airfoil Theory / 1.3 12 effects on the overall flow are small. Multi-element airfoil configurations are specifically designed to minimize boundary layer growth while operating with greater mean line cur-vature, and so it is not surprising that exact potential flow predictions of their forces also give good agreement with experiment, but for even higher lift configurations. It is reason-able to anticipate that the consistent relationship between one-element thin airfoil theory and exact potential flow theory would extend to multi-element configurations where one could then anticipate that a multi-element thin airfoil theory would give better predictions of the real flow forces than exact potential theory does, and that a close correlation with the real flow would apply for even higher lift configurations than it does with one-element thin airfoil theory. 1.3 T H E LINEARIZING P R O C E D U R E Consider the N airfoils in the steady, two-dimensional, incompressible, potential flow of Figure 2. The airfoils all have arbitrary positions, chord lengths, angles of attack, thickness, and camber. The boundary of airfoil j, say, is given by yy(x), its chord by Cy, and its angle of attack to the horizontal flow at infinity, u^ ,, by cry. The complex velocity, w[z) = u — iv, representing the exact potential flow about these airfoils can be written in terms of the flow at infinity and a nondimensional complex disturbance velocity, xv'(z) = u' — iv': w(z) = u 0 O(l + V J \ Z ) ) . The conditions which determine w\z) exactly and uniquely are: 1) tangent flow on all airfoil boundaries v dx 1 + u' 2) V J ' ( Z ) vanishes at infinity. 3) the Kutta condition applied to the airfoil trailing edges. The forces on the airfoils can be obtained through their surface pressures which are known via Bernoulli's equation and the pressure coefficient: Classical Thin Airfoil Theory / 1.3 13 z-PLANE FIGURE 2 N arbitrary airfoils. where p is the fluid density. In multi-element thin airfoil theory it is now assumed that the airfoils are thin and that they operate at small angles of attack. Thus, if yJa is some average horizontal line through a i r fo i las in Figure 2 , it is assumed that: < I. (i.i) Classical thin airfoil theory then makes the following corollary assumption: u ' l .k l du' d(y/cj) du' dv' < IT (1.2) d(x/cj) Although there are localized violations of this last assumption (at stagnation points and leading edges for example) that can lead to localized bad predictions (infinite pressures at stagnation points and leading edges), overall the assumption is good so that, as was previously stated, the overall force predictions and the overall qualitative nature of the Classical Thin Airfoil Theory / 1.3 14 pressure distribution have close relationships with the exact theory. It is to be expected, however, that the linearized theory's predictions of more localized characteristics will be less accurate. A good example of a badly modelled localized characteristic is the Kutta condition, which is needed to determine the circulation around an airfoil in potential flow. Descriptions of it that are often used in exact potential flow are: a) finite velocities are required at trailing edges. b) finite pressures are required at trailing edges. and for exact-numerical potential flow a favourite is: c) there can be no load on the trailing edge. In exact potential flow these descriptions all amount to the same thing. In thin airfoil theory they can all be different and, in fact, none of them are correct Kutta conditions for all situations. This is a consequence of applying Equation 1.2 to the pressure coefficient and tangent flow boundary condition. Neglecting second order products in the disturbance velocities, as thin airfoil theory does, one gets: Cp = -2u' (1.3) (M. so that on the surface of airfoil ; : W'(z) = u ' - i V = - 3 » - A . (i.5) V ' 2 dx v ' Thus, even though Equation 1.1 may be properly satisfied, in thin airfoil theory: A) if the airfoil is a thin curved plate with a weak infinite slope at the trailing edge, then a) and c) are violated while b) is true. Here, a logarithmic singularity at the trailing edge satisfies the boundary condition on v' but makes u' discontinuous across the trailing edge—hence the load. An example of this is the uniform load camber line used for some of the NACA 6-series airfoils. Classical Thin Airfoil Theory / 1.3 15 B) if the airfoil is a long thin ellipse and so has an infinite slope at the trailing edge, then only a) is violated. Here, thin airfoil theory requires a simple pole at the trailing edge. The pole is entirely contained within the imaginary part of w'{z) and, in fact, u' is constant everywhere on the surface of the ellipse. C) if the airfoil is conventional and, due to thickness, has an acute nonzero trailing edge angle, then a) and b) are violated while c) is true. In this case the boundary condition requires that v' have a discrete jump at the trailing edge. This is provided by a logarithmic singularity which results in an infinite but continuous u\ D) if the airfoil's trailing edge angle is zero and if it has a non-infinite slope at the trailing edge, as for a flat plate or the well known Joukowski airfoils, then a), b), and c) are all true. There are no singularities at the trailing edge. Perhaps the only statement that can be made to describe a general linearized Kutta con-dition is that there can be no infinite loads at the trailing edge. Of course, this statement would not prevent violation of the Kutta condition in an exact potential flow. Consider now the Taylor series expansion of u' and v' about y = yy for a given z: ^ ' J ^ ^ J + ^ - ' i . ) ^ ) * - ( 1 6 ) Thus, a further consequence of the small deviation assumption is that the boundary con-dition for airfoil dx/j/dx = can be applied on the horizontal slit y = y J a to within an error of order |(yy - yJa)/cj\2 (using Eqs. 1.1 and 1.2). Similarly, the pressure on the airfoil's surface, Cp = —2u', is taken as equivalent to the pressure on this slit to within the same small error. In this multi-element linearization each airfoil elementhas its own independent yJa. As in one-element thin airfoil theory this procedure breaks down at only a few localized points where u' or v' are singular. Also, in practice it is found that the Classical Thin Airfoil Theory / 1.3 16 linearized theory will often give sensible answers even though the deviations are not very small (in the strict mathematical sense). Now, if y'ej[x') and yjy(as') give the camber line and thickness distribution respectively for airfoil j (written in terms of a localized coordinate system, Fig. 2), then the airfoil's upper and lower surfaces are: y'u = y'c + y't (1.7) I I I VL = yc-vt so that the linearized boundary condition on the surface of airfoil j is: v ' - ^ - - a + ^ ± ^ i (18) V ~ dx ~ + dx' ± dx' • ( L 8 ) This shows that amplitude parameters for angle of attack, camber, and thickness that ap-pear linearly in the boundary condition will also appear linearly in the flow solution, w'(z). In fact, if w'aj(z), w'ej-(z)t and w'tj(z) represent functions that solve the three problems for airfoil j independently of both each other and the other airfoils, then the linearity of Laplace's equation ensures that the effect of airfoil j on the overall flow is simply: Wji*) = <M) + W'cj<Z> + Wh<Z) (L9) and that the overall flow solution will then be: N u/(z) = ] > > ; ( * ) . ( I . I O ) The linearity of Equation 1.10 is transferred through to the airfoil forces as well. Neglecting second order products in the disturbance velocities, the Blasius equations for lift (L), drag {D)t and moment (Af) become: D-iL }/> t too -i  n . / . , - j — — = 2i <b w{z) dz M n =2*< j> zw'{z) dz C where MQ is the moment about the origin of the coordinate system and C is a curve enclosing either the airfoil element of interest for localized forces or the configuration of interest for overall forces. Classical Thin Airfoil Theory / 1.3 IT However, from exact potential flow theory it is known that the circulation (r) around some closed curve C and the net source strength (Q) within it is given by: -r + iQ = u00j> w\z) dz C so that in multi-element thin airfoil theory the drag on airfoil j is zero, assuming that it is a closed body. The lift and moment are: CV-UA.)* ( U I , The overall lift and moment expressions are the same as Equations 1.11 and 1.12 except that the nondimensionalizing length is taken instead as the sum of the individual chord lengths. It is interesting that in thin airfoil theory the lift and drag on an airfoil, regardless of the presence of another airfoil, are just proportional to its circulation and zero, respectively: D3 = 0 if Qj = 0 so that the Kutta-Joukowski law applies on a localized basis. In an exact potential flow, however, this is only true for an isolated body or for the overall forces on an isolated group of bodies, as can be seen from the localized lift and drag expressions developed in Appendix A. Perhaps the greatest advantage of thin airfoil theory is its ability to describe complicated general problems in terms of a linear superposition of independently solved simpler problems, each of which deals with only one aspect of the boundary condition. The individual linearly superposable force solutions are, for a given orientation of the airfoils of Classical Thin Airfoil Theory / 1.4 18 Figure 2, simply constants multiplied by their respective geometric amplitude parameters. This permits a rapid investigation and easy understanding of the relative effects of these geometries on the forces. An understanding of the effects of the relative positions of airfoil elements in a configuration (interference effects) is obtained by simply looking at the variation of these force solution "constants" with position. For example, the variation of Ce/a with gap for a slotted flap airfoil would be of interest. Investigations of these types are presented in Chapter 4. 1.4 O N E - E L E M E N T T H I N AIRFOIL T H E O R Y F U N C T I O N S A N D FORCES What follows is a brief summary of the one-element thin airfoil theory problem. The func-tions used to solve the multi-element problems are just extensions of the simple functions presented here. Figure 3a shows the slit in the z-plane on which the boundary conditions for the airfoil of Figure 1 would be applied. Some problems are more easily solved in the f-plane where the flow about the slit has been conformally mapped into the flow about a unit circle using the well-known Joukowski transformation: 4z-2 = c + ^. (1.13) On the surface of the slit f = et9 and so: 2 z - l = cos0. (1.14) This transformation separates the upper and lower surfaces of the slit (they become, respectively, the upper and lower semi-circular arcs of the unit circle) with the result that simpler functions will solve a problem in the f-plane than would be possible in the z-plane. Classical Thin Airfoil Theory / 1.4 19 z-PLANE i f- PLANE , dyu dx f 1 dyL dx a) Linearized physical plane. b) Solution plane. F I G U R E 3 Joukowski transformation for the one-element problem. As previously mentioned, the general problem can be broken up and solved as a series of independent simpler problems: The a-Problem If a is the angle of attack of the airfoil, then: dx dx = —a and the solution is: w'Q{z) = ta which is correct and unique since: 2ia 7+T 1) on the slit surfaces y/z — 1 is imaginary so that v' = —a. 2) w'a(z) -* 0 as z —* oo. 3) the Kutta condition is satisfied. On the surface of the slit: w'a(z) = a tan - + ia It (1.15) so that the pressure is singular at the leading edge—positive infinite on the bottom sur-face [9 = — ir) and negative infinite on the top surface (9 = TT). Classical Thin Airfoil Theory / 1.4 20 The lift and moment can be evaluated using the residue theorem and the expansion: , , x iot ia ^ ( 1 \ as z —• co. . ia t'a _ /  \ Letting the closed curve C expand out to infinity in Equations 1.11 and 1.12 gives: Ct = 2ira ir (1-16) The aerodynamic centre of the airfoil, z a c , is that point about which the pitching moment does not vary with a. That is, dCmac/da = 0; and since: Cmac — Cm0 + Ct xac (l-l 7) t h e n : _ d C m J d a _ 1 X a c ~ ~ dCt/da ~~A ( L 1 8 ) The r;-Problem When w'(z) is written in terms of the f rather than the z variable it can be much simpler, as seen in Eq. 1.15. This is useful when writing down the solution to the more complicated flap deflection problem, where: v' = 0 for 0 < x < f v' = —T) for / < x < 1. The solution is: IT 2i9f c-ei0f (1.19) j.+ l ' $-e-i0J. which satisfies all the necessary conditions. The discrete jump in v' at the flap leading edge, z = / , is provided by logarithmic singularities in u' on the airfoil's top and bottom surfaces at this location. Note that when the flap is deflected the strongest resulting singularity in the flow is at the airfoil leading edge and not the flap leading edge, so that most of the lift from the flap deflection is borne by the front of the airfoil. Classical Thin Airfoil Theory / 1.4 21 Now: and: so that: ri+°(i) • as f —• oo as z —• co Ct = 2(9 f + sin9 f)t} = 2 [cos-1(2/ - 1) + 2^/(1 - /)] r,. (1.20) Similarly, using higher order terms from the above expansions: Cmo = -^{9f + 2sm9f + lsm29f) (1.21) so that: Cmac = -\^9f{\ + cos9f) = -2r//v7(l - f) . (1.22) The Camber and Thickness Problems Both camber lines and thickness distributions can be represented in a general fashion through the use of Fourier series. The problems are orthogonal in the sense that where the camber boundary condition is even in 9: dyv dyc dyL dyc dx dx ' dx dx resulting in pressures that are odd in 9, the thickness boundary condition is odd in 9: dyv _ dyt _ dy^ _ _dyL dx dx ' dx dx which results in pressures being even in 9, so that thickness has no effect on lift. If for camber: tt—1 then the boundary condition becomes: ^ = f + f > " C 0 S f l * U-24) n=l so that: n Bn = - f ^cosn9d9 ; n = 0,l ,2, . . . (1.25) ir J dx Classical Thin Airfoil Theory / 1.4 22 This Laurent expansion accommodates virtually all conventional camber lines, including those requiring logarithmic singularities in w'c(z). Equation 1.19 is just a special case of Equation 1.23. Lift and moment are: ACt = -ff(i? 0 + tfj (1.26) ^ m f l c = 4(^1 + ^2). (1.27) If the thickness flow function is: 0 0 A, n = l then the boundary condition is: *) = E^ (1-28) ±-f- = ^Ansmn& (1.29) 1 r » = l with: K •"1/ , sinnfldtf ; n = 1,2,3,... (1.30) dx The net source strength contributed to the flow by a closed body is zero so that, from Eq. 1.28, Ai = 0. Equation 1.28 is not completely general since it will not account for strong singularities in dyjdx which are usually present at the leading edge (at the trailing edge as well for an ellipse). These singularities are added if necessary but they make the evaluation of the A n 's somewhat more difficult. Since thickness has no effect on lift in the one-element theory, however, modelling it is usually unnecessary. Note that with the camber and thickness problems it is just as easy to specify a pressure distribution and work out what the corresponding boundary should be as it is to do the converse. This is particularly true when the specified u' function is simple, as it usually is, because this allows use of closed form analytic functions such as were used in Equation 1.19. Classical Thin Airfoil Theory / 1.4 23 forces: and: Finally, then, the total force is just a linear superposition of all the individual dC, dC, d a dV (1.31) = -n(B0 + B{) + lira + 2(9 f + sin 9f)rj C L = A C L + dCmaen mac mac ' a _ ' dj> (1.32) = ^ B 1 + j 9 2 ) - 2fvm-f) ri. The Munk Integrals If only lift and moment are required then it turns out that one can bypass finding the general flow solution and solve directly for the forces in terms of the bound-ary condition, as first presented by Munk (1922a). Using Eq. 1.25 in Eqs. 1.26 and 1.27 and changing variables one gets: l 0 0 Note that these expressions give the general one-element thin airfoil theory force solutions, including the a and r) problem solutions, since to£(z) (Eq. 1.23) is a general flow function capable of handling the a and 17 problem boundary conditions. It is by understanding and generalizing these simple expressions that thin airfoil theory is successfully extended to the multi-element problem. CHAPTER 2 Tandem Symmetrical Airfoils This multi-element airfoil linearization is initially restricted to the simple geometry of tandem airfoils, as in Figure 4a. In this chapter only the effects of angle of attack and flap deflection are considered. Camber and thickness are dealt with in later chapters. The linearized physical plane, Fig. 4b, has the airfoil and flap boundary con-ditions applied on the same average line which is taken as the real axis. The airfoil slit stretches between 0 and s and the flap slit between / and 1. The only conditions on a and / are: s and / are real ; 0 < 5 < / < 1 so that this could also be the problem of an airfoil with a leading edge slotted slat. This linearization will not model the effects of overlap. Note that for in-line airfoils, thickness will again have no effect on lift. This is because source strength distributed along the slits, with which thickness could be modelled, would have exactly the same effect on u' (the pressure) on the tops of the slits as it would on the bottoms. This would not be true for staggered slits. 2.1 T H E M A P P I N G S The conformal mapping from the z-plane to the A-plane, Fig. 4c, is a special case of a more general staggered slit mapping used by Garrick (1936) for developing biplane airfoil 24 Tandem Symmetrical Airfoils / 2.1 25 V k a) Physical plane z - PLANE 5 / b) Linearized physical plane 0) -PLANE (c) ' i oi - PLANE (d) The rectangle solution planes. (e) f - PLANE f) The annulus solution plane. F I G U R E 4 The tandem airfoil mappings. Tandem Symmetrical Airfoils / 2.1 26 potential flow solutions. Consider the small rectangle A ^ , iK', 0, K, K + iK'y A ^ of the A-plane. A Schwarz-Christoffel transformation is used to map the interior of this rectangle to the upper half of the z-plane, to corresponding points —oo, 0, a, / , 1, +oo. The rectangle below the /i-axis then automatically transforms into the bottom half of the z-plane in a similar way. The left and right sides of the big rectangle form both surfaces of the airfoil and flap, respectively. The mapping is given by the elliptic integral: A = _ y 7 ( i ^ ) f dz /dz x/z(z-s)(z-f)(z-l) ' ( 2 - 1 ) 2 J ^ zi -sXz-fKz-l) ' » It can be reduced to a standard form by the transformation: o f(z — s) • ° = <f I 2.2 z(f-s) so that: a. ^ A = / ° =sn~1(o,fc) (2.3) or. z = 7Z-3 • (2-4) l_L—Bn2{Xfk) k is the modulus of the Jacobian elliptic function sn(A,A;): * 2 = 7[[_1) > 0 < * < 1 . (2.5) K is a complete elliptic integral of the first kind: K=f , fa — . (2.6) If k' is the complementary modulus: fc'2 = l - * 2 = i ^ j ; 0<Jfc'<l (2.7) then K' is the same function of k' as K is of k. When z goes to infinity A = A ^ , and from Eq. 2.4: Sn2 A°° = J~ (2-8) Tandem Symmetrica] Airfoils / 2.1 27 where snA is written for sn(A,A:). If A,^ = /x ,^ ± iK' then the periodicity of the elliptic function gives: » 1 snA^ = so that, using Eqs. 2.5 and 2.8: snVoo = l - s 5 c n 2 = « J dnVoo = y (2.9) where, in general, the Jacobian elliptic functions, sn, cn, and dn, are related via: sn2 A + cn2 A = 1 ; dn2 A + k2 sn2 A = 1. (2.10) Thus: The w-plane, Fig. 4d, is a simple scaling of the A-plane: u = x + i1 = ^ (2.12) ITT Woo = Xco ± y (2-13) iK' , \ T = — . (2.14) Similarly, the w'-plane, Fig. 4e, is just: u' = x' + ii = OJT' (2.15) a 4 = ± ! + »7*o (2.16) T ' = % ' ( 2 - 1 7 ) By definition the Jacobian elliptic functions use the A variable as argument. However, theta functions, which will be used later and whose properties are summarized in Appendix B, are more naturally defined with the u and w' variables. The rectangle is unusual as a solution plane since it separates not only the upper and lower surfaces of the slits as desired, but also regions in the flow across which the Tandem Symmetrical Airfoils / 2.1 28 flow characteristics must be continuous. For example, the real axis in the z-plane (z < 0 , z > 1) occurs twice in the rectangle, along u = ±iK'. Therefore, the single valuedness of the flow must be ensured by requiring of any flow function, Ft that: F{\) = F(\ ± 2iK'). (2.18) This periodicity is consistent with the mapping, Eq. 2.4. Thus, the flow about the slits in the z-plane is entirely contained within the infinite strip 0 < /i < K of the A-plane where it is periodic in 2iK'. Although the Jacobian elliptic functions are doubly periodic functions, so that the mapping is also periodic in the fi direction, there is no reason the flow should be doubly periodic and, as will be seen, in general it is not. Any problem with maintaining the flow's single valuedness is avoided in the f-plane, Fig. 4f, where: c = g ' V * " (2.19) R = e*K/K' (2.20) and letting ^ = — d: d = t ^ t K ' . (2.21) It is particularly convenient that: f = e~2,w'. (2.22) Here the entire flow is in the annulus 1 < r < R. The airfoil slit has now become the unit circle with its leading and trailing edges at f = — 1 and f = 1 respectively; the slot s < x < / corresponds to 1 < $ < R; ( = R and f = —R are, respectively, the leading and trailing edges of the flap; 1 < z < oo corresponds to —R < f < — d and —oo < z < 0 to — d < f < — 1. The upper half-plane semi-circular arcs are the upper surfaces of the airfoil and flap slits. The £-plane will be the starting point for constructing the necessary two-element flow functions from the elementary functions of one-element thin airfoil theory. Tandem Symmetrical Airfoils / 2.2 29 2.2 T H E F L O W F U N C T I O N S Tangent flow on the airfoil and flap surfaces requires that: v' = —a on the airfoil slit (2.23) v = —a — rj on the flap slit. From the one-element theory it is known that the function i/(f + 1) has its imaginary part constant on the airfoil. It is not, however, constant on the circle r = R. This unwanted effect on the flap can be negated by using Milne-Thomson's circle theorem to add an image singularity at f = — R2. The function: ( + 1 $ + R2 . has its imaginary part constant (and, in fact, zero) on the flap but the new singularity has now created an unwanted effect on the airfoil — which is again negated using an image singularity, this time at $ = — l/R2. This process continues indefinitely and leads to an infinite series which converges everywhere in the annulus. The function: ' « = M {?TT + §[s^TT-?+W]} (2-24) has its imaginary part constant and equal to 1 on the airfoil and constant and equal to. 0 on the flap. A similar function, F2l can be constructed to deal with the flap boundary con-dition while having no effect on the airfoil boundary condition. Since w'(z) must be zero a * f = $<x» these functions are not yet flow functions. However, this is easily remedied by subtracting appropriate constants. Rather than working with infinite series, such as Eq. 2.24, it is more con-venient to work with conventional elliptic functions. Rewriting Equation 2.24 using the u/ variable, it becomes: ;2n • _ i q sin 2or ^-f 1 + 2q'2n cos 2w' + q fizz 1 * where: q' = e"iT' = l/R. (2.26) Fi=i- t a n a ; ' - 4 ^ (2-25) 1 ^-f l + 2g'2ncos2w' + ( 7 ' 4 n Tandem Symmetrical Airfoils / 2.2 30 This can now be summed in terms of the second theta function and its derivative: * i = « + £V,g'). (2.27) An invaluable aid to the user of elliptic functions is Whittaker and Wat-son (1927) where the four theta functions, their properties, and their relationships with the Jacobian elliptic functions are clearly presented. A brief summary is provided in Ap-pendix B and this also serves to define the elliptic functions as they are used here. 0' One of the important properties of the -^(w',?') function is that, while its v2 series representations, Eqs. 2.24 and 2.25, converge rapidly as the slot (/—s) becomes large (so that R —*• oo, q' —• 0), they lose their ability to converge as the slot becomes small (R,q' —* 1). This characteristic is common to the flow functions used in other analytical theories (Williams, 1973; Halsey, 1979), where it is overcome by using the computer to evaluate a large enough number of terms. However, the problem can be avoided entirely through the use of Jacobi's imaginary transformation, Eqs. B6, which allow F1 to be rewritten: where: 2u K'0\, , (2.28) ir K *i e*iT = e~*K'lK. (2.29) As the slot goes to zero, K —* JT/2, K' —* oo, and q —* 0 so that the series representation of Fi in terms of the u variable and q parameter is ideal for small slot configurations. The four theta functions are very similar to the trigonometric functions sinw and cosy, which and i?2(w) behave like as q —» 0. Indeed, — #i(w + |) as well. i?i(w) has a simple zero at the origin in the w-plane. t?2(w) has this same zero moved to a/ = rr/2. $3(0/) = i?4(w + f) has a simple zero at w = | + and t?4(u/) moves it to w = jrr/2. Thus, the -^(w) functions are analogous to tanw and cotw. They have simple poles in thew-plane at the four critical points of the tandem slit mapping. In the Tandem Symmetrical Airfoils / 2.2 31 z-plane they have square-root singularities at these locations. Also, as can be seen from Equations B l and B7, all four of these functions are purely imaginary on the surfaces of both the airfoil and flap (i.e. for u = »7 or w = | + »7), so that it's the second term of Equation B4 that enables the imaginary part of Fx, Eq. 2.28, to change from 1 on the airfoil to 0 on the flap. It is now possible to find the flow functions entirely in terms of theta functions in the w-plane. F2 has a singularity at the leading edge of the flap; its imaginary part is equal to 0 on the airfoil and equal to 1 on the flap: 2w K' ; (2.30) The a-Problem va = -a on the surfaces of both slits. As will be seen, a simpler solution is obtained by considering the overall angle of attack of the configuration as a single problem rather than by dealing separately with each airfoil element's boundary condition. The form of the solution will be: <{*) = Cx [Fx + F2- [Fx + F2)z=00] where Cx is some constant. But, using Eqs. B15: K' Fx+F2 = i + i— # 2 , ^ ^ 4 , ^ #2 V = 1 .2K' snAdnA i :— JT cn A and using Eqs. 2.4 and 2.10: Fx + F2 = i - i 2K' 1 - 3 z{z - f) (2.31) The square-bracketed expression is purely imaginary on the surface of either slit, so that: i i / 0 ( 2 ) = i a j l -\z-s)(z-\) (2.32) z{z - f) This satisfies all the required conditions and, as the slot goes to zero, it reduces in a simple manner to the corresponding one-element flow function, Eq. 1.15. In the last section of this chapter, this flow function is extended to include an arbitrary number of in-line airfoils. Tandem Symmetrical Airfoils / 2.3 32 The rj' Problem v; = o on the airfoil on the flap. The flow function will have the form: « ; ( * ) = C 2 ( F X + F2) + CZF2 - [C^Fi + F2) + C 3 F 21 If: _ 2 X 3 = ~jT ~K T3{Xoo) then, using Equations 87 and B3, it can be shown that F^u^) = l^3> 8 0 that: •(»-,)(«-I)" <{*) ^ ( 7 + I ^ z{z - f) n (2.33) (2.34) 2.3 T H E FORCES Lift and moment are obtained with Blasius' equations, Eqs. 1.11 and 1.12. In order to evaluate the integrals using the residue theorem the following expansions are necessary. As z —• 00: z{z - f) 1 I 3 + 1 - f l - ( / - s ) (3 /+s -2 ) t 2z 8z2 •(A) (2.35) and using Equations B17 and 2.1: 2u MK'0'2, , G2-Gl , G 2 ( 3 + l + / ) - G 1 ( 3 / + 3 - l ) , 8z2 where: „ l t S2K' rr (2.36) (2.37) E' is a complete elliptic integral of the second kind (Appendix B). Calculating the overall forces is now straightforward. For the forces on the airfoil alone, an airfoil selector function, A{u), is used: 2u K' Alui) = 1 — - r M w ) = 1 + t'odd(7) on the airfoil 1: K V (2.33) = 0 +1 odd(7) on the flap. Tandem Symmetrical Airfoils / 2.3 33 The notation simply indicates that the imaginary part of A(u>) is odd in 7 on the slit surfaces. As z —• 00: ( ^ ) G I - O J (~) G,(33+/+l) - G,(«+/+l) ^ , ~ I - B 4 + W J L _ + 0Q_J (2.39) where: , , * » ^ + f ^ ( X o o ) = E 3 + ^ . (2.40) Now, for any function F(z), Cauchy's Theorem gives: jF{z)A{u)dz = j F(z)dz + jF(z)[A(u)-l]dz + j F{z)A{u)dz (2.41) C Ct Ci Cf where C encloses the tandem airfoil configuration, Ct encloses just the airfoil, and Cy encloses just the flap. For the a-problem the airfoil lift is: Ct, = --Jw'a(z)dz=2-^.JF{z)dz c. c, where: F(z) z(z - f) = 0 + 1 odd(7) on the airfoil (2.42) = 0 + »odd(7) on the flap. Since A(OJ) has a square-root pole only at z = s, each contour integral in Equation 2.41 can collapse onto the slit surfaces — i.e. there are no simple poles in the integrands at the slit leading or trailing edges (or anywhere else) that need to be accounted for. Therefore, since dz is also odd in 7 on the slits, the last two integrals of Equation 2.41 are zero due to the oddness of their integrands and: *-*/™">* C which is easily evaluated using Equations 2.35 and 2.39. CmiQ is handled in the same way by replacing F(z) with zF[z) in the above procedure. Tandem Symmetrical Airfoils / 2.3 34 For the ^-problem this procedure will not work because the real part of V — -r-[u) = say, is not zero on both slit surfaces. By integrating by parts first, however, a function is obtained which has this necessary characteristic. Since 7(w) is periodic in JTT, one gets: j 7(u) dz = - j(z - f)^- dz C. Ct where / dz is taken as z - f to handle the singularity in 7(u) at z = / . Now: ( z - f ) ^ = - g ^ * " ^ " ^  " 7 ) = 0 + iodd(7) on the airfoil k "dz 2s/z(z - s){z - f)(z -1) K l ) (2.44) = 0 + iodd(7) on the flap. Identifying this function with F(z) from Equation 2.41 results in the last contour integral of that equation being zero. The second last integral is not zero since the square-root pole at z = s in A(ui) combines with that in Equation 2.44 to form a simple pole with residue: '2K' 2E'' v , M ^d7 A l , f-s 2K' ^(z-s)(z-f)-A(u) = - — — IT (2.45) so that for the r/-problem: a s [J ^ (z-1)- G2(z - f) (z - s)(z - f)(z - 1) A(UJ) dz + 2iri(f-s) 2K' 2K' 2E' } (2.46) which is now solved by residues. Cmso is evaluated the same way by taking / zdz as (z2 — f2)/2 when integrating by parts. Note that A(ui) is not the only airfoil selector that could be used in Equa-tion 2.41. The function: 1 2UJ IT K' - M also has the necessary characteristics and has a square-root pole at z = 1. Analogous functions constructed from t ? 2 ( w ) a n < ^ $i(u) a r e n o * used, however, since they have square-root poles at the flap and airfoil leading edges, respectively, which would combine with those in F(z) (Eq. 2.42) to produce simple poles in the £-plane. Tandem Symmetrica/ Airfoils / 2.3 35 a) b) c) d) e) 0 g) h) i) j) 1) dCt da d(h dr, da = 2TT - ^ 3 da dr, dC, da dCt dr, da m o da dC, m o dr, dCm da dC„ dr, da da dC, dr, S3 + G2 - G\ «+W.. (s+l-f){l-Vz) + G2-'^-J-Gl j _ da i _ E 3 — ^ + — oa a 3 + 1 - / dCt 1-f da ' 1-f da 3 + 1 - / dCt 3 dCtt 1-f dr,~T=J dr, ( G 2 - G 1 ) ( 1 - E 3 ) - 7 ^ G 2 t. 2 4 ( / - 3 ) ( l - / ) + (s+1-/) 2 [(3+l + / ) G 2 - ( 3 / + 3 - l ) C ? 1 ] »0 _ 3 da 2 ( 3 + l - / ) 2 ^ { [ 1 - ( / - a ) (3 /+3 -2)] ( l - E 3) + (3 /+3 -5 )G l + ( 3 / - S - l ) G 2 } dC„ (2.47) m /o _ 5a 2 dc, -^0\-{G2-Gif) 1 - / 5a / 3 + l - A 2 5 G m o / 3 \ V 1 - / ; dr, \ l - f j dr, • 1-f dr, m » 0 _j_ J v Equations 2.47 summarize the forces for the a and r, problems. They are all in terms of the three constants Gj, G 2 , and E 3 which are easily calculated with Equa-tions 2.48. G m / o is the flap moment about its own leading edge. Equations 2.48 are found in or derived from Whittaker and Watson (1927) and are all presented in Appendix B. As the slot in Figure 4b goes to zero, so too does q. For most airfoil flap configurations, q < 0.1 and the qA terms in Equations 2.48 can be Tandem Symmetrical Airfoils / 2.3 36 a) b) c) d) e) f) i) j) k) k2 = Jfc2-f-*'2 = l 2eft = it2 (l + ^ C l + N/ib7)2 g = e~*£- = c0 + 2eo + 15*0 + 150e^ 3 + O(l000cJ7) = (1 + 2q + 2g4 + 2g9 + 2?16 + • • ) 2 ir n \ n J q - | ( a ) + T : k - ( s ) v , ( « + v + v + ^ + - ) 2K 22?' 2 g) e = i V/ifc' 1 ( 2 . 4 8 ) cos 2XCO = * [ l - ? 4 ( 2 - 4 ^ 2 ) + <? 8 (3 - 2 0 ^ 2 + 3 2 ^ ) - qn(6 - 7 6 £ 2 + 272cT 4 ~ 3 2 0 £ 6 ) + 0(g 1 6)] 2K' 0<2Xoo<n 1-5 2 F"' Jr 2Xc /VV7A 7 + 7 + g 1 2 s i n l 2 X o o + 0( g 1 6)] neglected. For large slot configurations, evaluating the series up to and including the q12 terms should give sufficient accuracy. If extremely large slots are of interest, then it would be better to replace the q series with either q' series or the series of Appendix B which converge in powers of q16. Equations 2 . 4 7 and 2 . 4 8 have been programmed into a TI Programmable 5 9 hand calculator. It takes only half a minute to evaluate them to within an error O(l00(V 6). The results for dCt/da and dCmo/da are particularly simple, in keeping Tandem Symmetrical Airfoils / 2.4 37 z - PLANE x„ x 2 X 2 2 * 3 X 3 3 NN F I G U R E 5 N in-line uncambered airfoils. with the simplicity of their flow function, Eq. 2 .32, and they reduce to the one-element thin airfoil theory predictions, section 1.4, when f — s = 0 . The expression for the tan-dem airfoils' aerodynamic centre: dCmJdat dCt/dc is also simple while the pitching moment about the aerodynamic centre: (2.49) ^mac ~ dCmo [ dCt xac dn dr, s+l-f -( 5 +i-/)2{G4^=7 f . - Gi 2s s+l-f - 1 (2.50) is independent of E 3 . The rest of Equations 2.47 have also been shown to reduce to the one-element theory predictions for both f—s = 0 and f—s= 1. Comparisons between Equations 2.47 and exact-numerical potential flow theory are presented in Chapter 4 . 2.4 N IN-LINE U N C A M B E R E D AIRFOILS The a-problem for the N in-line airfoils of Figure 5 is solved by: w'a(z) = ta j ; (z - xn){z - z 2 2) • • [z - xNN) i } (2.51) (z-xl){z-x2)---(z-xN) since, again, the square bracketed expression is purely imaginary on the slit surfaces, it goes to 1 as z —*• oo, and it is zero at all the trailing edges. Tandem Symmetrica] Airfoils / 2.4 38 The overall lift is: // — 7 < Ct = — r coefficient of - in w'(z) as z —> oo z which gives: Ct = 2rca. (2.52) The overall pitching moment (about the origin) coefficient is given by: r - 4 n i ° m o _ (c1 + ...+c„)» coefficient of -=• in w'(z) as z —• oo If the slots are defined by: s0 = xx s1 = x2 -xn s2 = x3 ~ x22 etc. then: Cm 0 = ~Y j 1 + (Ci + .,4 + C j y ) 2 [ 8 0 C l + ( « 0 + Sl)c2 + • • • + ( « 0 + * • - + S i V - l ) C / v ] J . (2.53) If the origin in Figure 5 is at the aerodynamic centre, then CmQ = Cmac = 0, a0 = — xoc> and: * a c = (ci+'-'+c )^ 1 3iC 2 + (3i+3 2 ) c 3 + - - - + (3i+--- + 3N - 1 )c^ 4 (ci+-+cNy (2.54) Surprisingly, these simple, general results for the overall lift and moment do not seem to have been presented before, although Glauert (1926) shows that the exact inviscid lift for tandem flat plates of equal chord is Ct = 2jrsina. Richardson (1981) also arrived at the results Equations 2.52 and 2.53 at about the same time as the group here at U.B.C. He also found A C ; for a simple parabolic camber line — a result which is duplicated and generalized in the next chapter. CHAPTER 3 General Tandem Thin Airfoil Theory For tandem airfoil elements, the thickness distribution analysis and camber line design problems are solved with the same elementary functions and transformations used in one-element thin airfoil theory. The first two sections of this chapter present this part of the theory. The camber line analysis and thickness distribution design problems, however, are not so easily solved. They require the use of elliptic integrals or functions for a satisfactory description of their general properties. This is dealt with in sections 3.3 through 3.5. In the final section of this chapter, the general camber functions of section 3.3 are used to obtain generalized expressions for overall lift and moment which are the tandem airfoil extensions of Munk's integrals. In the sections that follow, basically two classifications of functions are pre-sented. The first consists of general functions which are analogous to the one-element camber and thickness functions of section 1.4 (Eqs. 1.23 and 1.28) in that they use simple Laurent expansions to describe the flow fields. They provide good qualitative descriptions and are useful for handling numerically specified airfoil profiles. Although they can be used for any type of airfoil profile, the necessity of evaluating a large number of coeffi-cients makes their use undesirable when a boundary condition is specified as an analytic function or, perhaps, as a group of analytic functions each of which applies over some dis-crete segment of the airfoil. Here, the second classification special functions is more useful. 39 Generai Tandem Thin Airfoil Theory / 3.0 40 F o r e x a m p l e , t he ry-problems of b o t h one-e lement a n d two-e lement t h i n a i r f o i l t heo ry are so l ved w i t h s p e c i a l f u n c t i o n s t ha t sa t i s fy one p a r t i c u l a r t y p e of b o u n d a r y c o n d i t i o n , w i t h the o n l y v a r i a b l e o r coef f ic ient b e i n g the a m p l i t u d e of tha t p a r t i c u l a r t y p e of d e v i a t i o n . O t h e r t h a n t r y i n g to s u m the ser ies of a genera l f u n c t i o n a f ter h a v i n g eva l -u a t e d i t s coef f ic ients , there is no gene ra l w a y of f i nd i ng the s p e c i a l f u n c t i o n fo r a pa r -t i c u l a r b o u n d a r y c o n d i t i o n . E x p e r i e n c e , s i m p l i f y i n g the p r o b l e m to t he ana logous one-e lement t h e o r y p r o b l e m , a n d , m o s t i m p o r t a n t l y , u n d e r s t a n d i n g the n a t u r e of the s i ngu la r -i t ies t h a t m u s t p r o d u c e t he des i red resu l t a re the best a ids t o d o i n g i t . T h e gene ra l r u le t o fo l low w h e n sea rch ing f o r a n a p p r o p r i a t e s i n g u l a r i t y is t ha t a d i s c o n t i n u i t y i n v' m u s t be assoc ia ted w i t h a s i n g u l a r i t y i n u ' , o r v i c e v e r s a . T h i s is a consequence of the a n a l y t i c i t y o f t he f low f ie ld . U s u a l l y , t he d i s c o n t i n u i t y i n v' c a n a lso be assoc ia ted w i t h a d i s c o n t i n u i t y i n the b o u n d a r y c o n d i t i o n , dy/dx, a n d the s i n g u l a r i t y i n u ' w i t h a s i n g u l a r i t y i n the a i r f o i l e l emen t ' s sur face p ressure . T h e i m p o r t a n t excep t i on to these a p p a r e n t l y s t r a i g h t f o r w a r d assoc ia t i ons o c c u r s at l e a d i n g a n d t r a i l i n g edges . Iu the a - p r o b l e m fo r e x a m p l e , E q . 1.15, v' — —a is pe r fec t l y c o n t i n u o u s e v e r y w h e r e o n the b o u n d a r y , even a r o u n d the l e a d i n g edge. H o w e v e r , u ' has a s i n g u l a r i t y at t he l e a d i n g edge. In th i s case , t he d i s c o n t i n u i t y i n v' o c c u r s be tween a p o i n t o n the a i r f o i l at the l e a d i n g edge a n d a p o i n t i m m e d i a t e l y i n f ron t o f the l e a d i n g edge, wh i l e the assoc ia ted s i n g u l a r i t y i n u ' is en t i r e l y o n the sur face of t he a i r f o i l . T h e r e i s , o f cou rse , a p h y s i c a l reason fo r the d i s c o n t i n u i t y i n v' a n d if t h i s c a n be a n t i c i p a t e d t h e n the p r o b l e m resolves i tsel f . T h i n a i r f o i l t h e o r y p ressure s i ngu la r i t i es c a n a lso be t hough t of as b e i n g assoc ia ted w i t h e x t r e m e s i n the exac t p o t e n t i a l f low pressure d i s t r i b u t i o n , s u c h as those w h i c h o c c u r at s u c t i o n p e a k s o r s t a g n a t i o n p o i n t s . O n e m u s t be ca re fu l here t o o , however , s ince no t a l l s t a g n a t i o n p o i n t s p r o d u c e p ressure s i ngu la r i t i es that exist on the airfoil's surface. A g o o d e x a m p l e of t h i s is seen i n the f low a r o u n d a n e l l i pse , as p resen ted i n the f o l l o w i n g s e c t i o n . General Tandem Thin Airfoil Theory / 3.1 41 iy n z - PLANE •PLANE F I G U R E 6 Solution planes for the thickness and camber problems. 3.1 T H E T H I C K N E S S DISTRIBUTION ANALYSIS F i g u r e 6 shows the m a p p i n g s to the a n d s o l u t i o n p lanes where the genera l t h i ckness p r o b l e m f u n c t i o n s are c o n s t r u c t e d . In t he ^ , - p l ane , a J o u k o w s k i t r a n s f o r m a t i o n opens the a i r f o i l s l i t u p i n t o the un i t c i r c le wh i l e l e a v i n g the flap a s t ra igh t l i ne s l i t o n the p o s i t i v e rea l a x i s : -(1 + cos 0S) o n t he a i r fo i l su r face . (3.1) (3.2) (3.3) S i m i l a r l y , i n the f r - p l a n e the flap becomes the un i t c i rc le wh i l e t he a i r f o i l is left a s t ra igh t l ine s l i t — th i s t i m e o n t he nega t i ve rea l ax i s : 1 + / 1-f f 1 \ ^ r = ,-I^ ±V(*-/)(*-l) x - / = 1 - / (1 + cos Of) on the flap su r face . (3.4) (3.5) (3.6) Genera/ Tandem Thin Airfoii Theory / 3 .1 42 The general thickness problem function is: w[(z) = A*0 ,X^A±L ( 4- 1 Cn + Afo , Afn + 1 (3.7) On the airfoil, the second group of functions in this equation is purely real so that the airfoil boundary condition is simply: ± ^ = ^ t a n ^ + £ A , n s i n n 0 . dx 2 2 ' 0. C O with: = i / te _ ir J [dx n=l sin n9t d9t (3.8) (3.9) exactly as in one-element thin airfoil theory when the leading edge singularity in dyt/dx is accounted for. AtQ must be determined from the nature of this singularity before the other Aln's can be calculated. If the airfoil elements are closed bodies, then the net source strength they produce is zero and A90 + Atl — 0 = Af0 -f A^, which is precisely what Eq. 3.9 gives for n = 1. The pressure on the airfoil is: C„ = "2 (3.10) Similar simple expressions are obtained for the flap. For conventional airfoil shapes, Equation 3.7 is completely general. w[(z) is just a linear superposition of the functions which satisfied each airfoil element's bound-ary condition in one-element thin airfoil theory. Since these functions are purely real everywhere on the real axis except on the slit whose boundary condition they are required to satisfy, any number of them can be linearly superposed to solve the general thickness problem for multi-element in-line airfoils. Note, too, that Equation 3.8 is odd in 9t since ±dyt/dx must be, and that Cpti Eq. 3.10, is even in 9t confirming that thickness does not affect lift for in-line airfoils. General Tandem Tbin Airfoil Theory / 3.1 43 In the following example, both leading and trailing edge singularities are used to illustrate the magnitude of thickness interference effects for tandem bodies. Consider the tandem ellipses: ytt = t9y/x(s — x) for the "airfoil" ytj = tfs/(x- / ) ( l - x ) for the "flap" where tj is the appropriate thickness to chord ratio. Now: (3.11) -t s X~2 and: dyt, dx dx V(* " / H I " * ) *\/x(s - x) so that the flow function is simply w[(z) = t. 1 -3 Z~2 \Jz(z - s) z — 1 + / 1 (3.12) which goes to zero as z —» oo. The singularities at the trailing edges of the ellipses are entirely contained within their boundary conditions, so that the pressures are: _ 1 + / C p ,= - 2 * , - 2 t ; Cp=-2tf-2t, 1 + 1 -V ( l - x ) ( / - x ) s X 2 \/x(x - s) 0 < x < s f< x< 1. (3.13) These equations are plotted in Figure 7 for a 33|% flap, for t„ = tj = 0.15, and for various slot sizes (an effective midpoint in slot sizes, between the one-element thin airfoil theory configurations at /—s = 0 and /—s = 1, is suggested by the rectangle geometry, Fig. 4c, when K = K'). In one-element thin airfoil theory, the pressure distribution over an ellipse is just Cp = — 2t, which is consistent with Equations 3.13 when f—s = l. It is the square bracketed expressions in these equations which describe the interference effects. General Tandem Thin Airfoil Theory / 3.1 44 F I G U R E 7 Thin airfoil theory predictions of the pressure distributions over tandem 15% thick ellipses, ( l - / ) / ( s + l - / ) = 1/3. When /—s = 0, the strong discontinuity in dyjdx at z = / results in the relatively strong singularities in Eqs. 3.13 at this location. Note that despite the fact that these ellipses have stagnation points at their leading and trailing edges, the pressures on the surfaces of the separated ellipses are not singular here. The singularities do in fact exist in u' but they are immediately ahead of or behind the ellipses, as can be seen from Equation 3.12 when, for example, z —> 1 along the real axis from z > 1. A more useful example is the thickness problem for in-line airfoils with the same thickness distributions as those of the NACA four and five digit airfoils (Abbott and von Doenhoff, 1959). That is: yt(x) = t (Ay/x + Bx + Cx2 + Dx3 + Ex*) , 0 < x < 1 (3.14) for an airfoil of chord length 1. If the conventional NACA distribution is adjusted so General Tandem Thin Airfoil Theory / 3.1 45 that yt(l) = 0 (instead of 0.01*), then A = 1.47792, B = -0.63000, C = -1.77370, D = 1.44700, and E = -0.52122. Now, scaling to the airfoil coordinate system: dyi \A Is r. / r \ 2 /r\3 dx and, from Equation 3.7, since: k . « . [ ^ + fl + 1 0 f + u , ( £ ) ' + „ ( £ ) ' ] ( , 1 5 ) then: as x —» 0 on the upper surface K = At, and the evaluation of the remaining A,n's can be accomplished with Equation 3.9. This is not, however, the most efficient method. Since Equation 3.14 is composed of simple algebraic functions, it is easiest to deal directly with the particular flow functions associated with each of those algebraic functions. For example, the function: 1 TT , , Z — 1 o Z 1 z 3 ln + z 2 + - + -z 2 3 has its imaginary part equal to +z3 on the upper and lower surfaces of a slit stretching between 0 and 1, it goes to zero as z —* oo, and, as a one-element thin airfoil theory thick-ness function, it cannot violate the Kutta condition. The flow function for the boundary of Equation 3.14 is: w\(z) = -± -4= In ^ -4 + (B + 2Cz + 3Dz2 + AEzz) In — 2y/z y/z+1 V ' Z + 2C + 3D (3.16) The first function in this equation is interesting. It was obtained in the y/z-plane where the necessary logarithmic singularities are compatible with the requirement that the flow func-tion be single-valued in the flow field. General Tandem Thin AirfoiJ Theory / 3.1 46 The thickness flow function for tandem NACA airfoils is obtained by scaling Equation 3.16 to the airfoil and flap geometries, and then linearly superposing the two resulting functions. The airfoil pressure distribution becomes: p* s/s — yfi + B + 2CX--rZD^)\AE(?)Z]\»S-X + 2C + ZD + 7r - 2tan 1 If-x 1 - / + In 1 - x ~x + 2C + 3D and the flap pressure is: x — s ir | 2 V * y/x + ^ /s [ s W \sJ J x *1 , n \fx\2 x 1 - + -\+AE\[-) + — + -s 2] [\3/ 2s 3 + 2C + 3D I A / ! - / y ^ T - y ^ T + . + 2 ^ + 3 . ( ^ )2 + 4 , ( ^ i ) 3 ] l n l ^ 1 - / + 2 j + 4 i i K l - / y 2(1-/) + 3 + 2C + 3D (3.17) (3.18) These pressure distributions are graphed in an example in the next section. General Tandem Thin Airfoil Theory / 3.2 47 3.2 T H E C A M B E R LINE DESIGN P R O B L E M Here, pressure distributions which are odd in 0, and 0j are specified and the camber lines needed to create them are then worked out. The general function to work with is, again, a simple linear combination of one-element flow functions: (3.19) The first grouping in this equation is purely imaginary everywhere on the real axis except on the airfoil slit and thus does not affect the flap's pressure distribution. Similarly, the second group of functions has no effect on the airfoil's pressure distribution. Therefore: 9 CP. = £ . o t a n V + 2 XX sinn*. n=l oo (3.20) CP, = Bfo^T + 2ZX 8 i n n*/ n=l Camber lines are usually designed to give optimum performance in a partic-ular situation; i.e., for given values of lift and, perhaps, moment. Since this ideal or design performance is achieved by minimizing adverse pressure gradients, Bj0 is usually chosen to be zero. Then: B,n ='iy Cp. An n0.de. B (3.21) sinntfy d0j and the lifts are simply: Ct=z+£j[»B.i+V-f)Bfl] Ctt = -nBtl ; Ctj = -*Bfl. Of course, each group in Equation 3.19 affects the forces on only one airfoil element since it affects the pressure on only one airfoil element. Again, these simple results and the simple construction of the camber design flow function are also applicable to multi-element in-line airfoils. (3.22) General Tandem Thin Airfoil Theory / 3.2 48 A better way of handling the camber design problem is to use closed form functions since camber pressure distributions are usually specified by simple algebraic functions. In the following example, tandem airfoil camber lines which combine uniform chordwise loading with zero pitching moment about the aerodynamic centre are designed. The camber function: . . . iff,. z — s iffr. z—1 , = I n — f i n — 3.23 IT Z IT Z — J gives constant airfoil and flap pressures which are odd in 6, and 6j (ff, and ffj are con-stants): CP. = T2ff, ; Cpj = T2fff. (3.24) The airfoil boundary is: dyc. o-,, s — x ffr, 1 - x - T T = " T m — 1 ~ ~ + " T M 7 — Z • (3-25 dX IT X IT J — X yc,(x) is now obtained by integration. The constant of integration is arbitrarily chosen so that yc,(s) =0: yCl(x) = f^- [s In s — (s — x) ln(s — x) — x In x] + — [ ( / - *) ln(/ - *) - (1 - x) ln(l - x) + {1-3) l n ( l - « ) - (/-a) ln(/-a)l . IT (3.26) Similarly, yeAx) is obtained by integration in such a way that ycAf) =0: yc ,(x) = — [ ( z - s ) l n ( x - s ) - x l n x - ( / - s ) l n ( / - s ) + / l n / j * IT + % ! - / ) l n ( l - / ) - (1 - x)ln(l - x) - (x - /)ln(x - /)]. (3.27) It is interesting that: = Mi l = in(l-a) + / l n / - ( / - a ) l n ( / - 3 ) ] . (3.28) ffj ff, IT The choice to zero the camber lines at x = 5 and x = / is arbitrary. Some stagger between the airfoil elements is acceptable as long as the linearizing requirement, that deviations from the average line be small, is kept in mind. General Tandem Thin Airfoil Theory / 3.2 49 Now: C<=7j^Z7k. + (W)<7] (3.29) and: c - o = (3+7-/)^sV«+^-f2>f\- (3-3°) If the flap chord to total chord ratio is fixed (so that only the slot size is variable), then it is convenient to define: = sTTJ ; S = f-s. (3.31) The lift is then independent of /—s: Ct = 4<x, + 4(ay - crt)Rf . (3.32) From Equation 2.49, the aerodynamic centre is: Z f l C = \ " 5 ( i " ( 3 ' 3 3 ) and its associated pitching moment is: [2^/(1 + S) - 3 - 5] , Cm„ c = ~<r. + (cr/ - O / L 7 V l _ j i . (3.34) If C ' m a c can be designed to be zero, then: CTf _ 1 - 5 T.=sl + Rf[2Rf{l + S)-3-S]' ( 3 ' 3 5 ) If 5 is small and the ratio o~f/o~t is required to be positive, then this last equation only has solutions for Rf > |. As an example, consider a tandem airfoil configuration with Rf — 0.625 and S = 0.2. A number of current tandem wing light aircraft designs use this type of arrangement. It should be noted, however, that many other considerations are important in the design of an aircraft wing and that this example is chosen mainly to exhibit the way in which camber design can be accomplished with thin airfoil theory. The slot size in this example is probably too small for a tandem wing aircraft and it is certainly too large for a slotted slat arrangement. It is chosen so that K M K'. General Tandem Thin Airfoil Theory / 3.2 50 If, in this example, a design lift coefficient of 0.6 is chosen and if Cmae is required to be zero, then a„ = 0.2833 and <Tj — 0.07. Figure 8a shows the camber lines that satisfy these requirements (Eqs. 3.26 and 3.27). If the NACA thickness distribution of the previous section (Eq. 3.14) is combined with these camber lines (according to Eqs. 1.7), then the airfoil shapes in Figure 8b are obtained. Figure 8a also compares the thin airfoil theory predictions for chordwise load-ing, Eqs. 3.24, with those which exact-numerical potential flow theory (Appendix A) would predict for the same geometry. The Becond order effects of thickness are shown by includ-ing exact-numerical computations for both 1% and 15% thick airfoils. Figure 8b shows the actual pressure distribution predictions over the 15% thick airfoils. The thin airfoil theory predictions of these pressures are obtained from Equations 3.24, 3.17, and 3.18. Table 1 compares the lifts and moments of the various computations for a = 0°. The linearized predictions in this table can be added to the a and n problem solutions of the previous chapter to solve the general force problem for this configuration. ACt A C , , A Q , A C m o &Cm.0 A C L mac 0.6 1.133 0.28 -0.244 -0.567 -0.14 0 0.585 1.087 0.284 -0.241 -0.542 -0.138 -0.003 0.623 1.154 0.305 -0.253 -0.553 -0.144 0.000 T . A . T Ex-num 1% thick Ex-num 15% thick T A B L E 1 Potential Sow predictions of the forces on the tandem airfoils of Figure 8. The overall thin airfoil theory force predictions agree well with the exact-numerical results, with the same trend as was seen in the one-element theory of the exact theory overpredicting the linearized theory for normal 15% thick airfoils. This accuracy is consistent with that of ACe in Figure 1. The individual airfoil element forces also show good agreement. As in one-element thin airfoil theory, pressure distribution predictions can General Tandem Thin Airtoil Theory / 3.2 51 0.8 0.4 1 • • • i i - 1" • • • 1 , •, 1 , — Thin airfoil theory. • • • • • Exact-numerical, 1% thick. • • Exact-numerical, 15% thick. • • • • • • r • —* " * —* • * • • (LC. 1 1 1 • 1 I I i • 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 x a) The camber lines and chordwise load distributions for 1% and 15% thick airfoils, a = 0°. A C p = C - Cp F I G U R E 8 Potential Bow predictions of tandem airfoils designed by thin airfoil the-ory to give uniform chordwise load distributions at a = 0°. f—s = 0.2, ( ! - / ) / ( « + ! - / ) = 0.625, K = 1.07K'-, xac = 0.325. General Tandem Thin Airfoil Theory / 3.3 52 be fairly inaccurate at localized points but, nevertheless, do show good overall qualitative agreement with the exact potential theory. The leading edge singularity in the thin airfoil theory thickness pressure dis-tributions of Figure 8b is an interesting irregularity. It is caused by a hidden discontinuity in ±dyjdx through x = 0, the second term in Equation 3.15. Since Cpt ~ O (-Bln^) as x —• 0 in Equation 3.17 and since B is negative, Cpi —* —oo. This is, perhaps, sur-prising because thickness functions usually give positive infinite pressure singularities at leading and trailing edges due to the presence of stagnation points. However, the rounded leading edge of an airfoil is like that of an ellipse and, as with an ellipse, the pressure sin-gularity associated with the stagnation point is not present on the airfoil's surface. 3.3 C A M B E R LINE ANALYSIS: G E N E R A L F U N C T I O N S This section presents two different camber functions which can be used for calculating pressures, lifts, and moments generated by arbitrary camber line shapes, including the simple shapes of the a and n problems. The first function is the simplest in that it uses the elementary functions of the previous two sections of this chapter and provides easily understood qualitative information. The coefficients of its Laurent series reduce to the one-element theory coefficients when f—s = l. However, since one set of these coefficients fails to converge when / — s = 0, this function will be of limited practical use for small slot configurations. The second function is also constructed from elementary functions but ones that are of a more complicated nature. It has the advantage that the coefficients of its series converge like the one-element theory coefficients at both slot extremes. Both of these general flow field descriptions are unsatisfactory in that the coefficients of their series must be recalculated each time the slot size changes, even though every other aspect of the geometry may stay the same. This problem is compounded by the fact that all the coefficients are calculated via elliptic integrals, and all of them are needed for calculating pressures or localized forces. The problem is less acute when only General Tandem Thin Airfoil Theory / 3.3 53 overall forces are being considered since, as with the one-element theory camber function, the overall forces from both functions are described with only the first two or three of the coefficients. Even so, the direct methods of the next chapter are more efficient when evaluating either localized or overall forces. Interestingly, these direct methods were first motivated by the qualitative information provided by the camber functions of this section, as presented in the last section of this chapter. If pressure distributions are required and the camber line is specified as an analytical function, then the methods of the next section of this chapter are the most efficient ones to use. If the camber lines are numerically specified, then the two functions of this section are probably as efficient a way of obtaining the pressure distributions as any other. If the slot is small, then the second camber function should be used. The First Camber function is: Bf0 , 1 n=l V (3.36) where the and planes are as shown in Figure 6 and as given in Equations 3.1-3.6. The first group of functions in this equation is purely real on the flap since the square-root expression in it is purely imaginary on the flap. Similarly, the second group of functions is purely real on the airfoil. Thus: ( dVcL= /T= L dx y f-x 2 ^ , L n=l cos nO. (3.37) and: . TT J dx V 1 - X ' ' IT J dx V Z ( l - : — x x)(s — x) cos n9g dx. (3.38) Similarly: and: dy, dx X — s B fo + J2BfnCOSTl9f n=l (3.39) 2 f dyc I ^ 2 t d]ic I ~~x Bfn = - -r^-J cosn9,dOf=: - - 7 A / 7 w H T ; rcosn0fdx. (3.40) /B KJ dx\x-s 1 1 irj dx V (x - s)(x - /)(1 - x) J v ' General Tandem Thin Airfoil Theory / 3.3 54 When f — 3 —• 1, Equation 3.36 reduces to the conventional one-element theory repre-sentation of camber for two airfoils infinitely far apart. There is no problem with the coefficients in this geometrical extreme. However, since the Bfn'a blow up when / —• s, this representation of camber is of little practical use for small slot configurations. The overall lift is: C t = Z ^ n B t o + B t l ) + (l-f)(Bfo + Bfl)] (3.41) and the overall moment is: C - o = 4 ( 3 + I _ / ) 2 [ 3 ( 3 - 2 + 2 / ) ^ o +2«(»-l+/)B.1 + s2B,2 + (l-/)(3/+l-2«)fl/o + 2(l-f)(l+f-s)Bfl + (l-f)2Bh]. (3.42) These expressions, through the Btn and Bfn coefficients, give the effects of the independent airfoil and flap boundary conditions on the overall forces. However, unlike Eqs. 3 .22 , the separate groupings of Btn and Bfn coefficients in the expressions do not give the effects of the localized airfoil and flap forces. In general, all of the Bln and Bjn coefficients from Equation 3.36 will have an effect on the localized forces. The Second Camber Function's Laurent series are defined in the f-plane which is arrived at through the elementary mappings illustrated in Figure 9. As f-s -* 0, this f-plane reduces to the f-plane of Figure 3 — the one-element theory's solution plane. When f—s —*• 1, the first Riemann surface of the f-plans becomes the f,-plane of Figure 6 , while the second surface becomes-the (f -plane. The intermediate z2-plane is arrived at through a simple Joukowski trans-formation with critical points at ZQ and S Q in the 2-plane, corresponding to Z2Q and ^ Q in the 2^-plane. Each side of the circle outlined in dots in the z-plane is transformed to the vertical line joining Z 2 Q and 22 0 ; the left side of the circle is in the first Riemann surface of the z 2 - p l a n e a n d the right side is in the second Riemann surface. One changes surfaces by crossing the branch cut joining ^ and Z 2 Q . The airfoil slit is also in the first Riemann sur-face and stretches between —2 and 2. It is superimposed exactly over the flap slit which General Tandem Thin Airfoil Theory / 3.3 55 2Q XO —•— z - PLANE • • • • • • 20 V2 -2 z2 -PLANE 2 •I 22o X2 -Branch cut joining Z 2 Q and z2g via oo. a) Linearized physical plane. b) Intermediate solution plane. $0 f -PLANE c) Solution plane with the airfoil in the first Riemann surface and the flap in the second. F I G U R E 9 The elementary mappings for the second camber function. General Tandem Thin Airtoil Theory / 3.3 56 is in the second Riemann surface. z - 2 z ( z - f ) + (z~s)(z-1) = 2 «(»-/) + («-«)(«-!)  2 z(z - f) - (z - s)(z - 1) a + 1 - / Z-XQ ZQ = X 0 + iyQ = a + ,V«(l-/)(/-«) 1 z — -8 3 + 1 - / (l + «-/)«2 + 2(1 +S + f ) - (l + S-f)yJ(z2-Z2Q)(z2-Z2Q) % = *2Q+mQ = 2[4s - (1 + s + f)(l + s-f)] + tW«(l-/)(/-«) («+l-/) a (3.43) (3.44) (3.45) (3.46) The f-plane is simply the 22-plane with the slit opened up to the unit circle: or: *2 = f + ~ /) + >/(ar-«)(ar-l) (3.47) (3.48) Since Equation 3.47 has the same effect in both Riemann surfaces, the opened up airfoil and flap slits are also superimposed in the f-plane—but still in different Riemann surfaces. Note that infinity in the z2 and f planes corresponds to z = oo or z = xQ) depending on which side of the branch cut and in which Riemann surface one is located. The second camber function is: ? +1 s" * - *o (+1 ~ r n—i n = l (3.49) Here, the function t'/(f + 1) solves the tandem airfoil a-problem and is equivalent to Equation 2.32. Since, in general, the disturbance velocity will be non-zero at z = x0 and since the functions 1/($ + !) and 1/f both have simple zeros here, the function 2yo/(2-Xo) helps ensure the generality of w'c(z) without introducing a singularity into the flow field. It also enables the second group of functions of Equation 3.49 to take on different values on the unit circle in the different Riemann surfaces of the f-plane for the same value of 9, whereas the first group has the same value in each Riemann surface for a given 9. General Tandem Thin Airfoil Theory / 3.3 57 Thus, using the notation of Figure 9a : 2yp _ [z - ZQ)(Z - zQ) - (z- xa)(z - xh) Z — XQ Z — XQ (3.50) so that: ^ = ^ +fXc°sng+(x~3g(*~It) dx 2 X - X n n=l x - x 0 ( X - Z 0 ) ( X - S 0 ) X-XQ c o + ^ Z?n cos n0 n=l OO Y + 52D»cosnd n=l (3.51) This separation is important since the function: (z-xa)(z-xb) = 1 + 3 - / Z-XQ A (z2 ~ * 2 n ) (3.52) takes on the same values in each Riemann surface—i.e. it is even about the branch cut. On the other hand, the function: ( Z - Z 0 ) ( Z - Z Q ) 1 + a - / Z - XQ •yJ{z2-z2o){z2-z2o) (3.53) is odd about the branch cut. Thus, the first line on the right hand side of Equation 3.51 takes on exactly the same values on both the airfoil and flap for given 0's, while the second line takes on values that are equal in magnitude but opposite in sign. Hence: dyc, = _ 2 (x - z0)(x - 2Q) -^ + J2Dncosn9 n = l (3.54) dx dx X — XQ This equation is only true in the first Riemann surface (the RHS would have to change sign in the second surface) so that 0 < x < s. The i?„'s, then, are given by: X - X 0 dx (X-ZQ)(X-ZQ) cos n9 dO dy, X-XQ dx (x-z0)(x-z0) —r cos n6d9 (3.55) o o where 0 < x < s for the first integral and / < x < 1 for the second integral. Since the second integral is evaluated on the flap, the sign of its integrand must be changed relative to what it is in Equation 3 .54. General Tandem Thin Airfoil Theory / 3.3 58 Continuing on: dx dx = 2 -^ + ]Tflncosn0 n=l oo + 2 so that: n=l n=l = dVc> dx (x-x a)(x-x 6) dyc, dyC} DQ dx dx + ^ DN cos nO n=l (x-x a)(x-x b) (X - Z Q ) ( X - ZQ) 1 + (x -x j (x -x 0 ) (x-z 0 )(x-5o) + <*y, dx j _ (x - xa)(x - X f r) (x -2 0 ) (x -2 0 ) . (3.56) with 0 < x < s. Then: 2 M2/c. ( * - * o ) 5 (ix (x-Zo)(x-5 0) cos n0 d9 + 2 ? rfye/ (x - x 0)2 T 7 dx (x-z 0)(x-2 0) cosnfldfl (3.57) o o where, again, 0 < x < s for the first integral and / < x < 1 for the second. This requires a change of the sign in the middle of the last square-bracketed expression of Equation 3.56 for the integrand of the second integral. Changing variables, the coefficients can be rewritten as: 2 f dyCt (x0 — x) cosnft dx >/x(3-x)( /-x)(i-x) dx + 2 f dyCf (x — xo)cosn0 1/ dx yJX(x- S){X- / ) ( 1 - l ) n i r j d x x/xTT^ cos n9 Vx(s -x ) ( / -x ) ( l -x ) dx 1 /"<tyc cosn0 dx >/x(x-s)(x-/)(l -x) dx (3.58) dx. (3.59) The behaviour of these coefficients as f—s —• 0 and f—s —• 1 is best examined for constant ify = (l-/)/(s+l-/). When / -a —• 0, the D n 's become 0(ln(/-s)). However, yl = /fy(i — Rj}(f—S) go that the second group of functions in Equation 3.49 disappears. In addition, the f?„'s reduce to the one-element coefficients, the Bn's of Equation 1.25, and, since the f-planes of the one-element and two-element mappings are equivalent for f—s = 0, the one-element camber function, Eq. 1.23, and those functions remaining in Equation 3.49 will also be equivalent. General Tandem Thin Airfoil Theory / 3.3 59 F o r t he in f in i te s lot e x t r e m e , where a —• 0 a n d / —• 1, a change t o the ( z ' , y ' ) va r i ab l es is m a d e so t h a t , i n the first i n teg ra ls o f E q u a t i o n s 3.58 a n d 3.59, x' = x/s a n d y ' = y/s. T h e second in teg ra ls a re dea l t w i t h i n a s i m i l a r way , r e s u l t i n g i n : Bn - (1 - Rf)Btn + RfBfn as f-s 1 1 1 (3.60) D n ^ ^ t n - - B f n a s / - a - + l ' whe re the Btn*s a n d i ? / n ' s are t he one-e lement t heo ry c a m b e r f u n c t i o n coef f ic ients fo r an in f i n i t e l y sepa ra ted a i r fo i l a n d f l ap , respec t i ve l y ( E q s . 3.38 a n d 3.40 w i t h / = 1, a = 0 ) . O n the a i r f o i l , as f-s -* 1 : —-° - -2Rf , 0 < x < a (3.61) Z-XQ w h i l e o n the flap: Z-XQ so tha t o n the a i r f o i l : ^ 2(1 -Rf) , f<x<\ (3.62) —iB 00 B <{*)-—r7-*E7? > 0 < z < a (3.63) A n=l a n d o n the flap: —iB 00 B u ' c W - 7 T f - 1 ' E - X . / < * < ! • (3-64) + i „ = i s/ T h u s , t he ser ies o f t he second c a m b e r f u n c t i o n converge at b o t h f—s ex t r emes i n t he s a m e way t h a t the one-e lement c a m b e r f u n c t i o n does . T h e r e f o r e , there s h o u l d a lso b e reasonab le convergence b e h a v i o u r f r o m these ser ies fo r i n t e r m e d i a t e s lo t s izes . T h e second c a m b e r f u n c t i o n ' s ove ra l l forces are eas i l y o b t a i n e d b y res idues : Ct = -7r (2? 0 + B{) (3.65) Cmn = . * _ f ) 2 { [1 - ( / - a ) ( 3 / + a - 2 ) ] Z ? 0 + 2 ( l + a 2 - / 2 ) B , + ( l + 5 - / ) 2 2 ? 2 - 8y2(l+*-/)(A> + £ > ! ) } (3.66) m ° 4 ( a + l - / ) 2 or : s+l-J (3.67) T h e exp ress ions fo r Ct a n d Cmae are m u c h s i m p l e r t h a n those fo r the first c a m b e r f u n c t i o n , w i t h Ct c o m p a r i n g f a v o u r a b l y w i t h E q u a t i o n 1.26 i n s i m p l i c i t y . General Tandem Thin Airfoil Theory / 3.4 60 The methods of the previous chapter can be used to obtain the localized forces from the first and second camber functions. However, all the coefficients in the series of these functions will contribute to the forces and, in order to "harvest" their contributions, one needs a recurrence relation to obtain all the terms in the expansion of the airfoil selector A[w)t Eq. 2.39. This is provided in Appendix C. General functions are not often used for actual calculations and no numerical examples have been worked out for either the first or second camber function. Their practical worths have yet to be proven. Qualitatively, however, they are important as relatively simple general descriptions of what has historically been considered a complicated flow problem. Note that the extension of these camber functions to the problem of an arbitrary number of in-line airfoils is straightforward, although the algebra for the second type camber function could get messy. 3.4 C A M B E R LINE ANALYSIS: SPECIAL F U N C T I O N S When camber lines are specified as analytical equations, the procedures of this section can be used to solve for pressure distributions, lifts, and moments. If only lifts and moments are required, the procedures of the next chapter are more direct. The special functions presented here solve for w'c(z) when dyc/dx is given as a simple algebraic function of x. The first function is simplest in that it solves a generalized o-problem using only elementary functions. It was first found by Richardson (1981) who used it in describing the aerodynamics of perforated bridge decks. The function takes dyc/dx to be an n t h order polynomial which is continuous over both the airfoil and flap. Its coefficients of xn are highly dependent on variations of f—s. The second function assumes dycJdx and dyCf/dx each have their own inde-pendent n t h order polynomial representation. As f—s varies, the coefficients of x" in these functions also vary, but in a simple manner. For example, if the airfoil's boundary condition Genera/ Tandem Thin Airfoil Theory / 3.4 61 were specified as a polynomial in (x/s)n, its coefficients would not vary at all with f—s and the coefficients multiplying its xn terms would only vary as l/sn. Note that the coefficients for these polynomials can be obtained if the one-element theory's Fourier series representa-tion of the airfoil element's camber line is known, Eqs. 1.24 and 1.14. In this case, the order of the polynomial will equal the order of the truncated Fourier series. The problem with the second function as a general representation of tandem airfoil camber is that the coefficients of the polynomials are 0(l/a n) and 0(l/(l—/)"). Thus, if f—s is large or if the airfoil or flap is relatively small, special precautions must be taken to handle the very large high or-der coefficients, which probably have the least effect on the aerodynamic characteristics. Now, for the first function, if: ^ = a0 + alx + a2x2+ a3x*+ --- + anxn ; 0 < z < s, / < z < l (3.68) ax then the following function is required: where the 5,'s are chosen so that Am(;z) —• 0 as z -* oo. That is: A m ( z ) ~ ^ + % - + % - + --- , a s z - o o . (3.70) Z JZ z° To find the A m . constants, the expansion for F(z) is needed: > - « ) ( * - 1 ) 1 F(z) = z{z - f) i F F 1 + -7- + + • • • , as z -> oo. (3.71) z z A recurrence relation for the F,'s is given in Appendix C. Then, the 5,'s are obtained via: 6i + Fi = 0 62 + 6^ + F2 = 0 63 + 62FX + 6{F2 + F3=0 (3-72) Sm + 6m_lFl+6m_2F2 + --- + Fm = 0 and the A m . ' s are: A m i = - [6mFi + 6m^Fi+l + ••• + « + m _ , + Fi+m]. (3.73) General Tandem Thin Airfoil Theory / 3.4 62 Thus, A m ( z ) has its real part equal to xm on the airfoil and flap, it goes to zero as z ~* co, and it satisfies the Kutta condition. The flow function for the boundary condition of Equation 3.68 is: = -» X) a m A m ( 2 ) . (3.74) m=0 The forces are: m=0 4TT " ^ f " 0 = ^ a + 1 _ ^ 2 YI a m ^ m a • (3.76) Richardson (1981) found this flow function, Eq. 3.74, for a parabolic cam-ber line and noted that its extension to include higher order terms was straightforward, as it is. He also extended the case of tandem airfoil parabolic camber to N in-line airfoils in a parabolic camber configuration. This latter case can be further extended to include camber lines with higher order terms by using the square-bracketed expression of Equation 2.51 to replace F(z) in Equation 3 .69 . The nature of Richardson's perforated bridge deck problem is such that Equation 3.68 is a practical description of the camber line. For multi-element airfoils, however, it is important to be able to use independent descriptions for each airfoil ele-ment's camber line. Thus, in the tandem airfoil case: % i = a0 + a{x + a2x2-\ + anxn ; 0 < z < s dx dyc-L = d0 + d1x + d2x2 + --- + dnzn ; / < x < l (3.77) dx for which the following function is required A,„M - « " { T + If §M " h + T + ? +'• •+ S FW} (3'78) which can be used to solve the flap boundary condition if: A / J z ) ~ ^ + ^  + ^ p- + --- , as*->oo. (3.79) General Tandem Thin Airfoil Theory / 3.4 63 The recurrence relation for the ^'s in the expansion: J(a;) = — + — E 3 + -^-+ - § + • . . , a s z - o o (3.80) is given in Appendix C, so that: 6h+SfJi+6fA + V3F3 = T3 (3.81) and the A r 's are: A / m < = Tm+i - [8fmFi + Sj^F^ + ••• + fyji^+m-i + E 3 F , + m ] . (3.S2) For the airfoil boundary condition, the required function is: A . m W = A m ( z ) - A / m ( z ) (3.83) and so: n = ~lJ2 [«m(Am(^) - A / m (z)) + dmAfJz)\ (3.84) m=0 solves the camber line problem of Equation 3.77. The forces are: 4JT N C ' = ^ pE[ f l «( A »..- A / m i ) + ( i A 1 ] (3-85) m=0 4JT " C m o ~~ (a+1-/) 2 ^ n v ' 7 m=0 ^ — 2 X [am(Am3 ~ A / m a ) + d m A / m J . ( 3 . 8 6 ) The pressures are obtained from the flow functions. For the case of indepen-dent airfoil and flap camber an elliptic function must be evaluated each time the pressure is evaluated. The procedures of Appendix B greatly facilitate this kind of calculation but the series presented in Whittaker and Watson (1927) could also be used. Other special functions dealing with other analytically specified boundary conditions could also be found. For example, the four and five digit series NACA cam-Genera/ Tandem Thin Airfoil Theory / 3.5 64 ber lines are specified with two polynomials in x, one applying over the forward segment of the airfoil element and the other over the back segment. This discretization of the bound-ary condition can also be handled with analytical functions but it gets so complicated that the general Laurent expansions of the previous section may be more efficient. If an analytical description of the flow fields around NACA type airfoil ele-ments were to be attempted, then the airfoil and flap segment selectors of the next chapter would be needed. These functions use logarithmic singularities of the type used in the one-element theory's ^-problem, Eq. 1.19, to obtain functions with imaginary parts zero everywhere on the airfoil and flap except over the desired segment of interest, where they are constant. Analytical expressions for the lifts and moments generated by the NACA camber lines are obtained in the next chapter using these functions. 3.5 T H I C K N E S S DISTRIBUTION DESIGN A N D O T H E R P R O B L E M S In the thickness design problem, desired pressure distributions are specified and the as-sociated thickness distributions then worked out. The functions that allow this are the same as the camber functions of the previous two sections, except that they are multiplied by ». This allows the real part of the functions on the airfoil and flap to be specified as the boundary condition. The functions of this chapter can be combined in other ways that allow the designer the versatility of solving mixed problems. For example, suppose an airfoil's shape is given and the designer wants a flap with a specified pressure distribution. The camber problem for this situation could be solved with the flow function: . (3.87) which uses the and (f planes of Figure 6. The first group of functions here provides a general camber line boundary condition for the airfoil and is purely imaginary on the flap. The real part of the second group of functions provides a general pressure distribution boundary condition for the flap and is purely real on the airfoil. General Tandem Thin Air/oil Theory / 3.6 65 3.6 M U N K ' S I N T E G R A L S E X T E N D E D In the same way that Munk's integrals were obtained from the general one-element theory camber function, Eqs. 1.33 and 1.34, one can use the general camber functions of section 3.3 to solve directly for the tandem airfoil forces in terms of the boundary conditions. The general camber functions' overall lifts and moments are given in terms of the first two or three coefficients from their Laurent expansions, Eqs. 3.41 it 3.42 and Eqs. 3.65 & 3.66. If, in these equations, the coefficients are replaced with their respective integral expressions, Eqs. 3.38 & 3.40 and Eqs. 3.58 U 3.59, then, after some simplification, one gets: C,= -4 s + 1 - / f x(/-x) ( 3 — x)(l - x) dx + fdy^ I x(x-f) J dx V (*-«)(!- dx (3.88) Ci 4 " 2 + (s+1-/) 2 'mo « / dyc. I x{f-x) dx V(a-x)(l-x) dx + J dx (x — s)(l — x) dx (3.89) By letting / —* 3, these expressions reduce in a very simple manner to Munk's original integrals. Obviously, if only lift and moment are desired, there is little reason to use the procedures of sections 3.3 or 3.4. Moreover, the simplicity of these expressions suggests that there must be some simpler way of deriving them — as, indeed, there is. This is presented in the next chapter along with efficient methods of evaluating the integrals. CHAPTER 4 Tandem Airfoil Forces an Easy Way At the end of the previous chapter, general integrals which give the overall lift and moment for any tandem airfoil camber line were presented. In this chapter, these integrals are rederived using a simpler, more direct method and the analogous integrals for localized lifts and moments are also obtained. Although all of these force integrals can be solved numerically, they can also be solved analytically by using selector functions. As an example, analytical expressions are found for the forces on tandem NACA airfoils. These thin airfoil theory predictions are then compared with exact-numerical potential flow theory predictions of the forces. 4.1 T H E O V E R A L L FORCES The general expression for overall lift is: C^j^-jfwMdz (4.1) c where w'c(z) is a general camber function that solves the a and v problems as well as problems involving conventional camber line shapes. The closed curve C in this integral can collapse onto the surfaces of the tandem slits since w'c(z) has no simple poles in the z-plane: 66 Tandem Airfoil Forces an Easy Way- / 4.1 67 However, on both the airfoil and flap, dye/dx is even in 7, Fig. 4d, and dz is odd in 7 so that the imaginary part of this integral is zero (i.e. the drag is zero, as proved in section 1.3). On the other hand, Cpdz is even in 7 which not surprisingly leads to the result: l a - " 2 s + 1 - / C„ dx (4.3) Thus, the problem with Equation 4.1 is that when C collapses onto the slit boundaries, the object of the integration becomes the unknown pressure distributions while the known boundary conditions are lost. This situation can be reversed in the following way. w'c(z) —• 0 as z —• 00: ^ ( 2 ) " - ^ - + -!+ - f + ' • • . a s z - 0 0 (4.4) z z so that: w'c(z) dz = 27rtiy1 . (4.5) c Now suppose there is an C(z) such that: j w'c(z)dz = j w'c(z)C(z)dz . (4.6) C C Then it must have the form: £ ( z ) ~ l + ^ + ^ + ^ + . . . , a s z - 0 0 (4.7) in order for the residue of the new integrand to be equal to wj, and since the u^'s are arbitrary while the £ n ' s will be fixed (zn terms, n > 0, are possible in this expansion of £(z) but, since the residue must remain equal to wlt the coefficients of zn would depend on the unknown tu„'s). Further, suppose £,(z) is purely imaginary on the slit surfaces and that w'c(z)C(z) has no simple poles in the z-plane. Then, the new integrand will collapse in a simple manner to the surfaces of the slits where the imaginary £(z) will combine with the imaginary part of w'c(z) to form the real part of the integrand, which will give the lift, while the imaginary part of the integrand containing the unknown pressure terms must integrate to zero. Tandem Airfoil Forces an Easy Way / 4.1 68 The conditions on £ ( z ) are very similar to those on the flow functions of the previous chapters: 1) the real part of C(z) is zero on the slit surfaces. 2) C{z) -* 1 as z -* oo 3) j£(z) must have its singularities at the trailing edges since the Kutta condition ensures that w'c(z) will not have any there. This prevents the singularities of L(z) from possibly combining with those of w'c(z) to form simple poles in the z-plane. Since the conditions on C[z) are as strong as those on the flow functions, it will be unique: m = and: r . I z(z - f) J . , v J > dz. (4.9) (4.8) ( z - s ) ( z - l ) Although it is not a requirement, it is not surprising that Z(z) is odd in 7 on the slit surfaces as this ensures that the imaginary part of Equation 4.9 is zero. On the slit surfaces, then, Equation 4.9 reduces to: -4 .+1-/ ( s - x ) ( l - x ) *</-> dx + j ^ . « < « - / ) .dx (4.10) 0 ' / which is identical to Equation 3.88. For the overall moment, let: j zw'c{z) dz = j zw'c(z)C'(z) dz . C C In order for the result of these integrals to be 2Kiw2, t'(z) must be of the form: (4.11) c c c Also, if L'{z) is linear in £ ( z ) : as z —• 00. L\z) = g{z)C{z) (4.12) (4.13) Tandem Airfoil Forces an Easy Way / 4.1 69 so that the boundary condition is selected when C collapses to the slit surfaces, then g(z) must be purely real on the slit surfaces. In order for £ \ to be zero: g { z ) ~ l - S + * ~ f + o(j£j , as 2 - o o . (4.14) Since any higher order terms in this expansion for g(z) create singularities which will either create unwanted simple poles in the z-plane or change the effect of £(z) on the integrand, it must be truncated after the second term. Thus: Ct 2 / , , , , , , / » ( « - / ) • ~T (7+wFJ"kW*V(.-.)(»-i)'b (4.15) and on the slit surfaces: _ Ct 4  m o ~ 2 + (s+1-/) 2 Mft . / x ( / - x ~ / x(x-/) ; rfx X]J ( S - x ) ( l - x ) a X + J dx I Y ( x - 3 ) ( l - . dx - x ) (4.16) which is identical to Equation 3.89. The imaginary part of the integrand again integrates to zero since it is odd in 7. Note that for in-line airfoil elements, the imaginary part of Equation 1.12 is also zero since w'(z) can be replaced with V J ' C ( Z ) and (dyc/dx)dz is odd across the slit surfaces. These integrals for lift and moment in terms of the boundary condition, Eqs. 4.10 and 4.16, can now be evaluated numerically using appropriate methods to take account of the integrable singularities at the trailing edges of the airfoil elements. If the airfoil elements' boundaries are specified numerically, then this is probably the easiest way to evaluate the camber line contributions to the forces, A C , and A C m o . However, if the boundary condition is specified analytically, then it may be simplest to carry out the integration analytically, as done in the remainder of this section. If dyjdx is given as an analytic function or as a group of analytic functions, then it is convenient to find a complex function, -y^(z), which is single-valued in the flow dx Tandem Airfoil Forces an Easy Way / 4.1 70 field and which: 1) on the slit surfaces has its real part even in 7 and equal to dyc/dx and has its imaginary part (if it is nonzero) odd in 7. 2) does not create any simple poles in the z-plane when combined with £(z) in the product ^ ( z ) £ ( z ) . dx Then: C t = - J L ; / fo.)./ Z i \ : f ) „ d z (4.17) 1 s+l-fj dxK ' y ( z - s ) ( z - l ) v 1 c Ct 2i fdyCf. I z(z - /) J c " ° - f - t j + w ? t li{z)2V ( « - . ) ( . - 1 ) i z • t 4 - 1 8 ) C Because of (1), these expressions are purely real and, because of (2), the curve C can col-lapse unhindered to the slit surfaces where the expressions are equivalent to Equations 4.9 and 4.15. Note that the conditions on -y^(z), while similar to those on u>c(z), are not d as strong. These conditions do not define a unique -r^(z) since there is no requirement dx ^ for it to meet as z —• 00. Of course, one obvious representation for -r^-(z) is to simply ax set it equal to iw'c(z), but this is a rather trivial use of the added flexibility the function provides. Consider the following examples. Suppose: ^Y± = a0 + alx + a2x2 + a3x3 + •••; 0 < x < s, / < x < 1 (4.19) dx then: ^ ( z ) = a0 + axz + a 2z 2 + a 3z 3 + • • • . (4.20) dx This allows the simple evaluation of Equations 4.17 and 4.18 by residues. A recurrence relation for the £„'s of Equation 4.7 is given in Appendix C, so that: Ct = s~**_f [apili + a, r2 + a 2 l 3 + • • •] (4.21) Tandem Airfoil Forces an Easy Way / 4.1 71 When o0 = - a , these last two equations solve the a-problem. Although they are similar to the special function lift and moment results, Eqs. 3.75 and 3.76, they are simpler to evaluate. Now suppose the airfoil and flap have their own independent camber line representations: ^ = aQ + axx + a2x2 + • • • , 0 < x < s dyct , (4.23) T-?- — d0 + diX + d2x -\ , / < x < 1 . dx Then: ^ ( z ) = (ao + aiZ + a^2+ •••)[!-?(u)] + {dQ + diz + d2z2+ ---)T{u) (4.24) where [l—7{UJ)] and 7(u) are airfoil and flap selectors with a square-root pole at the flap leading edge in the z-plane so they will not combine with £(z) to form any simple poles: J(w) = h — -^(w) = 0 +1 odd(7) on the airfoil * K * 2 (4.25) = l + i odd(7) on the flap. 7 7 7 7(UJ) ~ 7Q + — + -^ + ~ + --- , as z oo. (4.26) Also: z z 2 z 3 Note that 7(u) could have been constructed from the i?4.(w) function which has its zero at the airfoil leading edge. The process of evaluating overall lift and moment using the residue theorem is now straightforward. A recurrence relation for the flap selector 7N'8 is given in Appendix C. The lift is: C< = - ^ j ^ { « o [ - * + ^ ( l - ^ + d0[7i + £J0]+dl[72 + f l J l + C270] + --- } (4.27) which reduces to Equation 4.21 when the an's equal the dn's. When d0 = —r), Equation 4.27 Tandem Airfoil Forces an Easy Way / 4.2 72 solves the rj-problem. A more general version of this equation and the results for overall moment are given in section 4.3, in Equations 4.60 and 4.61. 4.2 T H E L O C A L I Z E D FORCES The airfoil lift is: Ctt = --sjw'c(z)dz. C Cauchy's theorem enabled the derivation of Equation 2.41 which is in terms of the airfoil selector A(u) (Eq. 2.38). It is convenient to replace F(z) in this equation with w'c(z) so that: j w'c(z)A{u)dz = j w'c(z)dz + j w'c[z)[A{u)-l]dz + j' vj'c(z)A{u)dz. (4.28) C C. C, Cf However, as z —• oo: ^ ( w ) ~ l - E 4 + o Q and V J ' C ( Z ) is O(lfz) so that, using the residue theorem: j w'c(z)A(u) d z = ( l - E 4) j w'c(z) dz (4.29) The airfoil lift, then, is: Ctt - £ 4 ) Q + l j w'c{z) [A{u)-l] dz+-sj w'c{z)A{u>) dz . (4.30) C, Cj If these integrals are evaluated on the surfaces of the slits they enclose, where [A(u>) — l] and A(OJ) are purely imaginary and odd in 7, then the boundary conditions are selected for the real parts of the integrations while the imaginary parts containing the unknown pressure terms integrate to zero: s+l-f, „ ^ . ^ ITT 0 < . 1 < y 0 / (4.31) Tandem Airfoil Forces an Easy Way / 4.2 73 Here, the choice has been made to integrate over the upper surfaces of the slits. This expression can be evaluated in this form if necessary but it is simpler to integrate by parts. The first integral is: / % LW , 'Tf)- 1 ] d * = »e.(*)W»l)-l] ~ / yci')^{il)-l]dx (4.32) o 0 0 where: * yCt(x) = j-^dx. (4.33) Since the simple zero in yc,(x) at z = s overpowers the square-root pole in A(v) at z = s and since [A{u)-l] \sO{zll2) at z = 0, the first term on the RHS of Equation 4.32 is zero. Also, using the relations from section C.2 (with G2 and Gj defined in Equations 2.37): G -G z(l~s) < 1 H i 2 ~ '/(*-*) ( 4 . 3 4 ) d z ' 2 y/z(z - s)(z - f)(z - 1) where, on the slit surfaces: y/z(z - s)(z - f)[z - 1) = ±i|x(x - s)(x - /)(z - 1)|* . (4.35) The following diagram shows which sign to use on which surface of which airfoil element: 0 For the second integral, it is convenient that A(u>) has square-root zeros at both z = / and z — 1 since this then allows yCj (x) to also be defined with a simple zero at z = s: dyc (4.36) Later, this consistency of definition (yCj{s) = 0, j = s,f) will facilitate the derivation of complex versions of the following two equations (note the singularities at x = s in the Tandem Airfoil Forces an Easy Way / 4.2 74 —A(u) terms of all the integrands): S 3J \/XlS — X)(f — X](l-i G - G x(1~3) Replacing tu£(z) with zw'c(z) in Equation 4.28 gives: c~* - f ^ ) 2 (i - * ) + ^  [G2 - 0,1=1] c£ . * / y - l x V x ( a - x ) ( / - x ) ( i - x ) d I + 5 2 ; c / C V « ( * - * ) ( « - / ) ( i - « ) (4.38) : .(x) = y x ^ L , j = S > / . (4.39) where: * . Equations 4 .37 and 4.38 are the localized force integrals analogous to the extended Munk integrals for the overall forces, Eqs. 3.88 and 3 .89 . They reduce to give the localized forces on a one-element airfoil as follows: 2 1 0 G i —• 0 , G2 -* -V / ( l - / ) , E 4 -> - c o s _ 1 ( 2 / - l ) = -i- ; as f-s — 0 (4.40) where 0j is as shown in Figure 3b. Therefore, as f—& —• 0: ^^V-TjJT + ^ — J + Jjd — J — , - , ^ . (4.42) These last two equations were also derived independently of Equations 4.37 and 4.38 using only elementary functions. Tandem Airfoil Forces an Easy Way / 4.2 75 Now, if dyjdx is given by Equations 4.23 (independent airfoil and flap cam-ber), then it is convenient to construct the complex functions: yc(z) = (a0(z - s) + ax + (d0{z -s)+dl Yc(z)= (oo z2-s2 2 z2 — s' + • i - + z2-s2 + (do 2 z2 - s'4 + « i — z — + + di Z * - 8 3 + ••)[i-n»)] (4.43) (4.44) Then: 't. ( G — G z(l~s) ^(i-s^+i(/»,(.) . . 'W 1V. 8 3 ^ > / z ( z - s ) ( ; Z - / ) ( * - ! ) — 2ir» (residue of integrand at a = /) (4.45) and: z(l-s) . ^ — 2;r»(residue at z = /) (4.46) which can be evaluated by residues. Note that since 7{u) i s O f t z - s ) 1 / 2 ) as z —* 8, the ?(U)-J-A(<JJ) product in these integrals has a term 0((z — s) - 1 ) , a simple pole which is dz eliminated by the zeros in the yCj(x) and Yc.(x) functions. However, the 7(u)/\/z — f quotient does produce a simple pole which contributes a residue to the contour integrals that must be subtracted. In the next section, a more complicated analytically specified boundary con-dition problem is solved. The results, Eqs. 4.69 and 4.70, reduce in a simple manner to give the results for the above expressions for Cit and CmtQ by choosing appropriate values for the polynomial coefficients. Tandem Airfoil Forces an Easy Way / 4.3 76 4.3 S E G M E N T E D BOUNDARIES: T H E S E G M E N T S E L E C T O R S Suppose both the airfoil and flap are composed of two segments, each of which has its own independent boundary condition: dy, dya i — = a0 + axx + o 2 i + ax d* )*pL = b Q + b l X + b2X2 + . dx Ct dyc dyd dx = do + dxx + d2x2 + dx I dy, o 1 -p = eo + eix + e2* + ax 0 < x < xx X\ < x < s f < x < x2 x2 < x < 1 (4.47) so that 0 < Xy < s and / < x 2 < 1. Then, in order to construct the associated complex function —(2), both airfoil and flap segment selectors are required. dx The airfoil segment selector is based on the one-element theory's r/-problem solution, w'n(z) (Eq. 1.19). Using the f-plane of Figure 4f and if ft = e , 9 i and ft correspond to z = Xj, then: 2ir f Xi < X < s - l - H o d d W for V | f I = 1 . (4.48) This function has an unwanted effect on the flap, |f| = R, which is negated by using Milne-Thomson's circle theorem to add a series of image singularities. This leads to the elliptic function: i f i ( f - f t ) a ( y - f i ) \ ^ - f t ) a ( y - f i ) 2 - - -2TT ft(<-fi)2(f-ft) (*f-fOa(f-ft) • . c? - ft) n [Rin -R2n &+7)+*] (4.49) T 1 1 1 (ft) (; - a) n [^ *" - - R 2 ' (^+7).+ i] Tandem Airfoil Forces an Easy Way / 4.3 77 Transforming to the w'-plane, Fig. 4e, using f = e 2 , w ' , R = l/q', and ft = e~2* x«, this can be rewritten as: i l n ^ - = V f r x K i ( « 0 ) * s i n ( . ' + y I ) n[»-^"-2 ( . ' + x ' 1 ) + ^ ] * ' l ( " + x l l r t which is periodic in ir in the w'-plane as required. Whittaker and Watson (1927, 21.3) gives the infinite products representation for t?j(w). The real part of Equation 4.50 still has the characteristics of the real part of Equation 4.48 on the airfoil. On the flap, however, its real part, while constant, is not equal to zero as required. This can be rectified by multiplying the flap selector, by the appropriate constant and then subtracting it from Equation 4.50. As well, it is convenient to convert to the w-plane, Fig. 4d, so that the airfoil segment selector is: * M " r t a * J ( = + ^ ) - - r ^ M - ( 4 - 5 1 ) If 0 < t'7x < ^ , then: S(u) = 0 + todd(7) , 0 < x < z i = 1 + iodd(7) , z, < x < s (4.52) = 0 + iodd(7) , / < z < l . The expansion coefficients for S (u/) are derived in Appendix C: c c S(w) ~ $o + — ' + - £ + •• • , asz->oo. (4.53) z z* 1?' Note that, since the -—-(UJ) function in Equation 4.51 is purely imaginary on both the airfoil and flap, the way it complements the logarithm function must be different from the way 7(u) was used to complement the logarithm function of Equation 4.50. When the latter combination was converted to Equation 4.51, terms linear in u/ from each of Equation 4.50 and /(w) cancelled. As a result, the logarithm function of Equation 4.51 has ail the required boundary condition characteristics on the airfoil and flap (Eq. 4.52) Tandem Aittoil Forces an Easy Way / 4.3 78 0' but not the required periodicity in JIT in the w-plane. This is corrected by the — (w) 1*2 function so that 5(w) is periodic in jrr. The flap segment selector can now be obtained directly from by replac-ing w-i'7! with u - (f-»72) a n ( * w+*7i with w - (f+»7 2). This preserves the relative orientation of the vectors w—1'71 and w-M'71 in the logarithm function. As decribed above, #7 0' the constant in front of -^(w) is chosen to ensure that T(u) is periodic in JTT: 7 » = - l n 2 1 2 +-22-i(c) . (4.54) JT t/2 ( W — I 7 2 J JT t/2 K 0 < »72 < ^ , then: T(w) = 0 + »odd(7) , 0 < z < s = 1 + *odd(7) , f < x < x 2 (4.55) = 0 + iodd(7) , x 2 < x < l . The Tn's are derived in Appendix C: T T 7(w) ~ 7 J + — + - f + --• , a s z - » c o . (4.56) 5(w) and T(w) are similar to elliptic integrals of the third kind (Whit-taker and Watson, 1927, 22.74). The -^(w) function, with its square-root pole at z — f, v2 was chosen for the selector functions since they will be used to construct a -j^-(z) function dx which should not combine with £(z) , in Eqs. 4.17 and 4.18, to form simple poles at z = s or z = 1. S(<J) and T(w), then, will satisfy the Kutta condition requirement on flow func-tions and so can be used to construct A m (z) functions, section 3.4, which could provide a closed form analytical description of the flow field for the segmented boundary problem and permit the calculation of pressure distributions therefrom. Now, the overall lift and moment for the segmented boundary problem are given by Equations 4.17 and 4.18 with: ^ ( z ) = {aQ + a l Z + a2z2+ • • • ) [ ! - T { O J ) - S { U ) ] ax + (60 + 61z + 6 2z 2+ •••)$(") + (dQ + dxz + d2z2 + ---)T(u) + (eQ + elz + e2z2 + ---)[?(u)-T{u)] . (4.57) Tandem Airfoil Forces an Easy Way / 4.3 79 In order to solve for the lift and moment using the residue theorem, the following expansions are needed, all as z —• oo: C7(z) = C(z)7(u>) ~ t 70 + ^ + ^ + • • • z z1, CS(z) = C(z)S(u>) ~ £S0 + — + ^ + • • • (4.58) z z JLT(Z) = £ ( * ) 7 » ~ £T0 + Q + ^  + • • • z zc where: CM0 = £0Mo\ £Mi = £0Mi + CiMo; CM2 = £0M2 +11^ +£2MQetc. (4.59) Then: —4TT f ^ = 7 H Z / I ao ( A - ^ i - W W i W * 2 - £ s 2 ) + ---+ bQCSi + b1CS2 + -- + dQ£Ti + &XLT2 + • • • + eQ{£7l-llTl) + el(L72-LT2) + --- } (4.60) and: C 4JT ( cm 0 = y + ( a + 1_ / ) 2{ M£2-£?2-£S2) + al(£3-C73-£S3) + ••• + 6 0 £ $ 2 + 6 X £ 5 3 + • • • + rf0£T2 + d^Ta + • • • + e0(£72-£T2)+el{C73-£T3) + --- }. (4.61) As usual, the localized forces are more difficult. The segmented boundary conditions, Eqs. 4.47, can be used directly in the expression for C(l} Eq. 4.31, but not in the integrated by parts version of that expression, Eq. 4.37, since yc(x) is no longer continuous over an airfoil element. If, for j = a, 6, d, and e: X X V j { x ) = J^dx and Yj(x) = f ^ d x (4.62) * • then the first integral of Equation 4.31 becomes: 0 0 *i with: x *i G G X ' Tandem Airfoil Forces an Easy Way / 4.3 8 0 since at z — 0. Hence, the segmented boundary condition version of Equation 4.37 is: 3 + 1 - / (l-^)Ce + j{[ya{xl)-yb(xl)][A{uJl)-l] + + Go — G i dx + G 2 - G 1/(*-•) V * ( « - * ) ( / - * ) ( i - « ) " 1 *y w v^-x)( / -x)( i -x) 0 * i ^ n xjl-s) j ~ ~ x(l-3) dx J G 2 - G, * 7 » G 2 - G , - - / &(*) ^ — u , * >da:~T / y e ( x ) A - ? — f r sy */x(x-s)(x-/)(l-x) 3 ^ T / X ( X - S ) ( : I x-/)(l-x) where and w 2 are on the upper surfaces of the airfoil and flap. However, if: dx (4.63) yc{z) 5 y . W [ l - 7 ( u ) - 5 ( « ) ] + y 6 ( « ) 5 (w) + y<,(*)T(w) + y.W[7(w)-T(w)] (4.64) then the integrals of Equation 4.63 become: z(l-s) t - < 3 Go — G i y yc(z) —7= w / ^ w ^ = ^ — 2TTI (residue of integrand at z = /) •. (4.65) Now, if: G i " G l / ( . - . ) as 2 —> oo then: 1 - 3 G.s n , n > 1 and it is convenient to define the »7m's and, for z —* oo, the Jmn's: 4(z)= £ ( * ) $ ( * ) ~ ? J2(z) = LT{z)Q(z) ~ 1 /3(z) = LZS{z)9(z) ~ 1 Z Mz) = rrwfl*) ~ 1 Z Z ^ 7 i ^ 4. ^ 2 , 3^0 + — + - y + z z* 7 J . ^ 1 X ^ ± JiO H 1 T + (4.66) (4.67) (4.68) Tandem Airfoil Forces an Easy Way / 4.3 81 The recurrence relation for the Jmn's in terms of the previously defined expansion coeffi-cients is similar to Equations 4.59. Thus, the airfoil lift is: + ^{[y«(*i)-y6(*i)][«^(wi)-i] + [ydM-yeM]iAM} - — | ao(Vio_^20-«^3o) + y(^n-^2i-^3i) + y(^i2 _^22-^32) H u r 1^ T 2^ T j w di _ (^ 2 T (4.69) + O(W30 + Y 31 + "3" 3 2 "* *" 0 4 0 + "2" + If ^ + go(^20-^4o) + y(^21 _^4l) + y(^22_^42) H + T " ( T " ' T - ) - [*(/)-*(/)]£- [»(/>-*(/)]£]}. Following similar procedures for the airfoil moment, one gets: 2 G 1 - 3 G i ~ T (4.70) - ^{[y.CxO-nMIWCwi)-!] + [Yd{x2)-Ye(x2)]iAM) + ^ 2 ~ | y ( ^ i i - ^ 2 i - ^ 3 i ) + j t ^ - ^ 2 2 - ^ 3 2 ) + - ^ ( ^ 1 3 - ^ 2 3 - ^ 3 3 ) + 4. b° 7 4. 61 7 4. 6 2 7 j . 4- d° 7 4. rfl 7 4. ^ 2 7 4. + y ^ l + + ~£JM • r y v / 4 1 + — */42 + "^ -'43 H + y(^2i-^4i) + - ^ ( ^ 2 2 - ^ 4 2 ) + - ^ ( ^ 2 3 - ^ 4 3 ) H + ir ( T - " T " ) M-W) - fc(/)-n(/)]£ - [W)-W)]£]} where, in these last two equations, ut = 171., w 2 = § +1'72, (and from Fig. 4c) Ax = iuu and A2 = i f + tV2 are all on the upper surfaces of the airfoil and flap. It is important to appreciate the generality and versatility of these last two equations and the analogous expressions for overall lift and moment, Eqs. 4.60 and 4.61. They contain the force solutions for all the analytical boundary conditions discussed so far. The a and i] problems are solved by simply choosing the appropriate values for the polynomial coefficients in the boundary condition, Eq. 4.47. If o„ = bn, dn = e„, or if an = bn = dn = en, then the force expressions simplify in a straightforward way. These Tandem Airfoil Forces an Easy Way / 4.3 82 coefficients are the "geometric amplitude parameters" mentioned in Chapter 1 and the coefficients from the selector function expansions, the "force solution constants", measure their relative effects on the forces. These expressions are exact analytical solutions to the linearized boundary condition problem. Note that the segment selector functions allow an airfoil element to be divided into any number of discrete segments. They allow, for example, the solution of a problem where both the airfoil and flap have attached flaps and/or slats at arbitrary deflections. Or, these selector functions could be used to describe numerically specified camber lines by fitting n t h order polynomials to any number of segments, each containing n + 1 camber line coordinates. In this case, a closed form analytical solution to a numerically specified problem would be achieved. It is questionable, however, as to whether the numerical problem is best handled by selector functions or by simply numerically integrating the force integrals of the previous section. It is convenient to have the a and n problem solutions in terms of the selector function expansion coefficients, as provided on the next page in Equations 4 . 7 1 . All these equations have been shown to be identical to those derived in Chapter 2 , Eqs. 2 . 4 7 . Rather than calculate leading edge moment variations with a and n, it is probably more meaningful to calculate aerodynamic centers and their associated pitching moments. If: dC defines xjaJcj which is associated with xac/(s+l—f), xSac/s} and (x/a c—/)/(l — f), then the aerodynamic center characteristics are as shown in Equations 4 . 7 3 . A C refers to the shift in a coefficient's value at a = n = 0° due to the presence of a camber line. In the next section, all of the above aerodynamic characteristics for tandem NACA airfoils are evaluated and the results compared with exact-numerical calculations. Tandem Airfoil Forces an Easy Way / 4.3 83 a) b) c) d) e) *) g) h) j) 1) dCt da dCt dr, dCt, da An s+l-f Air S+l-f S+l-f Hi s oa s da dr, LL -da da 3Ctf dr, da i _ s or, s s+l-f dCt s dCla 1-f da 1-f da s+l-f dCt s dCe> 2K'f2K' 2 E ' \ , f : da "»0  1-f dr, 1-f dr, Alt 7T = j r - ( s + 1 _ / ) 2 ^ = - 2 (4.71) 1 + 4 ( / - S ) ( l - / ) {s+l-f? da mo ldC, Air dr, 2 dr, {s+l-f)*Zl2 0 ^ ^ ^ M , da da mp S + l-f da m * 0 dr, (1 - E 4 ) 2s2 s+1-. _ 1 — 3 ^ G 2  2JT JH ~da ~ a2 ~2~ / \ r i - v 1 a c< - [G2 - — — dC„ 7o da dC, mfo _ dr, 2TT |V 2 1 2K' f2K' _ 2E'\ 3 2 [ 2 ir \ ir ir J = (S+1~f)2 dgmo - (-^y) 2 d C m $ 0 ' f d C t j ( W ) 2 ^ f - f e ) / 2 - 3 2 1-f da 1-f dr, da X N R . mo da »ac l™± . xfac-f = dCmfo /dCtj / da * 1 - / da / da dC, ac dr, da dCm- dC.x-mJO _j_ c j Jac dr, dr, A C m • = ACL,- + A C / . m3ac mJo ^  ?j (4.73) Tandem Airfoil Forces an Easy Way / 4.4 84 4.4 T A N D E M N A C A AIRFOILS, A N E X A M P L E The NACA 4 and 5 digit airfoils have camber lines (Abbott and von Doenhoff, 1959, Chapter 6) that are described by two different polynomials: ^ = a'0 + oil* + o'2x'2 , 0 < x' < x\ d x (4.74) = 6Q + b\x' , xi < x' < 1 . Although it is not a requirement of thin airfoil theory, these polynomials are matched so that the magnitude of dy'c/dx' is continuous through x' = x\. The coefficients in the polynomial boundary conditions (Eqs. 4.47) for tandem NACA airfoils become: a 0 = a'Q , O i = o'./s , a 2 = a 2/s 2 ; bQ = b'Q , 6i = b\/s A a' f a' + ( f \ \ ' A - 1 ,' 2 f ' ,' A - 1 J d0 = a 0 - YZ7al + \JZ~f) °2 » dl ~ Y—jal ~ (1=7)202 ' d2 ~ (IZ7)2a2 eo = 6d - jzy^i ' c i = JZ/ 6 ' 1 X i = a x i , x2 = / + ( l - / ) x i (4.75) These coefficients illustrate what could become a serious problem when they are used in the expressions for the aerodynamic forces of the previous section. That is, when the slot is large or one of the airfoil element chord lengths is small, the coefficients become large. If cy is the smallest of s and 1 — / , then the largest coefficient in Equation 4.75 is 0(Vcymln)* Thus, if £ is the error in the selector function expansion coefficients, then the maximum error in an aerodynamic coefficient will be: •(f) (4.76) since the pitching moment coefficients are nondimensionalized by Cy. Further, for the NACA 23012 airfoil, A C m o c is O( l0 - 2 ) and so the percentage error in its value would be: \ J m i n / It is therefore important to keep £ as small as possible. Tandem Airfoil Forces an Easy Way / 4.4 85 The segment selector expansion coefficients, Sn and T„, are different from the other expansion coefficients, An and 7n (which solve the a and n problems in Chapter 2), in requiring the evaluation of elliptic functions and integrals with imaginary arguments (such as when ui = = The procedures of section B.2, which were sufficiently accurate when evaluating elliptic functions with real arguments, lose accuracy when the arguments are imaginary. For example, the first truncated term in the expression for X)3, Eqs. 2.48k and B26, is: 0 ( g 1 6 s i n l 6 X o o ) - 0 ( g 1 6 ) . Replacing Xoo :'7i makes this error: and if x± is close to zero, ifi will be close to and the error will be 0(q&). This loss of accuracy, combined with the previously mentioned need for greater accuracy, motivated the development of the highly accurate procedures in section B.3 for calculating Wj, u2, ^(wi)> ^( w2)» SO^oo)* a n c * 7"( woo) 3 8 w e ^ 3 8 * n e usual constants q, Glt G 2 , and E 3 . The following is a step-by-step procedure by which the aerodynamic coeffi-cients of the previous section can be evaluated for polynomial boundary conditions of the type given in Equation 4.47. At the end of each step, the order of magnitude of the error in that step, £,, is noted. Given s and / , calculate: 1) k and e0 from Eqs. 2.48a &b. £i = 0 2) /tj and q from Eqs. B27,28,<k29, using only the first two terms of Eq. B28. 2K 2K1 2E1 3) — , , , and then G x and G 2 using Eqs. 2.48d, e, f, and i & j . £z = 0(Z2qZ2) IT IT IT 4) 2cXoo, dn(/iO 0 2, k2), and Xoo fr°m Eq 8- B23,30,31,32,«k33, using only the first two terms of Eq. B32. £ 4 = 0(qZ2) 5) - ^ ( 2 X 0 0 , q % - r-(Xoo), a n d E 3 u s i n & E c l s - B37,38,&26. Only the term 0(qiC) from v& t / 3 the series in Eq. B38 is included. £h = 0(l6q32) Tan Jem Airfoil Forces an Easy Way / 4.4 86 6) eWl, eW2, dn(A l 2,A:2), dn(A22,fc2), (2M, 7i,and 72 from Eqs. B35,30,31,32, &36, using only the first two terms of Eq. B36. If £ 2 m a x is the maximum of £2(wi) and £ 2 (w 2 ) , each of which is always positive, then £ 6 = <2(32£ 2 m i x <j 3 2 ) 7) i3-(2w1,g4) and -r3-(2w2,g4) from Eqs. B38,39, & 25, truncating the series in Eq. B38 after the 0(<?16) term. Then, Eqs. B15 are used to get the expression: . , 2w K ' # l f , , 2w JiT'i?,, , 2tf' cnA „ M . , _ _ _ _ _ i M . , _ _ _ _ j M _ _ _ ( 4 . 7 7 ) and using Eq. B25, one can show that — l] and i>t(w2) are equal to: 2Ji £ _ _ G / _ x . 5 V 1 _ xj + \]~7u~xj K' OL 1/Zy + V - ^ l 1 ~> xj) K ^ 3 3 — Xj A , ' " (1-3)(1-/) where; = 1,2. £7 = 0(l6^mixq32) 8) 5 0 and T0 from Eqs. C8 & 16 and B40&41. £ s = 0(^|m i xg3 6) 9) 5 X and Tx from Eqs. C12& 17. £ 9 = 0 ( l 6 £ 2 m a x g 3 2 ) 10) -Cm $nt and 7J, for n = 0 to 4 for the NACA camber lines, using the recurrence relations of Appendix C. £ l 0 = 0 11) £7n, LSni and LTn from Eqs. 4.58&59. Jmn from Eqs. 4.67&63. m = 1 to 4 and n — 0 to 3 for the NACA camber lines. £u = 0 12) all the aerodynamic characteristics from Eqs. 4.60,61,69,70,71, & 73. £ 1 2 = 0 13) The maximum error in the above quantities will be: (4.79) where p is the order of the highest order polynomial in the boundary condition. Of course, this does not include the roundoff error associated with these computa-tions, determined by the number of digits the computer or calculator uses. The above procedure has been programmed into Texas Instrument's Com-pact Computer 40 (CC-40), a textbook size portable microcomputer (34K of ROM and Tandem Airfoil Forces an Easy Way / 4.4 87 6K of RAM) using TPs enhanced basic language. Calculations use a minimum of 13 digits per number. The program, which is listed in Appendix D, utilizes most of the available RAM memory and takes a little over half a minute to solve for all the aerodynamic char-acteristics. As a comparison, the procedures of Chapter 2, which solve only the a and TJ problems, use about IK of memory when programmed into this computer and are evalu-ated in only 6 or 7 seconds. Thus, in order to have the additional capability of calculating A C ^ and A C m j a c , all of which are relatively small forces in most situations, about five times as much memory and computation time are needed. The thin airfoil theory predictions for the variation of the aerodynamic char-acteristics with geometry for tandem NACA 23012 airfoils are presented in the next six figures. The figures also make comparisons with exact-numerical potential flow theory (Appendix A). For the numerical theory, the variations of the coefficients with a and rj are approximated by assuming that they are constant over a 1° interval, beginning at a = T) = 0°. Also for the numerical theory, the aerodynamic center functions are calcu-lated using the exact version of Equation 4.72: % ' Cmjac = Cmj0 + ( Q y C O S <* + Cdj « H <*) (4.80) where, again, dCmj /da = 0 defines X y /cy. The exact-numerical predictions of the aero-dynamic centers, Figure 13, are calculated in this way. For a proper comparison of Cm • in Figures 14 and 15, the exact-numerical predictions of Equation 4.80 are calculated using thin airfoil theory values for x- . Thus, the figures compare moment predictions about lac the same point on the airfoil elements. In all cases, the error in the numerical calculations is estimated to be within the size of the symbol representing the point on the graph. In some instances, this required representing each airfoil element by as many as 100 straight line segments — as described in Appendix A. Parts (a) of Figures 10 to 15 show how the aerodynamic characteristics vary with slot size, with the flap chord to total chord ratio (Rf) held constant at a typical value Tandem Airfoil Forces an Easy Way / 4.4 88 of 23%. The thin airfoil theory predictions, as indicated by the solid line curves, vary monotonically (except for 4 localized prediction curves) between the one-element theory's predictions at f — s = 0 (an airfoil with a simple flap) and f-s = 1 (two airfoils infinitely far apart). The small horizontal arrows in the figures indicate the predictions of the one-element theory. Of course, all of the integral expressions for the tandem thin airfoil theory force characteristics have been shown to reduce to the one-element predictions at f—s — 0 and, with the TI CC-40 able to accurately calculate values for the tandem airfoil predictions at f—s values as small as 10~6 where the predictions are virtually identical to those of the one-element theory, there is now the necessary confirmation that the analytical solutions to the force integrals are also behaving properly at this important limit. For f — s values approaching 1, the calculations are limited in their accuracy by the error £ m a x , Eq. 4.79. For Figures 12 and 15, p = 2 and the calculations are only accurate up to / — s « 0.93, although there is little doubt that the curves are approaching the one-element prediction at f — s = 1. For the other figures, the a and rj problem results, | ^ 2 m a x l — P = ®> an<^ roundoff error limits the accuracy of the calculations before £ m a x does. As a final indication that the present theory is indeed a correct representation of the flow about tandem thin airfoils, the exact-numerical calculations were also performed for 1% thick airfoils (23001 airfoils) and in all cases the numerical predictions collapsed to the linearized predictions (approximately 90% closer than the 23012 numerical predictions). This also happened when these "thin" exact-numerical calculations were performed for the curves in figures (b). Parts (b) of Figures 10 to 15 show the variations of the same aerodynamic characteristics with flap chord to total chord ratio, while keeping the ratio K'/K constant at 1.3. This is accomplished by simply keeping the modulus k constant: This value was chosen from figures (a), where f — s = 0.05 is a slot size sufficiently far from the one-element extreme at f—s = 0 to make the comparisons worthwhile yet not so Tandem Airfoil Forces an Easy Way / 4.4 F I G U R E 10 Lift curve slope characteristics of tandem NACA 23012 airfoils. linearized theory; •, o, • exact-numerical theory; j-* one-element TAT prediction for jth airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 F I G U R E 11 Flap defection effects on lift for tandem NACA 23012 airfoils. linearized theory; •, o, • exact-numerical theory; j-+ one-element TAT prediction for jth airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 0.16 0.14 -0.12 0.08 0.06 0.04 a) ACt. vs. f-s; Rf = (l - / )/ (a+l-/) = 0.2308 0.16 0.12 ' — -0.08 0.06 0.04 1 1 i i i 0 6 i _ _ _ J - o -o ° tc,. ^ \ • • ' o I „ o \oo ° m m m . o \ • * V J * • \ • \ • Aq • o \ 0 \ - • o 1 -• **/ o • 7 i i i i i • i i (> D 0.2 M R, 0.6 0.8 1.0 b) ACtj vs. Rf] f—s = 0.05, K' = l.ZK -0.2 AC. -0.6 -0.8 F I G U R E 12 Camber line lift characteristics of tandem NACA 23012 airfoils. linearized theory; •, o, • exact-numerical theory; j—* one-element TAT prediction f o r ; t h airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 F I G U R E 13 The aerodynamic centers of tandem NACA 23012 airfoils. linearized theory; °, • exact-numericai theory; j-* one-element TAT prediction for 3 t h airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 93 F I G U R E 14 Flap deflection effects on moment for tandem NACA 23012 airfoils. linearized theory; •, °> • exact-numerical theory; j—* one-element TAT prediction for ;th airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 94 4.02 -0.01 1.01' I-1 1 - i 1 — — i r i - 1 1 — 9 S o • I / O If* o • • • a • • • *> 3 1 1 1 1 1 1 1 • i -0 0.2 0.4 0.6 / • J 0.8 1.0 vs.f-3; = (!-/)/(,+ !-/) = 0.2308 - 1 . / •<f r- \© * * S -0.02 AC, mlcc 0.01 0.03 _1 l _ © i • oc « r-«« 2 « n — o o o o o b o 6 o o o _ J I • I b) A C m y a c vs . RF; / - s = 0 .05, K' = 1.3iT F I G U R E 15 Camber line moment characteristics of tandem NACA 23012 airfoils. linearized theory; •, o, • exact-numerical theory; j—* one-element TAT prediction for j1^ airfoil element. Tandem Airfoil Forces an Easy Way / 4.4 95 large that it is too far from typical slot sizes for slotted slat or flap airfoils. The functions shown in parts (a) and (b) of the figures, then, have common values at f — s = 0.05 and Rf = 0.2308. Note that for figures (b), f-s is minimum at Rf = 0 and Rf = 1 (where it is zero) and maximum at Rf = 0.5. Figure 15b includes a scale showing the variation in f-s over the range of Rf values. Where appropriate, figures (b) also include the horizontal arrows indicating the one-element theory's prediction of the limit point at Rf = 0 or 1. When calculating the tandem thin airfoil theory predictions for figures (b), the largest £ m a x was O ( l 0 - 3 1 ) . However, roundoff error became a problem on the CC-40 for extreme Rf values (Rf < 0.01 and Rf > 0.99). Figures 10 to 15 show that tandem thin airfoil theory provides good quanti-tative predictions of the aerodynamic characteristics of the exact potential flow. Also, the linearized theory's predictions of the qualitative aspects of these characteristics is excellent, with the only discrepancies occurring in extreme geometries which are of little practical interest in aeronautics. It is particularly encouraging to see the consistency with which the overall forces are modelled, as this is a good indication that the linearized theory's predictions will correlate well with real flow forces. For figures (a), the quantitative relationship between the overall force predictions of the exact and linearized theories at the one-element extremes is maintained throughout the range of f—s values. This relationship is also maintained for the overall lift variations of figures (b). The overall moment characteristics of figures (b) do exhibit this consistent relationship to some extent, but with a great deal of sensitivity to the overall thickness of the configuration (the overall thickness is greater at Rf = 0 and Rf = 1 than it is at Rf = 0.5). Not surprisingly, the localized forces are less consistent than the overall forces when thin airfoil theory predictions are compared with exact potential theory, although this does not necessarily mean that the linearized predictions will be equally inconsistent when Tandem Airfoil Forces an Easy Way / 4.4 96 compared with the real flow forces. Indeed, Foster et al (1970) include lift comparisons for their 31% slotted flap configuration which show that the experimental values of the airfoil's localized lift are overpredicted by exact potential theory by about the same amount as the configuration's overall lift is overpredicted (note the linearized and exact theories' relationships for Ct and Ctt in Figures 10 and 11), whereas experimental values of the flap's lift were very closely predicted by the exact potential theory (which is, again, compatible with the relationships of Figures 10 and 11). The camber line force prediction comparisons (Figs. 12 and 15) are appar-ently the least accurate of all the figures. However, these forces tend to be small in comparison to the total forces on the airfoil, as shown in Figures 16 and 17. Also, it must be remembered that the localized force coefficients have been nondimensionalized using the localized chord length so that, for example, if Rf —* 1 (so that s —• 0), then A C ^ , A C m # a c , and their errors are magnified relative to the actual forces. Figures 16 and 17 show the actual nondimensionalized forces and their vari-ations with a and ij for Rf = 0.2308 and for f-s = 0.025 (figures (a), a realistic slot size) and /—s = 0.1333 (figures (b), an interesting midpoint between f—s = 0 and f—s = 1 where K = K'). For Figures 17, is again calculated from Equation 4 .80 using the thin airfoil theory values for x- . Except for high values of a+rj, the linearized predictions Jac of the overall forces agree well with the exact potential flow forces, with the relationship between the exact and linearized theories being much the same as for the one-element 23012 airfoil of Figure 1. Although some of the localized linearized moment predictions show poor agreement with the exact predictions, it i3 in proportion to their decreasing magnitudes and, therefore, importance. Thus there is strong evidence that the consistent close relationship between the linearized and exact overall force predictions seen in one-element airfoil theory is main-tained in the tandem two-element regime as well, so that one would expect the tandem thin airfoil theory overall force predictions to again be closer to the real flow overall forces F I G U R E 16 C( vs. a for different n for tandem NACA 23012 airfoils. Rf = 0.2308 linearized theory; •--<>--• exact-numerical theory J 1 " 1* I 1 1 1 L*. I 1 1 ( !_». | | | | • I • • • I • I I, ' 5 10 IS -5 0 5 10 15 -5 0 5 10 15 -5 0 5 10 15 -5 0 5 10 15 -5 0 5 10 15 a a° o° a° o° a° a) f-s = 0.025 b) f-s = 0.133 F I G U R E 17 Cm , vs. a for different TJ for tandem NACA 23012 airfoils. R, = 0.2308 <P linearized theory; «--o--• exact-numerical theory Tandem Airfoil Forces an Easy Way / 4.4 99 than are exact potential flow theory predictions. Furthermore, the linearized predictions should correlate better with the real flow forces than they did in the one-element regime since properly designed multi-element airfoils reduce boundary layer effects on the overall flow. This completes the discussion of tandem thin airfoil theory. There is much else that could be done with it, such as finding the most efficient way of handling numer-ically specified airfoil profiles, or conducting a series of experiments on tandem airfoils in order to directly compare the linearized predictions and real flow results. However, since the tandem airfoil configuration is not the most common of configurations, there is little motivation for giving it more attention. The present theory has served its purpose by show-ing what can be accomplished by exploiting the theory of elliptic functions in conjunction with the use of modern hand calculators or microcomputers. In the next chapter, a beginning is made on the general linearized theory for staggered airfoils. CHAPTER 5 Staggered Two-Element Thin Airfoil Theory In this chapter, expressions for the general, two-element, linearized airfoil theory forces are derived. Overlap is accounted for so that the two airfoil elements of arbitrary chord length can be arbitrarily positioned relative to one another, as long as each airfoil element's chord line does not deviate appreciably from an orientation parallel to the direction of the flow at infinity — the linearization requirement. This flexibilty comes at a cost. As previous investigators have discovered (Munk, 1922b; Glauert, 1926; Millikan, 1930; Garrick, 1936), the staggered slit mapping is much more difficult to solve than the tandem slit mapping. Solving for its parameters requires a numerical approach that is not pursued to completion in this thesis. The force solutions are in terms of these mapping parameters. 5.1 T H E S T A G G E R E D P A R A L L E L SLIT M A P P I N G This mapping is shown in Figure 18. The linearized physical plane is again the z-plane, Fig. 18a, with the airfoil slit on the real axis stretching between xx = 0 and xn. The flap slit is displaced downwards a distance h from the real axis, with its leading edge at z2 = X2 — ih and its trailing edge at z 22 = X22 — ih. When the plane potential flow about these slits is mapped into the infinite strip in the A-plane (0 < / i < K), Fig. 18c, the leading and trailing edges of the airfoil and flap no longer coincide with the corners of 100 Staggered Two-Element Thin Airfoil Theory / 5.1 101 V A z -PLANE XQ xii x 22 222 (a) X -PLANE t -PLANE U hi h t22 Cb) F I G U R E 18 The staggered parallel slit mappings. the rectangle as in the tandem airfoil mapping. The airfoil slit still opens to form the left side of the rectangle but its leading edge now maps to the point Aj and its trailing edge to A n . The flap forms the right side of the rectangle, with its leading and trailing edges at A 2 and A 2 2 . The f-plane, Fig. 18b, is a convenient secondary solution plane which helps with some of the mathematical formulations that follow. The leading and trailing edges of the airfoil and flap correspond to the points tu tiu t2, and * 2 2 . If h is positive, as shown in Figure 18, then it turns out that t± and t2 are on the bottom surfaces of the slits in the 4-plane while tn and t22 are on the top surfaces. The mapping from the <-plane to the A-plane is the same tandem slit mapping used in Chapter 2, Eq. 2.4, with A = ±iK\ 0, K, and K ± iK' corresponding to t = 0, a, / , and 1 as before. In the z-plane, these points correspond to z = x0, xt, Zj — Xj — ih, and ze — xc — ih. For the geometry shown, xQ and Zj will be on the top surfaces of the airfoil and flap slits while xt and zc will be on the bottom surfaces. The mapping from the z-plane to the A-plane is obtained by considering the Staggered Two-Element Thin Airfoil Theory / 5.1 102 FIGURE 19 Doublet How in the X-plane corresponding to zero incident uniform Bow in the z-plane. exact formulation for the zero incident uniform flow, {7, about the slits in the z-plane: w(z) = U . (5.1) This is correct since the slits do not obstruct or redirect the flow in any way. Integrating tv(z) gives the exact complex potential, fl(z), which has the same value at corresponding points of a conformal mapping: n(a) = Uz = n(z(A)). (5.2) In the A-plane, the corresponding flow is certainly not uniform as there must be a doublet at A = A^, corresponding to z — oo. This is the only singularity that can be present in the flow in the rectangle. The lines fi = 0 and p = K are solid boundaries across which there is no flow, a condition which is satisfied by placing an infinite series of image doublets about these lines, as shown in Figure 19. Since the flow must also be periodic in 2iK', this pattern of singularities continuously repeats itself up and down the imaginary axis. The generality of the flow is ensured by allowing the doublets to be oriented at any angle /?, and allowing for a superposed vertical flow over the plane. Staggered Two-Element Thin Airfoil Theory / 5.1 103 Consider the complex velocity: w(X) = C [eif} sn 2(A+/*«,) - s n 2 ( A - + tfc] (5.3) where C and b are unknown real constants. The singularities of the sn function are located along A = fi ± i K ' , v/here sn(jt ± i K ' ) = 1/k sn p: w(fi ± iK') = C The expansion of sn A for small A is: sn A ~ A + O (A3) , as A -* 0 so that: c ^ + ib (5.4) (5.5) These are the complex velocities for four doublets of equal strength symmetrically placed about the imaginary axis in the A-plane and, since sn2(A db 2K ± 2iK') = sn2 A, w(X) is indeed the complex velocity representing the doublet flow of Figure 19. As a further check, one can use the addition theorem for the sn function to show that to (A) is purely imaginary along the lines ft = 0 and fi — K so that, as anticipated, the flow does not cross the surfaces of the opened airfoil and flap slits. Integrating to(A) gives fl(A) which must be equivalent to fl(z), Eq. 5.2, at corresponding points. Upon choosing an appropriate constant of integration, the staggered slit mapping is: A *(A) = + \eiffsn2(X+fL00) - e - ^ s n 2 ( A - M o o ) + ib] dX . (5.6) o As in Whittaker and Watson (1927, 22.73), it is convenient to choose: A E{X) = Jdn2XdX (5.7) Staggered Two-Element Thin Airtoil Theory / 5.1 104 as the fundamental elliptic integral of the second kind. Then, using the relation between the sn and dn functions (Eqs. 2.10) and absorbing terms into new real constants Cj and blt z(X) becomes: z{\) = x, + d { c o s / ? [ £ ( A - / 0 - ^ ( A + M o o ) + 2£(/0] + ibl\-i8\nP[E{\-li00) + E{\+ii00)]}. (5.8) The first line on the RHS of Equation 5.8 is periodic in 2tK': F(\ n \ F(\±,l ^ _ 2cn/x00dn/xo0 A; 2snVcom 2A while the second line can be made so by choosing an appropriate value for 6^ Since: % - / » v _^./vx 2cnAdnA k2 sn2 « „ sn2 A , . £ ( A - ^ ) + £ ( A + ^ ) = 2 £ ( A ) m X L f J . ^ A ( ' - 1 0 ) and: JE(A ± 2ii)C) = £(A) ± 2i(K' - E'), (5.11) then 6, must be: 6i = 2 s i n / ? ( l - | Q . (5.12) Also, on the flap where A = K + iv, the imaginary part of Equation 5.8 must equal — ih. On the flap, Equation 5.9 is purely real and Equation 5.10 is purely imaginary except for one term: E(K + iv) = E + imaginary terms so that, using Legendre's relation, Eq. B16: Cx = — • (5.13) 1 TT sin/3 v ' Thus, the mapping is: 2K' h i acn dn fi^ k2 sn2 fi^ sn2 A Z(AJ = X -i — - < C O S/7 TTj 5 T ~ T i smjJI sn Poo l - k l sn* y.^ sn2 A • • * 17, ^ x n / n cnAdnA A2 sn2 j i -sn 2 A l ) + , s m / ? [ ( l - - J A - S ( A ) + ^J^^J} (5.14) Staggered Two-Element Thin Airfoil Theory / 5.1 105 which, for /? ^ 0, is only quasi-doubly periodic. This is more easily seen when the doubly periodic components of this equation are written in terms of the ^-variable and the quasi-doubly periodic term is written in terms of a -^ -(w) function. One gets: Z = XQ + H sin/? t cos 0 - i sin /? ^(t-s)(t-f)(t-l) -ih ir K tV } where: H 2K' (5.15) (5.16) Of course, z(t) is not infinite at f = 0. The square-root pole in the first square-bracketed term of Equation 5.15 is cancelled by that in the -^(w) function. When h and /? go to zero, this mapping reduces to the tandem slit mapping, although it is not yet obvious that iT/sin/? reduces to the correct value. Note that when h —* 0 with overlap (/? •/+ 0), none of the terms in Equation 5.15 disappear since K' co. When /? = TT/2, the airfoil is in an unstaggered biplane configuration. The eight unknowns in the z(X) mapping: x»> P> k> /^ooi A i » *2i A 22 ( 5 - 1 7 ) are determined by simultaneously solving eight equations: * dz z(\j) = zj, ^ ( A ; ) = 0; ; = 1,11,2,22. (5.18) In the f-plane, the k, fi^, and Xj variables are replaced with / , s, and tj. When z is written entirely in terms of the f-variable, it must be written as an integral, which can be conveniently put in the form: Z(t) = XQ + dz H sin/? *cos/? — i sin/? / « 2 * 2 - ( l + s+/)* + / 1-<i-.>! y/t{t-a){t-f)(t-l) dt } . (5.19) The -r-[tj) = 0 requirement becomes: dt where: cos{3^/tj(tj-s)(tj-f)(tj-l) = »sin/? 2 l+s+/ 6 j 2 >'+2 6s f F' 1 - il-»)Y, (5.20) (5.21) Staggered Two-Element Thin Airfoil Theory / 5.1 106 Squaring each side of Equation 5.20 produces the final relation for the iy's: <J.-(l+a+/)*J.+ ( /+ 3 + 3 /)cos 2 /? + 8 i n 2 / 3 ( ^ 1 + s + ^ + f » sfcos2p + 8in2p^-^-6 tj + s'm2 p— = 0 . (5.22) In the f-plane, the uniform flow of the z-plane, Eq. 5.1, is transformed into a flow incident to the tandem slits at an angle of attack a = p. The above equation for the tj's gives the stagnation points of that flow. For the general problem, this quartic polynomial does not have simple factors but can be solved analytically using a method given in most mathematical handbooks. However, this first requires knowledge of the first four variables of Equation 5.17. These variables can be solved only through a trial and error process involving the repetitive evaluation of elliptic functions (see the last term of Eq. 5.15). Although the methods of section B.3 would greatly facilitate this calculation, the calculation time needed on a small microcomputer would still be relatively long. The problem could be easily handled on a large mainframe computer but then it can be argued that one might just as easily solve the exact-numerical problem. In the remainder of this section, alternative ways of posing the staggered parallel slit mapping problem are considered in the hope that one might simplify the problem's solution. Suppose the angles (3 of the doublets in Figure 19 were set to zero. The stagnation points of the flow would then move to the corners of the rectangle, A = 0, and A = K, and the airfoil and flap boundaries would be enclosed by the same streamline. In the *-plane, the angle of attack of this new flow relative to the tandem slits would be zero. In the z-plane, the angle of attack would be a = —/? and the stagnation points in Staggered Two-Element Thin Airfoil Theory / 5.1 107 this flow would be at z = xQ} xt, Xj-ih> and xc-ih, as shown in Figure 18a. Using the mapping, Eq. 5.15, one can show that: X ' - X o = ^ 3 ' X f - X ° = T*Vpf' X c - X 0 = T*h- (5-23) Then, with k given by: ' .2 = /-« = (l - 0)(/-3) /(!_,) ( / _ o ) ( l - s ) ' the corresponding points to-0, s, / , and 1 in the z-plane also give: [xf-x0)(xc-x,) and: s n 2 = l-s = X° ~ X" . (5.25) xc - x0 Similar relationships for c n 2 ^ and dn 2 /x^ are obtained. J^Note that the last relation of Equations 5.23 can be rewritten as: 2K' h = > / ( * / - * < > ) ( * « - * . ) • (5-26) K tan/3 It is. now easy to see, in Equation 5.15, that this constant reduces to the correct value needed for the tandem slit mapping as /3, h —> 0 .j Thus, one way of obtaining the first four variables of Equation 5.17, without having to deal directly with the last four, is to solve for that unique circulationless flow in the z-plane which encloses both the airfoil and flap slits within the same streamline. Although this would still require a numerical approach, perhaps this could be more easily handled in the physical plane where all the geometry is known. A method for confor-mally mapping general airfoil shapes to circles, such as by Ives (1976), might prove useful, especially since the staggered parallel slit geometry is so simple. Another approach to solving for the mapping parameters is one which takes advantage of the properties of the theta functions. Since E(X) is closely related to Staggered Two-Element Thin Airfoil Theory / 5.1 108 the -r^(w,g) function, the mapping, Eq. 5.8, can be rewritten: #4 z(u) - xt 2 7 i h b l n 9 { cos/3 t?' ^ ( w - X o o . 9) - ^(w+Xoo , 9) + 2^(Xoo>q) + xsin/3 > 9 t?4 or, using the u'-plane and q' parameter (Fig. 4e): } (5.27) cos/3 t sin (3 fa-Woo, fl + fa+i-LJ) 2 2 (5.28) This last equation is the simplest form the mapping can take on, with the mq terms and the term linear in w from Equation 5.27 being absent. For large gaps, q' —* 0 and it should be possible to invert the q' series representation of Equation 5.28 and obtain u' as a function of z. However, since slotted flap airfoils have very small gaps, these q' series would be inefficient (q' —» 1 as the gap goes to zero). This problem is further compounded by the fact that, for a given separation distance of the airfoil elements, q' is closer to 1 in a staggered slit than in a tandem slit configuration. The corollary of this is that q will be closer to 0 — so the q series representations of Equation 5.27 should be very efficient indeed. Since Equation 5.27 is in terms of u, \nq, and series of sinw and q terms, a trial and error or double series solution is unavoidable (the familiar x = In i , solve for x problem). Nevertheless, a procedure for solving Equations 5.18 by taking advantage of the q series expansion of Equation 5.27 has been developed for the biplane configuration (/3 = 7r/2 => 4 equations in 4 unknowns, including a trial and error solution in 2 of the unknowns). The procedure uses Newton's method of successive approximations and occupies virtually all the available memory in a TI programmable 59 hand calculator, which takes from l\ to 2 minutes to solve for the four parameters k, fi^, A n ( = —Ax), and A22( = A 2). Although it neglects terms O(g4), the procedure is accurate for gap sizes as large as the airfoil elements themselves. Staggered Two-Element Thin Airfoil Theory / 5 .1 1 0 9 Besides showing the inadequacy of today's hand calculators for solving the general staggered slit thin airfoil theory problem, the biplane mapping solution produced some hard information on the values of the important parameters of the staggered slit mapping at this, the opposite extreme to the tandem slit mapping. Table 2 shows some of these values for a 23% "flap". Note the extremely small values of both f-s and q for typical slotted flap gap sizes of from 1% to 4%, although in a true slotted flap configuration P would likely be closer to 0 than JT/2. Nevertheless, f—s and q appear to be consistently the smallest of all the staggered slit mapping parameters and an efficient general solution of this mapping must surely take advantage of this. f-s — : — ; O R h 50% 15.4% 4% 1% s+l-f Ci + C2 f 0.846 0.800 0.778 0.772 Tandem slit f-s 0.333 0.133 0.038 0.010 mapping q 0.102 0.043 0.013 0.003 parameters, /? = 0 q' 0.013 0.043 0.103 0.175 0.369 0.343 0.326 0.321 f 0.834 0.890 0.951 0.970 f-s 0.152 0.007 1.4(10-6) 4.1(10"19) Q 0.053 0.004 1.8(10-6) 8.6(10"19) Biplane slit q' 0.035 0.163 0.474 0.789 mapping parameters, 0.323 0.219 0.142 0.112 un/K' 0.706 0.875 0.966 0.991 V22/K' 0.560 0.625 0.765 0.896 E'/K' 0.772 0.389 0.151 0.048 T A B L E 2 Comparisons of staggered slit mapping parameter values at the tandem and biplane slit extremes, C2/(c1+c2) = 0.2308. Staggered Two-Element Thin Airfoil Theory / 5.1 UO Although q appears to be the smallest parameter for the greatest range of P values and gap/slot sizes, f—s is sufficiently small for typical slotted flap configurations that one is tempted to work directly with Equations 5.19 and 5.22 to obtain solutions for the mapping parameters by expanding these equations in powers of f—s. However: jgt 2 K'~Hf=s) a3/-a-*° and so the 5 term (Eq. 5.21) in these equations creates the same problem for a series expansion solution of z(t) in powers of f—s as the lncj term does in the g-series expansion solution of z(u), Eq. 5.27. E'/K' - » O a s / — s -* 0 but at a very slow rate, as shown in Table 2. Still, it is useful to at least consider the zeroth order term of such an expansion. The following relations are obtained from Equation 5.19 in the limit that h —• 0 with overlap. They are not completely general since they assume x2 > 0 and z 22 > a typical slotted flap arrangement. As f—s —• 0: *(0 -* 1 ~ l ° S ^ X 2 2 + x22(tcoaP - isinpy/t^t-l)) (5.29) Kz) ~* ^—- + —(zcosP+ ia'mP\/z{z-x22)) . (5.30) 2 3C22 These are just simple Joukowski transformations of the single slit that results from the staggered slits coming together. The surfaces of the airfoil and flap that come together in the overlap region (x2 < x < xn) a r e l o s * *° the external flow. On the single slit in the <-plane, the airfoil surface ends at t = / , with the top of the slit corresponding to r = i n at this point and the bottom to x = x2. Thus, P is determined by requiring that t(xn) = t(x2): t^p-*-]— z " - * 2 a s / - s - > 0 . (5.31) \/X2(x22-X2) +\yxn(x22-Xn) For a small gap staggered slit mapping, the differences between the above predictions of the mapping, Eqs. 5.29 to 5.31, and the true predictions will be 0(s'mP(E'/K')). Staggered Two-Element Thin Airfoil Theory / 5.2 111 Finally, using the expressions for z(t) and z(A), it can be shown that: He-*. z ~ sin/? •t + Z0 + -j- + -jjt + '-' > as*->oo (5.32) where: and: Z0 = x0 + » ( j r ^ y - ^ - f c E 8 ) (5.33) * , = - i f f 4 i f + 3 + S } ) ~ 4 S . (5.34) 8 E 3 is as defined in Chapter 2. In the following sections, this expansion is used for evaluating the forces by residues. A recurrence relation for higher order terms can be obtained from z(t), Eq. 5.19, by considering dz/dt zst-*oo. 5.2 T H E O V E R A L L FORCES In the remainder of this chapter, general expressions for the aerodynamic forces are ob-tained by generalizing the selector functions of the previous chapter to apply to staggered slit configurations. No doubt the flow itself could be solved for by finding the "general" and "special" functions analogous to those described in Chapter 3. Indeed, some of these tandem slit functions could be used on the "tandemized" staggered slits in the i-plane, with appropriate adjustments to account for the change in position of the leading and trailing edges and the fact that camber and thickness are no longer purely even or odd in any parameter. Also, the initial complicated mapping from the z-plane to the i-plane cannot be avoided. Although a general solution to the linearized staggered two-element airfoil flow field is not pursued further, the selector functions presented in this chapter are easily rearranged to give the flow field solutions for those simple geometries of the a and n problems. Now, if the overall lift is: Ct = I w'(z) dz = I w'(z)C(z) dz , (5.35) ci+c2 j c 1+c 2y c c where u)'(z) is a general flow function that solves the a, r/, camber, and thickness prob-Staggered Two-Element Thin Airfoil Theory / 5.2 112 lems, then the conditions on £{z), the generalized staggered slit version of the boundary condition selector, are: 1) its real part is zero on all airfoil element surfaces. 2) £(z) —* 1 as z —• oo 3) the poles in £(z) must be at the airfoil and flap trailing edges so that w'(z)£(z) has no simple poles in the z-plane. Although not a problem for aeronautical applications, it now appears that thickness distributions must be restricted to those without infinite slope trailing edges. Otherwise, w[(z) will have a square-root pole at the trailing edge (see Eq. 3.12 for tandem ellipses) which would combine with a square-root pole in £(z) to form a simple pole, preventing the necessary unhindered collapse of C to the slit surfaces. This is not a problem in the tandem theory since the residue of this pole is real and so does not affect lift. For staggered thin airfoil theory, it is not obvious this would still be true. Again, £(z) is unique: £ { z ) = Vt(t-s)(t-f)(t-l) + (t-tn)(t-t22) 1 T22~tll \ /<22(<22-*)(<22-/) ( t22-l) \All(*ll ~ *)(*!! ~ 1) t — t22 '""^11 . (5.36) This function is obtained through a linear superposition of the -^ -(w-Wn) and -r^(w—w22) functions which put simple poles at the airfoil and flap trailing edges in the w-plane. These singularities also occur as simple poles in the f-plane but become square-root poles in the z-plane since the trailing edges are critical points of the z(t) mapping. In the f-plane, it is important to distinguish between tj and tj, the upper and lower surfaces of the tandem slits at t — tj. When h > 0 (0 < /? < TT), tn and t22 are more properly written as txl and t22. Using Equation 4.35, it can be shown that the second line on the RHS of Equation 5.36 combines with the first line to keep £(z) finite at t = t^ and t = t22. Alternatively, when h < 0, tj = t^t^ft^ ,kt22 and £{z) is finite at txl & t22. Staggered Two-Element Thin Airtoil Theory / 5.2 H3 To calculate the forces, the large z expansion of £(z) is needed. If: + + ^ + + , a s * - o o (5.37) defines the A„'s, then the z(t) expansion, Eq. 5.32, can be used to obtain the £ n ' s : n f> p £{z) ~l + -± + ±l + Ji + ... , as z-> oo (5.38) Z Z Z> and: Cl==sw£l>> i2 = ZoCl + -^T£t>> etc- ^ Taking advantage of the relations (from the stagnation point equations, Eqs. 5.20 and 5.22): l+*+/ = *i + *n + *2 + *22 (5.40) *11 ~ <1 + *22 ~ h = ~i \ Z * 2 2 ( * 2 2 - s ) ( * 2 2 - / ) ( * 2 2 - l ) ~ V^llC^ll ~ s)(*U-f){t n - 1) 2 cos/? sin/? '22 — (5.41) * l l + * 2 2 - ( * l + *2)(*ll+<22) + g 2 cos/? •-» < 2 2 \ / < 2 2 ( ^ 2 - s ) ( < 2 2 - / ) ( * 2 2 - l ) ~ hi\A11 - / ) ( ' 11 ~ 1) sin/3 <22 -<n (5.42) it can be shown that: £ *n-ti+ti2-hja t H t n - t 1 + t 2 2 - t 2 1 2 cos/? sin/? 2 cos/? v ' and: 4 ( /+s+ 3 / ) - ( l + 3 + / ) 2 - 4 5 ^ 1 + e 2 2 - (f 1 +f 2 ) (f u +f 2 2 ) + g ^ = 8 + W 6 ' ( 5 - 4 4 ) When /? -» 0, #/sin/? is finite (Eq. 5.26) as are Equations 5.41 and 5.42. The LHS's of these latter two equations are used in Ctl and £, t 2 in order to eliminate the indeterminacy of the RHS's at /? = 0. However, when P —* n/2, the LHS's become indeterminate and need to be replaced by the RHS's. Now, the contour integral giving the overall lift can be evaluated on the surfaces of the staggered slits, where: Staggered Two-Element Thin Airfoil Theory / 5.2 114 The overall moment analogue to this equation is somewhat more difficult to get. Although it is true that: j> zw'(z) dz ( ' i + c 2 ) 2 tol 4(*-£Ju/(*K(*)<fe when C collapses to the slit surfaces z = x—ih on the flap and t(z) does not select the boundary condition in one of the resulting terms, namely: ihjiw\z)Z(z) dz flap The following procedure avoids this problem. C m o can be rewritten as: C_„ — m ° ( c i + c 2 ) 2 " ) / V sin/? • C He-* t + Z, ) tv'(z) dz (5.46) since the difference between z and the term replacing it in Equation 5.46 is 0(l/z) as z—>Q (Eq. 5.32) and, by the residue theorem, this will make no difference to the result of the integration. This substitution is useful because t is purely real on both the airfoil and flap. Thus: ' m o ( C l + C 2 ) 2 5R Z0 j w'(z) dz - iH j tw'(z) dz + H cot /? j tw'(z) dz C C C But, with H' = tfe-^/sin/?, and since dz/dt = H' + 0(l/t2): (5.47) Also: so that: j tw'(z) dz = j H'tw'(z) dt. C C ( l - ^ £ ( z ) ~ l + o(j^ , a s f ^ o o jtw\z)d* = f(f-jjiS) dz (5.48) (5.49) Staggered Two-Element Thin Airfoil Theory / 5.2 115 Finally, using Equation 5.35, and the fact that the second integral in Equation 5.47 already selects the boundary condition when C collapses to the slit surfaces, one gets: Cl cos2/? - x0 ° m o ~ c 1 + c 2 i c ' - T ^ * k i z M { * ) d * (5'50) where M(z) is purely imaginary on the slit surfaces: M{z) = Ht{cotl3C{z)-i) ~ * + ( t f c o t / ?A 1 - Z 0 ) + ^ + ^ + z z and, like Ci, Mi is purely real: (5.51) as 2 —•co f+S + sf - ( l + S 4 + / ) 2 + < U + *22 - (<l+<2)(<ll+*22) (5.52) 2 sin'/? A recurrence relation for the Mn's can be obtained from those for the £ n ' s and Z n 's. Although dy/dx can be easily broken up into components which are even and odd about the surfaces of the staggered slits: dy _ dyc dyt dx dx dx ' C(z) and M(z) are not so readily separated. Equations 5.45 and 5.50 are probably the simplest forms of these two-element staggered slit versions of Munk's integrals for the overall forces. They could each be evaluated numerically as the sum of four line integrals but this would require as many evaluations (or inversions) of the complicated z(t) function as there were points in the numerical discretization of the integrand. Complex versions of these equations which enabled their evaluation by residues would be highly desirable. For this, it is convenient to again find a complex function, -7^(2), which is single-valued in the dx flow field and which: 1) has its real part equal to dy/dx on the slit surfaces. 2) preferably does not combine with £(z) or M{z) to form any simple poles in the z-plane. This condition is not strictly necessary because the residues of any simple poles that are created can be accounted for. Staggered Two-Element Thin Airfoil Theory / 5.2 UG Since the oddness or evenness of the boundary condition can no longer be taken advantage of (as with Eqs. 4.17 and 4.18), the overall lift and moment become: mo ( c i + c 2 ) 2 The simplest problem is the o-problem: dyai dz (5.53) (5.54) dx (z) = -a (5.55) and, using the above equations, one can easily obtain the linearized predictions for the lift curve slope and aerodynamic center of staggered two-element airfoils — results which do not appear to have been previously presented: Q _ — ji _ — -n -i • - a 1 cx+c2 1 cx+c2 TT ^ / ( l - s ) sin/? cos/? 4na 2ira 2K' hi _ ' l + 2^2 — h (5.56) d C m o / d a ( c l + c 2 ) = x0-£1 cos2/? + 9± = xQ + dCt/da {2K'/n)h Ci \/f{l-s) tan/? f+s+sf (1+s+f)2 , ,2 (5.57) + tn +122 - {ti+t2){tn+t22) t n — ti + f 2 2 — t hi — h + '22 — h C( and xac] both reduce to the correct limit when h —* 0, with or without overlap. It is informative to look at the former case in more detail. When h —• 0 for /? ^  0: 8 —• /, f -* s, and the stagnation point equation (Eq. 5.22) gives (for 0 < /? < TT/2): ^ _> (1 _ cos/?)/2 , hi-*f, h~*f, * 2 2 - ( l + cos/?)/2. Also, from the last of Equations 5.23: 2K' h x22 sin/? \ Thin airfoil theory makes no prediction of the vertical position of the aerodynamic center since it predicts each airfoil element's drag to be zero. Staggered Two-Element Thin Airfoil Theory / 5.2 117 since xc —• z22(l+cos/?)/2 a n c * %o -* x2 2(l-cos/?)/2 (Eq. 5-29). Thus: Ct-*2irax22/(c1+c2) and xac -+ 1/4 as required. (For a single slit, of course, x2 2/(ci+c2) is taken to be 1.) When (3 -* TT/2, the indeterminate terms in C( and xae can be replaced with the RHS of Equation 5.41. Glauert (1926) solves for the exact overall lift on an unstag-gered biplane arrangement of flat plates of equal chord length. The coefficient of a in Equation 5.56 reduces to give the coefficient of sin a in Glauert's solution at this extreme. For the r;-problem, the generalized staggered slit flap selector is: r f , 2w K'v'2, s 2K'/n » ff*2 V ? ( l - S ) t2-f lt(t-s)(t-l) + \/t2(t2-s)(t2-f)(t2-l) t-t2 v t-f t-t2 = 0 + imaginary terms on the airfoil = 1 4- imaginary terms on the flap. (5.58) 2u K10' 7(oj) is just the 1- —-^-(w-u/2) function with an imaginary constant subtracted so that ?{woo) is purely real: ?o + ?Jr+ + , a s * - o o I 1 (5-59) T , *i , *2 • ~ /o H 1—~ H , as 2r —• oo. z z Using Equation 2.36 and taking advantage of the stagnation point equation, Eq. 5.20, and the relation: r one gets: , _ 2K' t2-f 2JT' 1 (2 1+s+f 6\ & (5.61) «• V 2 2 / c 0 8 ^ G (1 + a + / _ 4 f 2 ) 2 ^ 1 t 2 ( i + / - a ) 2 - / ( l + / - a ) , 4 T = 8 + « 2 l i ' Ti and 72 are obtained by replacing £ l n with #n in Equations 5.39. Although TQ and 7\ are purely real, in general the higher order T^s are not. Staggered Two-Element Thin Airfoil Theory / 5.2 118 For the ^-problem, then: ^W = -?W ( 5 - 6 2 ) and Equations 5.53 and 5.54 enable a straightforward solution of the flap effectiveness for a slotted flap airfoil. Again, this appears to be a new result: Ci+C2 4*1 H J tn -ti + t22 -t2 | 1 G2-Gi 2K'(ti-f)(t2-f) 2 n y/fil^s) > . Ci+c2sin/3 j 2cos/3 cos/3 (5.63) The last line of Equation 5.63 clearly shows how this expression reduces to the tandem airfoil configuration predictions for dCt/dr], Eq. 2.47b. However, when /? —• TT/2, the first line must be used in conjunction with Equation 5.20 to remove the indeterminacy in the expression. Just as one would expect, this linearized prediction of dC^/dr] for staggered airfoil elements has the same value regardless of h being positive or negative. The expression for Cmac is relatively long: C m = 4*i » m« (ci+c2r {7lg_cos^,,)_^e-,%j (, 6 4 ) but it is independent of both E 3 and x0. The method of solution of the a and r) problems via Equations 5 . 5 3 & 5 4 is more complicated than necessary. If C{z) is written as Z{z\ txx,t22)t then the flow functions that solve these problems are: < ( * ) = ia[l-£(*;*i»<2)] ( 5 . 6 5 ) v'n(z) = iri[7{u)-7QC{z-)tut2)\ and the large z expansion coefficients of these functions give the lift and moment di-rectly. The main reason for developing Equations 5 . 5 3 and 5 . 5 4 is so the forces for the general camber and thickness problems can be easily solved. This requires finding -j^-{z) dx and ^j^{z) functions capable of solving polynomial boundary conditions such as those in dx Staggered Two-Element Thin Airtoil Theory / 5.3 119 Equations 3.14 and 4.47. No doubt staggered slit versions of the segment selectors of Chap-ter 4 would play a part in these formulations. However, finding the —~{z) functions for dx other than the zeroth order term of a polynomial boundary condition is relatively difficult and is not pursued further. 5.3 T H E L O C A L I Z E D FORCES The airfoil lift is again calculated using Equation 4.28, but with u)'c(z) replaced with w'(z) and using the staggered slit version of the airfoil selector: A(U) = 1 - _ - _ . - ! ( „ ) + hi-s t{t-f)(t-l) t-tn\ t-3 , V'II('II-«)(«II-/)('U-1) = 1 + imaginary terms on the airfoil = 0 + imaginary terms on the flap. (5.66) 2LJ K't?' This is the 1 j " r ( u _ w u ) function less an imaginary constant so that Afa^) is IT K Vy purely real. Thus: j w'{z) dz = AQ j w'(z) dz- j w\z) [X (w) - l]dz - j w'(z)A{u)) dz . (5.67) C\ C Cy c% On the slit surfaces, the unknown pressure terms in w'(z) are completely contained within the imaginary parts of the RHS of this equation (and they must sum to zero), so that the airfoil lift is: C t i = h ± £ l A o C l - * £^[A{u)-i]d*-££%lAMdz. (5.68) Cy Cy J CiX Cy j dx airfoil flap If yi(x) and y2{x) a r e continuous functions over their respective domains (including around the leading and trailing edges) and if yy(x), at least, has a simple zero at z — Xyy, then, upon integrating by parts, becomes: „ c i + c 2 A „ 2 * X , ^A(UJ) j 2i t . sdA{u) , , r . Cel = -L^-AQCe + - ^yy(x)-^dz+-^y2{x)-j-J-dz (5.69) airfoil flap Staggered Two-Element Thin Airfoil Theory / 5.3 120 where: 2 2K'/n u 2-dA(u) sin/? ^t(t-3)(tn-f)(tn-l) + ^/t[Jtn-S)(t-f)(t-l) t-tn Ctl = —— dz H 2\/t(t — s)(t—f)(t — l) cos/? —tsin (5.70) which is purely imaginary on the slit surfaces and reduces to Equation 4.34 when /? —> 0. If a y(z) function with the necessary characteristics can be found, Ctl could be rewritten: A0Ce + 3i j j p j y dz - 27ri(residues of integrand) j (5.71) and then evaluated by residues. Note that dA{u)Jdz has a pole 0((z—xn)~3/2) at the airfoil's trailing edge. Also, for the a and n problems, yi{x) and y2{x) are polynomials linear in x. The localized airfoil moment is a particularly difficult problem. u>'(z) cannot simply be replaced with zw'(z) in the above procedure since z is no longer purely real on both slit surfaces. One way of overcoming the problem is to introduce the complementary selector function A(u>) which is just a flap selector function with its singularity at the flap trailing edge. If A(OJ) is written as A{u}\Un), then: A(u) = 1 - A(u;u22) = 0 + imaginary terms on the airfoil (5.72) = 1 + imaginary terms on the flap and ^(Woo) is also purely real. Note, from Eq. 5.36, that £(z) oc 1 — [>l(w)+>l(w)]. Now, writing A and A for A{u)) and A(UJ), consider: dtlf[z{A+l-A) + ih{A-l)A]w'(z)dz\ Kc J (5.73) = SR j 2 j zw'{z) dz - ih j w'(z) dz\-i £ [x{A -1 - #) + ih(A - 1)A] ^  dz Ci Ci airfoil - i £[x{A+l-$) - ihA{l-A)}^}dz . flap Staggered Two-Element Thin Airfoil Theory / 5.3 121 The real part of the integral on the LHS of this equation can be evaluated (by residues) in terms of the overall lift and moment (the real part of the second term in the integrand integrates to zero). The real part of the first integral on the RHS gives the airfoil moment, while that of the next integral is zero. The last tv/o integrals on the RHS can be evaluated numerically, with all the problems that entails, but it would be more convenient to convert them to a form that is more easily solved by residues. Until now, this was accomplished by integrating by parts so that the resulting integrands consisted of a boundary condition multiplied by a function whose complex form was purely imaginary on both slit surfaces — a characteristic which greatly simplified the formulation of the complex version of the boundary condition. However, the A A products in the above integrands negate the effectiveness of this process. The problem can be seen in a different way, as suggested by the forms of the mapping function, Eq. 5.15, and the A(OJ) and A(OJ) functions. If: H Z = x0 + sin/3 and, writing Z as Z(t;tn), if: then one can show that: i r n * ,-n:n fl>/*('-')(«-7)('-l) + >/*ii(«n-«)(*n-/)(*n-l) i cos p — \ sin p t-hi (5.74) 2 = Z(t;t22), (5.75) z - Z = ih{A-\) - (5.76) z - Z = -ihA . Z and 2 are purely real on the surfaces of the slits. The integrands of the last two integrals of Equation 5.73 can now be rewritten: 2 7 9 x{A-l-$) + ih{A-l)A = x{A-l) - ZA = X ~ 2 (5-77) x{A+l-A) - ihA{l-A) = xA + Z{l-A) = x--Z2 ih The Z2 product has only square-root poles at the airfoil and flap trailing edges. The RHS's of Equations 5.77 probably should not be used for numerical calculations involving Staggered Two-Element Thin Airtoil Theory / 5.3 122 small gap configurations since they become indeterminate as h —• 0. Nevertheless, they are informative. One way of writing the airfoil moment, then, is: ( c y + c 2 \ 2 + ( C i + C 2 \ Ay-fit m i o \ Cy J 2 M° V c\ ) 2 1 airfoil flap The Z f - ^ integrands can be handled by converting to one complex integral: ax ij-»yzzV(*)<fej. SR<J-»|> ZZw'{z)dz\. (5.79) The difficulty is with the remaining x — terms. They show that the problem of turning dx the integrals of Equation 5.78 into complex integrals i3 the same one encountered in the previous section. That is, since zn is not purely real on both slit surfaces, finding the complex versions of polynomial boundary conditions requires the development of new pro-cedures and/or functions. No doubt there is a solution to this problem, but it is left to be dealt with in some later work. CHAPTER 6 Conclusions and Recommendations The tandem thin airfoil theory of Chapters 2, 3, and 4 has shown how successful a lin-earization of the multi-element airfoil flow field can be. The theory's predictions of the aerodynamic characteristics are all analytical expressions, some of which are simple alge-braic functions of the initial geometry while the rest are calculable on a small hand held computer. It has been shown that the consistent relationship between the exact potential and linearized theories' overall force predictions extends, at least, into this region of the multi-element regime. This, in conjunction with the arguments of Chapter 1, suggests that a thin airfoil theory analysis of a properly designed multi-element airfoil configura-tion would give overall force predictions which correlate with the real flow forces as well as and probably better than is the case in one-element theory. In addition, for a + r\ values less than 20° or 30°, these predictions should be better than those of exact potential flow theory. i As mentioned in Chapter 1, however, the breakdown of the correlation be-tween thin airfoil theory and the real flow can be abrupt and come without warning since, of course, the linearized theory does not model the boundary layer effects that are respon-sible for this breakdown. In addition, the user must have other input in order to judge just what constitutes a "properly designed configuration". While not insignificant, these 123 Conclusions and Recommendations / 6 124 disadvantages are the same ones with which the successful exact potential and one-element linearized theories must also contend. The above expectations of the correlation between the multi-element lin-earized and real flow results have yet to be fully proved. This will require calculations of the overall force predictions for the staggered airfoil elements of Chapter 5 in conjunction with a solution of the staggered slit mapping. Comparisons with experiments on realis-tic slotted flap configurations, such as presented by Foster et al (1970), should then be possible. It is unfortunate that the simplicity of the tandem slit mapping is lost when the more realistic staggered slit geometry is accounted for. However, this author does not see this as a major problem and suggests that a form of Newton's method could be success-fully applied to Equation 5.15 to obtain the mapping parameters. The assembly language programming capabilities of a small computer such as the TI CC-40 might be useful for achieving the necessary reduction in computation time. Given a solution to the mapping, the staggered two-element thin airfoil the-ory force predictions can be evaluated from the expressions given in Chapter 5, with the overall forces for the problems of incidence and flap deflection having simple analytical solutions. The general effects of mean line curvature and thickness on the overall forces are summarized in the integrals of Equations 5.45 and 5.50. These equations are proba-bly as efficient a way as any of handling numerically specified boundary conditions. The analogous expression for the localized airfoil lift is given in Equation 5.69. The airfoil moment integral (Eq. 5.78 with the integrands replaced with the LHS's of Eqs. 5.77 for small gaps) can certainly be used as presented here, but it would be preferable to put it in a form where the non-boundary condition factor in the integrand was at least expressable as an algebraic function of the f-variable—for any gap size. It would also be desirable to obtain analytical solutions for the incidence and flap deflection problems from each of the localized force integrals. Conclusions and Recommendations / 6 125 A time constraint has prevented the completion of the staggered slit problem to the degree desired. However, it is felt that there are no major obstacles to this completion and that, if done, the results might very well suggest a simplification of the general theory as presented in Chapter 5. With modern computational hardware, engineers today tend to concentrate on the numerical approach, using powerful computers to get quick numerical answers. The approach used in-this thesis has been analytical, making extensive use of conventional elliptic functions and their supporting documentation. This has led to direct expressions for a comprehensive set of doubly connected flow field characteristics, as well as very efficient procedures for evaluating some of these characteristics. In this case, at least, using modern computational hardware together with powerful analytical techniques has provided readily accessible numerical answers, as well as a better overall understanding of the problem. 126 References Abbott, I.H. and Greenberg, H. (1939) Tests in the Variable-Density Wind Tunnel of the N.A.C.A. 23012 Airfoil with Plain and Split Flaps. NACA Report No. 661 Abbott, I.H. and von Doenhoff, A .E . (1959) Theory of Wing Sections. Dover Publications, New York Birnbaum, W. (1923) Die Tragende Wirbelfache als Hilfsmittel zur Behandlung des ebenen Problems der Trag&ugeltheorie. ZAMM, vol. 3, p. 290 Foster, D.N., Irwin, H.P.A.H., and Williams, B.R. (1970) The Two-Dimensional Flow Around a Slotted Flap. ARC R M 3681 Garrick, I.E. (1936) Potential Flow About Arbitrary Biplane Wing Sections. NACA Report No. 542 Glauert, H. (1926) The Elements of Aerofoil and Airscrew Theory. Cambridge University Press Halsey, N.D. (1979) Potential Flow Analysis of Multielement Airfoils Using Conformal Map-ping. AIAA Journal, Vol. 17, No. 12, p. 1281-8 Hess, J.L. and Smith, A.M.O. (1967) Calculation of Potential Flow About Arbitrary Bodies. Progress in Aeronautical Sciences, Vol. 8, Pergamon Press Ives, D.C. (1976) A Modern Look at Conformal Mapping Including Multiply Connected „ Regions. AIAA Journal, Vol. 14, No. 8, p. 100G-11 • Kennedy, J.L. (1977) The Design and Analysis of Airfoil Sections. Ph.D. Thesis, University of Alberta Millikan, C B . (1930) An Extended Theory of Thin Airfoils and Its Application to the Biplane Problem. NACA Report No. 362 Munk, M . M . (1922a) General Theory of Thin Wing Sections. NACA T.R. No. 142 Munk, M . M . (1922b) General Biplane Theory. NACA Report No. 151 Richardson, J.R. (1981) Aerodynamic Forces on Perforated Bridge Decks. National Maritime Institute, England, R 118 Stewart, H.J. (1942) A Simplified Two-Dimensional Theory of Thin Airfoils. Journal of Aeronautical Sciences, Vol. 9, No. 12, p. 452-456 Theodorsen, T. (1931) Theory of Wing Sections of Arbitrary Shape. NACA Report No. 411 Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis. 4 t h edition, Cambridge University Press Williams, B.R. (1971) An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils. ARC R M No. 3717 Woods, L.C. (1961) The Theory of Subsonic Plane Flow. Cambridge University Press APPENDIX A Exact-Numerical Potential Flow Computations This appendix describes the numerical procedure used in this thesis to obtain predictions of the exact, two-dimensional, incompressible, potential flow about multi-element airfoil configurations. The procedure is a slightly modified version of one of the more efficient surface singularity methods, by Kennedy (1977). A . l T H E F L O W FIELD F O R M U L A T I O N In Kennedy's method, airfoil surfaces are represented by a series of straight line segments over which vorticity is distributed. The ends of these segments are the specified coordinates of the airfoil while the middle of each segment is a control point at which the tangential flow boundary condition is applied. This boundary condition can be accurately satisfied by using the stream function formulation for the flow and requiring the value of the stream function at each control point of the A;th airfoil element to be the same (rpk say). The Kutta condition is satisfied by requiring this V"jt streamline to also go through a point just off the trailing edge, on the bisector of the trailing edge angle. The general stream function formulation for an arbitrary point (x, y) in the flow field around some multi-element airfoil configuration is: ^(x, y) = ycosa — xsina + / l{S) ln(r(x, y\S)) dS . (Al) 27T J S 127 Exact-Numerical Potential Flow Computations / A.1 128 Here, the flow at infinity is at an angle a to the z-axis; S is all the surface in the flow over which the dimensionless vorticity, 7(5), is distributed (the equation has units of length, each side having been divided by the magnitude of the velocity at infinity); and r(z, y;S) is the distance from a point on S to [x,y). This integral equation is solved numerically by dividing surfaces S into N straight line segments and then assuming that 7(5) is constant over each segment. Thus: 1 N f V'K y) » y cos ct - z sin a + — 7y / In (r(z, y; 5y)) dSj (A2) S} so that at the i t h control point on the fcth airfoil element, the boundary condition becomes: N ipk = y,- cos a — Xf sin a + ^ IjKij > t = l,2, . . . , i V (A3) 3 = 1 where the Jf.-y's are the influence coefficients and are calculated from the geometry of the airfoil configuration. They are given in Kennedy's thesis. In this problem, there are N+M unknowns: N 7y's and M V'jt's, where there are M airfoil elements in the configuration. The additional M equations are the Kutta conditions. They are the same as Equations A3 but with (z,-, y,) replaced with (z t p, y t p), the coordinates of the trailing points. The N + M equations are linear in the N+M unknowns and so can be solved through a solution of an (N + M) x(N + M) matrix. This formulation has the added advantage that the matrix solution gives the 7y'8 directly and, since it can be shown that the velocity of the flow inside the bound-aries of the airfoil elements is everywhere zero, 7(5) is also the velocity distribution over the surfaces of the airfoil elements. Pressures are then easily calculated and, as Kennedy shows, if Cp. = 1 - |7y|2 is taken to be the pressure at the control point (z;-,yy), then it will closely predict the true value of the pressure at that point if each airfoil element is represented by a sufficient number of segments (about 30 according to Kennedy). Also, since the sign of 7y gives the direction of the velocity over the jth segment, it is simple to find the stagnation points and verify that the Kutta condition is working. Exact-Numerical Potential Flow Computations / A.2 129 The approach used in this thesis differs from the above approach only in the way in which the Kutta condition is applied. Since airfoil surfaces are discontinuous at the trailing edge, it is inevitable that the segmentation of the boundary results in the trailing edge being the junction of two straight line segments. Hence, it is not a control point and, therefore, it is not redundant to specify as the Kutta condition that the ipk streamline go exactly through the trailing edge. This simplification eliminates the need to calculate trailing point locations and gives essentially the same numerical answers as Kennedy's method. Care must be taken, however, in evaluating 2M of the Jf,-y influence coefficients since each of the M trailing edge control points is exactly on the ends of two of the Sj segments, resulting in OlnO and Otan-1(0/0) terms in the expression for This trailing edge Kutta condition has been found to work well for slotted flap airfoil configurations at angles of attack up to a = 90°. In one instance at this high angle of attack, it was found that the flow around the deflected flap had actually reversed, with the ipf streamline first encountering the flap at the trailing edge and leaving it somewhere on its upper surface. Despite this, the circulation determining condition, that requires the ipli streamline to branch into or out of the body contour at the trailing edge, still worked. A.2 T H E FORCES The remainder of this appendix deals with the overall and localized force calculations. Ob-taining the overall forces is a reasonably straightforward procedure. The Kutta-Joukowsky law, for example, gives the overall lift in terms of the total circulation around the con-figuration, which is just the sum of all 7y£y products (£y being the length of segment j). Kennedy only needed to calculate the overall lift and did so in this manner. The localized forces are more difficult to calculate. They are usually obtained through a numerical inte-gration of the surface pressures including, perhaps, a curve fit of the pressure distribution to improve accuracy. Exact-Numerical Potential Flow Computations / A.2 130 In this thesis, the forces are all calculated the same way, by using the Blasius equations to solve for the exact expressions for the forces in terms of the 7y's, so that no error is introduced beyond that which already exists in the approximation of Equation A2. To begin, the exact formulation for the flow, Eq. A l , is rewritten in its complex form. If the stream function is combined with its harmonic conjugate, the velocity potential, to form the complex potential fl(z) = <f> + iipt then: 0(«) = ze~ia + ± J 7(5) ln(* - zs) dS (A4) s where zs is on surface 5 in the complex plane z — x+iy. The segmentation approximation is: j 7(5)ln(* - zs)dS*Yl7y Jln(z - zs.)dSj (A5) S i=l Sj and, using the notation of Figure 20, the complex potential is: 0(«) « ze~ia + JL £ Tj j Hz - z() d£ . (A6) J o Note that this approximation does not necessarily require 7,- to be constant over all of the j t h segment. The mean value theorem, for example, allows Equation A6 to be exact for a given z. It is by dealing directly with Equation A6 and avoiding unnecessary assumptions about the physical meaning of 7y that the expression's accuracy is fully retained in the following force calculations. Performing the integration yields: 0(*) = ze~ia + JL £ 7y[(*-*i>-''> ln(*-*iy) - ) « - ' ' > l n ( * - « * , ) - (,•]. (A7) J The real part of the square-bracketed expression gives the matrix influence coefficients, the if,y's, when z = z,-. The dimensionless complex velocity is: " ( * ) = ^ = e-' a + ^ E V " J " (A8) y Exact-Numerical Potential Flow Computations / A.2 131 2 U z - PLANE FIGURE 20 The control point z}- in the middle of the jth segment The Blasius equations for lift and moment are: D-iL=^\w0o\2eia fw2{z)dz M0 = £\tvQO\2Vl{ j>zw2(z)dz |y  dz^ (A9) (A10) where the drag is taken to be parallel and the lift perpendicular to the flow at infinity. These expressions can be evaluated by residues using the expansion, as z —» oo: real terms + Thus: (All) (A12) (A13) ( c i + ' - H w ) 2 ^ ' ^ V - J — • ( A H ) Calculating the localized forces on the k^ airfoil element is less straightfor-ward. For the lift and drag: cA = o C, = 1 ( c i + - •+cM)Zr - j w2{z) Ck dz (A15) Exact-Numerical Potential Flow Computations / A.2 132 This expression is broken up into two integrals, as suggested by the form of Equation A8 in the integrand. The first integral is evaluated by residues: 7i = / Ev " " ' M z- 2ii) - ln(*-J&,)]<fa = 2 i r t £ ; 7 / y . (A16) This is correct since, for j £ kf there are no singularities within C/. and so no contribu-tions to the integral; for j € k, can expand out to infinity, allowing the expansion of Equation A l l to be used in an application of the residue theorem. The remaining integral is: /2 = /EEw-i,w In z-zl}. In z-zX} dz. (A17) It is convenient to break up the double summation as follows: (A18) £X>££+E£+££+££ • j J jek Jek jekHk &k Jek j& J$k The last double summation term on the RHS of this equation does not contribute to I2 since it has no singularities within the Ck contour. Neither does the first term make a contribution since: real terms \ E£~°(J as z —• oo \ z* J jek Jek The middle two double summation terms are equal. Thus, I2 can be rewritten J 2 = 2EE^ / G W N * - * i i ) - M * - * 2 , ) ] dz (A19) ftk Jek lk where G(z) is regular within and on Ck but has singularities elsewhere which prevent Ck from expanding out to infinity. The integration is carried out by joining zlf and z2j with a branch cut and integrating around it. One gets: *2j j v7(s)[ln(*-*1(,) - m{z-z2])] dz = 2JT» j G{z) dz . (A20) This expression can now be evaluated conventionally: *2, 2fft<T'(«/+M j [ln(z-z l y) - ln{z-z2.)]dz = 2iri(Zj3- - Zjj) (A21) Exact-Numerical Potential Flow Computations / A.2 133 where: Zjj = Xjj - iYjj = e -i(Oj+°j) z2j In - Z l In- - ± (A22) Thus, the forces are: C*. = ^ LI1IJ\EV[™«(XJJ-XJJ) - 8 i n a ( y j y - F y j ) ] } (A23) ^ = f E { 2£j - ^ £ 7y [cos a (y j y - Y}J) + sin a ( X j y - JTyj)]}. (A24) k Jek K j&k J The localized moment is calculated in the same manner. If: W j ^ U j j - i V j ^ e - ^ ' * ) 2 ^ In - zf, In zlj-Z2} (A25) then: C.,., — (A26) The localized force "influence coefficients", the Zjj's and Wjj'a, are most easily calculated simultaneously with the matrix influence coefficients, the iif.y's, since they are all calculated from common geometrical parameters. The efficiency of this calculation is improved if one takes advantage of the fact that, if: HJJ = In ^ and ujj = zXj In (A27) then: and: Zjj ~ ZjJ = ~ Pjj ~ A*y+i,J + h j ] e ~ i [ 0 3 + 9 j ) (A28) (A29) Note the J *-» j symmetry in these equations. This is why, when all the localized force coefficients are added together to form an overall coefficient, these terms all sum to zero. Exact-Numerical Potential Flow Computations / A.3 134 A.3 C O M P U T A T I O N A L A C C U R A C Y The numerical calculations are performed on UBC's Amdahl 470 mainframe computer. All calculations are made in double precision (numbers are represented with 16 significant figures). An indication of the accuracy of the present method (for which forces are cal-culated using Eqs. A12, 13, 14, 23, 24, and 26) is obtained by comparing its overall lift predictions to Williams' (1971) exact solutions for the potential flow around a slotted flap airfoil configuration. This is done in Table 3, where the numerical predictions are made by 1) using all the coordinates specified by Williams (61 segments per airfoil element) and 2) only using every other of the coordinates (31 segments per airfoil element). Ci error Williams, exact a = 0° 61 segments/element 31 segments/element 2.7230 2.7180 0.18% 2.7240 0.04% Williams, exact a = 10° 61 segments/element 31 segments/element 3.7440 3.7389 0.14% 3.7434 0.02% TABLE 3 Comparisons of the exact and present numerical predictions of the lift on Williams' conBguration A slotted Bap arrangement. « The accuracy of the present method is seen to be quite good. However, its predictions do not appear to be converging to the exact values as the number of segments per airfoil element (Nk say) increases. An explanation for this anomaly is found in the following investigation. Figures 21 to 25 show in greater detail how the numerical force predictions on a slotted flap airfoil configuration vary with Nk. The configuration consists of NACA 23012 airfoil elements in an arrangement similar to that of Williams' configuration A, with the Exact-Numerical Potential Flow Computations / A.3 135 airfoil's chord line stretching between 0 and 1 on the z-axis, the leading edge of the flap located at (.99,-.02), the flap chord length equal to 0.373, and the flap deflected 30°. a = 0° for these calculations. The analytical expressions for the 23012 profiles given in Abbott and von Doenhoff (1959) make it easy to segment the airfoil boundaries at will. Throughout this thesis this segmentation is done in the same way, by dividing the chord line up according to equal increments in 6 (Eq. 1.14) and then finding the profile coordinates opposite those divisions. This provides a concentration of smaller segments around the leading and trailing edges of the airfoil element boundaries. Figures 21 to 25 show that the force predictions of the present theory do indeed converge to a constant value as Nk becomes very large but that, initially, the predictions have a tendency to oscillate. This is the probable explanation for what occurs in Table 3. That is, the good agreement between the numerical and exact lifts at Nk = 31 is only coincidental and, as can be seen from Figure 21a, cannot be counted on to repeat itself in other situations or with other profiles. Note that in the interpretation of Figures 21 to 25, the not unreasonable assumption is made that the asymptotes to which the forces are converging are identical to the values of the exact forces. These figures also show the force predictions of a simpler but cruder analysis in which all the forces are calculated by integrating the pressure distributions and assuming Cp. = 1 — |7y|2 is constant over all of segment j. These force predictions appear to converge to the values of the present theory as Nk becomes large but are generally less accurate for a given Nk, especially for the drag calculations. All of the "present method" force predictions for Nk > 60 in Figures 21 to 25 are within 0.2% of their values at Nk = 100. Throughout the thesis, this is the method and degree of segmentation used to calculate exact-numerical potential flow forces. The accuracy is needed for making accurate predictions of the variation of these forces with small increments of a and n. ExactrNumeric&l Potential Flow Computations / A.3 136 2.22 C , 0% 2.20 1 1 • • • • -1% 2.18 2.16 20 _1 L_ 40 60 80 100 a) Overall lift vs. Nf.. 0.08 • p r e s e n t t h e o r y • CPj a s s u m e d c o n s t a n t o v e r a l l o f s e g m e n t /. 0.06 0.04 0.02 • i T • *• 20 40 60 80 100 b) Overall drag vs. Nk. F I G U R E 21 The accuracy of the overall lift and drag exact potential flow numerical calculations as a function of the number of segments per airfoil element. NACA 23012 airfoil elements in a slotted Bap conGguration at a = 0°. Exact-Numerical Potential Flow Computations / A.S 137 2.44 0% "1 • 1 2.42 2.40 1.64 +1% -/2 1.62 1.60 1.58 -1% 1.56 1.54 20 40 60 a) Airfoil lift vs. Nk 80 100 •i S 1 1 1 p r e s e n t t h e o r y CPj a s s u m e d c o n s t a n t o v e r a l l o f s e g m e n t / F I G U R E 22 0 20 40 60 80 100 b) Flap lift vs. Nk The accuracy of the localized lift numerical calculations. NACA 23012 airfoil elements in a slotted flap configuration at a = 0°. Exact-Numerical Potential Flow Computations / A.3 138 -0.26 -1% +1% • " - - - -• • w w z 1 1 • -0.24 •0.22 -0.20 • present theory • CPj assumed constant over all of segment /. 0 20 40 60 80 100 a) Airfoil drag vs. Nk. 0.72 0.70 0.68+1% •1% 0.66 • m -I 1 1 • • - • 20 40 60 80 100 b) Flap drag vs. Nk. FIGURE 23 The accuracy of the localized drag numerical calculations. NACA 23012 airfoil elements in a slotted Sap configuration at a = 0°. Exact-Numerical Potential Flow Computations / A.3 139 • present theory • CPj assumed constant over all of segment / -0.92 -0.90 -1% •0.88 +1% -* "I 1" »--0.86 20 40 60 80 100 F I G U R E 24 The accuracy of the overall leading edge moment numerical calcula-tions. NACA 23012 airfoil elements in a slotted Bap conBguration ata = 0°. Exact-Numerical Potential Flow Computations / A.3 140 •1.02 -m -1.00 10 -0.98 •1% +1% •0.96 • present theory. • Cp j assumed constant over all of segment /. 1 1 1 * 20 40 60 80 100 Nk a) Airfoil moment vs. iVj.. -0.58 •0.56 - • -m 20 -0.54 +1% -0.52 - • • • 1 _L _L 20 40 60 Nk 80 100 b) Flap moment vs. N^. F I G U R E 25 The accuracy of the localized leading edge moment numerical calcu-lations. NACA 23012 airfoil elements in a slotted flap conBguration at a = 0°. APPENDIX B Elliptic Functions In this appendix, the reader is presented with the various properties of elliptic functions needed for the present theory. In the first section, some of these properties are summarized from Chapters 21 and 22 of Whittaker and Watson (1927) (often referred to as "WW" fol-lowed by a section number) and they are then used to develop further simple relationships. In the remaining sections, more complicated relationships are developed for the evaluation of the two-element thin airfoil theory lift and pitching moment coefficients. In the tandem airfoil mapping, Eq. 2.3, elliptic integrals of the first kind, which can be written in terms of an inverse Jacobian elliptic function, are encountered. The Jacobian elliptic functions, sn A, cn A, and dn A, are doubly periodic with respect to their argument A, leading to cells of periodicity in the A-plane. These periods are described in terms of K and K' are determined via their moduli, k and k', for which numerical values are known in the tandem airfoil theory. The parameters of the rectangle, Fig. 4c, can all be calculated by using either conventional series in k or A;', tables, or Landen's transformation which is a repetitive way of reducing an elliptic integral to an elementary function. None of these procedures, however, are as efficient as the ones described in WW and those developed in this appendix. B . l G E N E R A L and K', the complete elliptic integrals of the first kind (; i.e. sn A = sn(A ±4K± UK')). K Ml Elliptic Functions / B.l 142 When numerical results are required, the elliptic functions to use are the four theta functions: t?1(ur, g) = 2q« sin u — 2q* sin 3w + 2q~* sin 5w i \ 1 9 21 V2[u,q) = 2q* cos a; + 2q* cos 3a; + 2g < cos5a; H ^3(w» q) = l + 2q cos 2a; + 2g4 cos 4a; + 2g9 cos 6a; H t?4(a;, g) = 1 - 2q cos 2a; - f 2q* cos 4a; - 2g9 cos 6a; H where, in keeping with most of the notation used in WW: TTA _*KL .K' It is interesting that: = i?3(2oM4) + t?2(2o;,g4) (Bl) (B2) vi{uiq) = #3{2uj,q*)-v2{2u,q4). The theta functions, while periodic in ir or 2TT in the ^-direction, are not perfectly periodic in the ^ -direction and so are referred to as quasi-doubly periodic func-tions. If i?'(o;) is the derivative of t? with respect to a;, then for any of the theta functions these periodicities give (WW21.11): 0' 0' tf' -(a;+*r) = -(a/) ; -{U+KT) = -2i + -(w) (B3) where t?(a;) is written for 0(utq) and " (^w) for . However, since the flow functions must be periodic in JTT, Eq. 2.18, the following function, which has this required periodicity, is of interest: 7 M + 5 i T - (W) Alternatively, if the theta functions are defined relative to the a/ variable, Fig. 4e, and the q' parameter, then: j(u'+ir,q') = (B5) Elliptic Functions / B.l 143 is also a possible flow function. In fact, Jacobi's imaginary transformation (WW21.51) gives: , „ 2u K'fi' , ( B 6 ) 0' The -r^ -(w) function has a simple pole at the origin of the w-plane which can 0 1 be transferred to the other corners of the rectangle using the half-period properties of the theta functions (WW21.11): Now, evaluating the theta functions at their zeros (WW21.61), gives: V 2 /oirxVa /ivic\W and: 0i(O) = 02(O)03(O)«/4(O). (B9) Using Equations BI and B8, then, K can be related to the q parameter through an infinite series. Also, since y/k1 = t/4(0)/t73(0) and using Equations B2, an expression for q in terms of a known parameter, e0, is developed (WW21.8): 9, = l~y/k~' - *»(o.g ) - *4(Q,q) _ 02(0,g4) m i m " - l + v/it7 U^) + U0tq) «/3(0,?4)' 1 ' This infinite series for e0 in powers of q can be inverted to give: q = c0 + 2e50 + lSe90 + 150e£3 + O (lOOOeJ7) . (Bll) Thus, q is calculated using Equation B l l , K is obtained as above, and: K' = -—lnq. (B12) Elliptic Functions / B.l 144 For typically small values of f—s, these are all very rapidly converging series. As f — s —* 0, k —> 0, k' —> 1, and q —» 0; for most configurations q < 0.1. However, even if A: is as large as 0.999 (so that q = 0.33), these series will give four significant figures. Note that for small slots, y/k1 will be very close to 1 and so the number of significant figures in a calculation will be reduced when evaluating Equation B10. This problem is avoided by using Equations 2.48a &b. The Jacobian elliptic functions can be written in terms of theta functions (WW22.11): °^  = OT; ^ =yfS; ™=^£R- (B13) Also: d d d , — sn A = cn A dn A ; — cn A = — sn A dn A ; — dn A = —k sn A cn A oA dX dX so that, taking the derivative of the logarithm of Equations B13: cn A dn A sn A sn A dn A (B14) cn A dnA 2K IK (B15) IT 2K —[to] • A suitably chosen combination of quasi-doubly periodic functions, then, will yield a doubly periodic function. Also, taking the derivative of a quasi-doubly periodic function can do the same. From WW22.73, Equations B15, and using Legendre's relation: EK' + E'K - KK' = - (B16) one can show that: d dX 2ui — + 2E' 2K' 1 TT sn2A d dX — + _ TC 2E' 2K'dn2X 7r cn2 A d dX '2ui — + TT - % ) 2E' 2K'k2cn2X v dn 2A d dX '2ui + 7T K^U\ 2E' n k sn A TT (B17) Elliptic Functions / B.2 145 Here, E and E' are complete elliptic integrals of the second kind and, as with K and K\ they are the same function of their respective moduli k and k': l E o The easiest way to evaluate E is through one of its three q series representa-tions. Using versions of the last three of Equations B17, Eqs. B l and B8, and evaluating the expressions at u = 0 gives: I = (2T)V2 7=k (2?1/4)(1+"2 + 2 5 , 6 + 4 " 1 2 + " 0 ( B 1 9 ) I = ^ + 8 ( 2^) V 2 (* + V + " 9 + 1 6 ^ C + * * ) ' E' is then obtained through Legendre's relation. B.2 ELLIPTIC F U N C T I O N SERIES REPRESENTATIONS IN POWERS OF q4 In Chapter 2, Eq3. 2.47 & 48, the a and rj problems are solved in terms of the three constants Gi, G2, and E 3 . Gx and G2 are simple functions of K' and E' and are evaluated with the series representations of section B . l . E 3 , however: _2Xco ,K'0' requires evaluating both an elliptic integral of the first kind (Eq. 2.11) to obtain x<x» an^ 0' ' then the - ^ ( X o o ) function, which is similar to an elliptic integral of the second kind. This 03 can be done conventionally, using Landen's transformation to get Xoo a n c * * n e n Fourier series for -^-(Xoo) (WW21, chapter problems), or the following more efficient procedures can be used. Inverting a Jacobian elliptic function, which is equivalent to evaluating an elliptic integral of the first kind, can be accomplished by exploiting the theta function Elliptic Functions f B.2 146 representation for dnA, Eq. B13, in conjunction with Equations B2. The first step is to obtain dnA, by using Equations 2.10 if necessary. Then: 2 e _ dnX-Vk1 _ 02(2u;,g4)  € w = dnA + VF " 03(2u/,g4) 2q cos 2w + 2g9 cos 6w + 2g25 cos lOw H 1 + 2g4 cos 4u/ + 2g16 cos 8w + or: cos 2w = — 1 + 2g4 (2 cos2 2u>-1) - g 8 (4 cos2 2w-3) ] 9 By continually resubstituting this equation into itself, a series for cos 2w in powers of g 4 and in terms of: is obtained: £ = ^ (B21) cos2w = tf[l - g 4 ( 2 - 4 £ 2 ) + g 8 ( 3 - 2 0 £ 2 + 3 2 £ 4 ) - g 1 2 ( 6 - 7 6 £ 2 + 2 7 2 £ 4 - 3 2 0 £ 6 ) +0(g 1 6)]. (B22) Note that u can be real or imaginary. For u real, it can be shown that |£| < 1 and, therefore, that the coefficients of powers of g 4 in Equation B22 are no worse than the coefficients of powers of CQ in the expression for g/c0, from Eq. B l l . To solve for u = x<x» o n e g e * s : *«» = 1r=— (B23> V F + 1 since dn fi^ — s/f. Using Equations B21&22 completes the procedure. 0' Evaluating the -^{v) functions is easily done using their Fourier series representations. For example: W (B24) n=l which converges in powers of g. However, since the Jacobian elliptic functions of Xoo 0' are known (Eqs. 2.9), - ^ ( X o o ) c a n D e evaluated with a series which converges in powers 03 Elliptic Functions / B.3 147 I?' of q4. Although the following procedure is applied to the -r^ (w) function, any of the other v3 —(u>) functions can be included by first converting them to the -r (^w) function using Equations B15. Equations B2 can be rewritten as : i/3(o;,q) + 04(w, q) = 2dz{2u, qA). Taking the derivative of the logarithm of this expression and rearranging gives: t?4(w,g) 2/f , snAcnA Ar or: = d n X - + 2 ^ ^ ' CB2B> For E 3 , then: 2 X o o 2*' , K'^{-Yq^smAnx n = l + 8 t f ^ f T ^ • (B26) Again, u can be complex in these equations but this would reduce the rate of convergence of the series representation for -r3-(2w,g4). This problem is overcome in the next section. v3 B.3 ELLIPTIC F U N C T I O N SERIES REPRESENTATIONS IN POWERS OF q16 In Chapter A and Appendix C, first, second, and third type elliptic functions with imaginary arguments are inverted and evaluated. While Equations B22 & 25 can be used for the first two of these computationsif f—s is small enough, they cannot be used over as large a range of f—s values as is required in this thesis. In this section, the procedures of section B.2 (that gave the q* series) are applied twice in succession to get series which converge in Elliptic Functions / B.3 148 powers of q16. This process begins with the derivation of a more efficient version of the series representations for qt Eq. B l l . In the same way that k determines q, Landen's transformation (WW22.42) shows that: 1 - k' ki = — determines q2 1 - k' k2 = JJ determines g 4 1 + kx etc. In fact: Now, if e02 is defined: ko — *4 (l + A ^ l + v/* 7) 4 ( 2 e ° ) 4 ' 2, _ l - V ^ = 0 2 ( O , g 1 6 ) then, from Eqs. BIO and B l l : (B27) so that: 94 = e02+2€g2 + 15eg2 + 150e05 + .- . 1/4 1 , 1^4 117^8 , n(„M\] 1 + 2 % - l 2 - £ 0 2 +0{q (B28) where: 1/4 2jo °2 ~ r . . . . . — x 2 T l / 4 (B29) [2(1 + k'2) (1 + y / % ) 2 ] As before, to invert a Jacobian elliptic function one first obtains dnA; 2ew is then given by Equation B20. However, since: 02{u,q) t/2(0,<7)cn(A,A:) 2cw can be rewritten as: t/3(0,9)dn(A,A:) ' cn(A2,fc2) 2ew = 2e0 which, when squared and rearranged gives: dn2(A2,A:2) dn(A2,fc2) L/2 1 - ( 4 * o 0 2 (B30) Elliptic Functions / B.3 149 Now, in the spirit of Equation B20: 2e = dn(A2,A;2) - \A£  w a " d n ( A 2 > A 2 ) + > / ^ so that, as in Equation B22: cos 4w = Lw2 1-? 1 6 |2 + However, cW2 and g 4 both have terms in them O(CQ) which should be cancelled before their ratio is calculated, g4 is given by Equations B28 & 29 and 2eW2 is rearranged as follows: 2e,.,„ = dn4(A2, A2) - k'2 with: so that if: " 2 [dn(A2,k2) + y/lcl]2 [dn 2(A 2 ,k 2) + k'2] dn4(A2, A2) - *22 = ^ M ( 2 e 0 ) 4 \% g^)' - 1 - ( 2 £ J 4 c2 = (1 + A2)(1 + V ^ ) 2 [ 2 ( ^ ) 2 - 1 - ( 2 6 W ) 4 1 + \ 2 > ^ ) J L + dn(A 2 ,* 2)J 2 (B31) then: and: 2f 2 9 dn 2 (A 2 ,^)J f ' ( 2, ? = £ 2 [ l - 2 g 1 6 + 5g32 + 0(g4 8)] cos4UJ = (2{l + 4g16[el-l] + 4 g 3 2 [ 3 - l l £ 2 + 8 £ 2 4 ] + O (lOO^g 4 8)}. (B32) An investigation of the characteristics of £ 2 m the rectangle of Figure 4d reveals that the only very active term in it is the expression: 2 GO 1 ( 2 0 4 When u = |, is zero. For w real, |ew| < e0 and |£2| < 1. However, from Equation B20 it can be seen that cw has a pole at u = | ± 27rr, so that in the rectangle (—^ < 17 < T^) it Elliptic Functions / B.3 150 will take on its maximum magnitudes along u = x± although this is still some distance from the pole. At w values of | ± ^ , and § ± y-, |ej = 1 so that £ 2 = 0{l~2)-Thus the maximum error in Equation B32 for u> anywhere in the rectangle is O(l00g3 6). To obtain XOOJ o n e again uses the fact that dn/x^ = s/f to calculate ew and then Equations B27&30 to get £ 2 and Equation B32 for cos4w. Since ew is odd in x about u — J, one can use its sign to determine which value to choose from the arccosine operation that gives u>: €w > 0 for 0 < Au < n for u real. (B33) €w < 0 for TC < Au) < lit To calculate u>i = t'71 and w2 = | +i*72, corresponding to xx (0 < x± < s) and x2 (/ < x2 < 1), one makes use of the fact that: s ( l - x t ) dnA>' = V ( i ^ ; f o r ° ^ x ^ 1 (B34) so that: _ y / A j l - x J - y / x j Sf For j = 1 or 2, cos4u>y = cosh 47y and, for 0 < »7y < ^ : cosh2 47y + yjcosh2 47y — 1 7y = ^ ln (B36) 0* When evaluating the •^ •(w) functions, the procedure to follow is the same as in the previous section up to and including Equation B25. Then: 2K2 2 sn(A2,A:2)cn(A2, k2) k2 where, from WW22.42: V ^ 2 (1 + Vk1)7 K2 = K . Also: dn(A2,A:2) 2e0 2 ' V 1 - ( 4 e o 0 2 Elliptic Functions / B.3 151 The sign of the sn function is determined by where w is in the rectangle. If: 2 then: ™ S ± V , _ ( & ) (B37) * J " ' IT 1 + I £3 dn(A2, fc2) When w is real, as when E 3 is being evaluated, and when 0 < x 5; §> as is always the case in the rectangle, then sn(A2,A;2) is real and positive and the + sign is chosen in T(u;). When u = t'7 or u = | +1'7: T M = ±n/(g2-i. sn(A2, k2) is odd in 7 for x — 0 a n d X = §• For the top surface of the airfoil it is positive imaginary and for the top of the flap it is negative imaginary. However, 2ew/2e0 is always positive on the airfoil and always negative on the flap, sb that in Equation B38: 2 , , M . \ 2 2eQ K J 2en I V V 2en J " 2 ~ ~ 2 This allows the evaluation of A{u{) and A(OJ2) in Chapter 4 and Appendix C. The elliptic functions of the third kind that are evaluated in Chapter 4 contain the function: » l n 04 (Xoo-Wy) This is of the form: - In n- = -(2t6) so that: i, 04(Xoo-^y) 2 , u n l n M x ^ ) ^ ^ T e { M x - + U j ) } ' Elliptic Functions / B.3 152 Now, from Equations B2: MXoo+Uj) = 0 3 ( 2 ( X o o + W y ) , 9 4 ) " ^ ( X o o + W y ) , ? 4 ) t> 3(2(Xoo+"y),<?4) x ^(2(Xoo+^y),94) ^( 2 (Xoo+"y),? 4 ) = r 2 % /p^(2(Xoo+^y) ,g 4 ) dnC/x^+Ay) + \/A7 ' The numerator in this function is: kx - Hi = 1 + 2$ 4cos4(Xoo+"y),+ 2g 1 6cos8(Xoo+Wy) + 2g36cosl2(x00+wy) + and using the addition theorem for the dn function, WW22.21, the denominator is: /—. dn /ioo dn Ay — k2 sn /iQQ cn [AQQ sn Ay cn Ay R 2 - t I 2 = Vk + — - ^ - — ^ Thus, if: a [ «/ ( l - « ) (W)] T + Vxy(l-xy) " (d=t)|(xy-a)(xy-/)|> £j = 1 + 2g4 cos 4Xoo cosh 47y + 2g16 cos 8Xoo cosh 87y + 0 ( ^ . g 3 6 ) Ji = 2g4 sin 4Xoo sinh 4 7 y + 2ql« sin 8Xoo sinh 8 7 y + O ^ f .g36) R2 = + Vxy(l-Xy) 72 = ± | (xy - s) (zy—/) | 5 , + for airfoil upper surface — for flap upper surface (B40) then: i m t ? 4 ( X o o - " y ) = 2 a r ( R j - i J A * 1 1 « ? 4 ( X o o + W y ) * \ R2 - il2 j where u>y = i~fi or u/y = | +t'72 a ud 0 < :*7y < (B41) APPENDIX C The Selector Function Expansion Coefficients This appendix gives the coefficients for the large z expansions of the selector functions L(z)} A(OJ), 7(u), 5(w), and T(w) as well as for the flow function F(z). C l £(z) A N D F(z) £(z) is first used in section 4.1: ^ 2 ) = y ( z _ a ) ( z _ 1 ) ~ i + T + ^ + ^ - + " If v = 1/z, then: = 1 + 2CiV + ( 2 £ 2 + C] )v2 + ( 2 £ 3 + 2£ii :2)w 3 + _ 1 . o / t . /0/1 > /i2^.2 (1 -5V ) (1 -V) = 1 + [ l - ( / - s ) ] v + [ l - ( / - s ) ( l+ 5 ) ]v 2 + [ l - ( / - s ) ( l + 5+a2)]v3 + and equating coefficients of vn: 2 ^ = 1 - ( / - « ) 2 £ 2 = l - ( / - s ) ( l + s ) - £ 2 2 £ 3 = 1 - (/-s)(l+s+a 2) - 2 t i t 2 (Cl) , _ n - l 2 £ » = 1 - 7 - ^ ( 1 - 0 - £ 4 » 4 . - m , » > m=l 153 The Selector Function Expansion Coefficients / C.2 154 Although F[z) is used rather loosely in Chapter 2, it is given a unique definition in section 3.4: l(z-s)(z-1) , F, F2 F 3 , as z —* oo. Of course, F[z) = l/C{z) which leads to a simple recurrence relation for the Fn's in terms of the £n's. However, if F{z) is being used, C(z) is probably not being used, or vice versa, so that a direct relationship for the F„'s is desirable: 2F1 = /-(l+a) n - l 2 ^ = [/2-/(l+s) + s ] / B _ 2 - £ FmK-m , n>2. ( C 2 ) m=l C.2 AND 7{u) The airfoil selector A{yj) is introduced as Equation 2.38 in section 2.3 and is used again in Chapter 4, beginning in section 4.2. „, , , 2UJ K'A\. . . A >f2 Jlw =1 - T T T 1 " ~ 4 + — + + JT A 1/j z zl as 2 —• oo. Using Equations B7 and since = Xoo ± y It is convenient to define: w K #, 4 .o (Xoo) .o (Xoo) and using Equations B15, 2.5, and 2.9: £ 4 = £3 + y (C3) where (?i (and the soon to be used G2) are defined in Equations 2.37. The Selector Function Expansion Coefficients / C.2 155 Now: dA{u) _ dAju) dX dv dX dv where, again, v = 1/z. From Equation 2.1: dX _ y/fjl-a) 1 dv 2 N / ( l - s v ) ( l - / t ; ) ( l - v ) so that, using Equations B17: dAju) dv 2K' z(f-s) 2E' 1-fv TT f(z — s) JT Ai + 2X2v + 3A3v2 + (1 -av)(l - v) 1-fv or: 1-s G^l+sv+aV+sV-r-- • •) - G2 2 L / (C4) [l + £ i V + Z2v2 + £3v3 + •••] = Ai + (2A2 - fA{)v + (3X3 - 2fA2)v2 + Thus: A0 = 1 - E 4 i l i s -J~G1 - G 2 2A2 = ( /+s) Ai + Ai Ci + (C5) sG nAn = ( / + « ) ( « - l ) / l n _ i - af{n-2)An-3 + + 5 » > 3. The flap selector is introduced in section 2.2 and is used throughout the thesis: , , , 2u K'$'2. . _ Ti 72 as z —• oo. In much the same way as the A„B were obtained, one gets: 2Xoo K'v", _ G2 ~ Gl  1 ~ 2 2^ 2 = M + ?id + 1JY~GI nTn = / ( n - 1 ) 7n-i + + ^Gi{Cn.2 + f£n-3 + / 2 £ n _ 4 + • • • + fn~2) n > 3. (C6) The Selector Function Expansion Coefficients / C.3 156 The recurrence relations for the AN'8 and ^,'s are certainly not unique. The relation for the An's has another version similar to that of the ^,'s, and vice versa. The methods of Appendix B ensure the fast and accurate evaluation of the constants Gj , G 2 , and E 3 . C.3 S(w) A N D T(w) The airfoil segment selector is derived in section 4.3: T ^(W + H l ) » *J , C 7 ) /. 5i 52 ~ 5 0 H T - -5 - -1 , as 2 —> 00. z z l Here, 7 j is on the upper surface of the airfoil (0 < t 7 l < 22L) in the w-plane, Fig. 4d, and it corresponds to the point X\ which is on the airfoil in the z-plane (0 < < s). Now, So = 5(w00), = X o o ^ a n c * periodicity of the theta function * « 7 4 U o o + * 7 l ) * t ? 3 which can be evaluated using the methods of section B.3. Taking the derivative of S(UJ) with respect to z, or v — 1/z as is done here, leads to a doubly periodic function which can be represented by algebraic functions of z: dS(uj) dX „ „ „ . o ,\ ' — ~ 5i + 252v + 3S3tT + • • • , ass->oo. aA av The following expression from dS{w)/dX (from the first term on the RHS of Eq. C7) can be rewritten as follows (WW21, problem 18): I K ^ • ( ( j - u / J - ^ - ( w + w j TT t?'4 2sn A t cn A t dn Ax = " 2 2 T ^ + s n ' A — n ' A , ( C Q ) 2iy/{s-Xl)(f-Xl) where: \ / l - x x - y/Ax[ N / X 1 ( 1 - . T 1 ) N/x7+N/Z(l^x7) Z — Xt The Selector Function Expansion Coefficients / C.3 157 The last line in Equation C9 is in its most desirable form since all of its terms take on only moderate values for any value of xx between 0 and s. This, despite the square-root poles at Xy — 0 in each of the terms on the RHS of the previous line (they have cancelled out). The other expression from dS(uj)/d\ that needs to be considered is: dX K #2 ' K' TT 2E' 2K'dn2X 1 K' TT TC cn2A K 2E' f-s2K'z-l 1 (Cll) JT 1 — 3 TT Z—f K where I/J is associated with 7j and is the variable shown in Figure 4c. dX/dv is given in Equation C4, and so: 1 -s/{3-xl){f-xl)-T=^-—7== 2 if ' G 2 -K (C12) Now, if: (C13) and since, as z —• oo: 1 ~ , *, (i + +i:2v2 + £3t;3 H — ) 1 - sv){l - fv){l - v) l - f v K 1 then: (1 _ fv)^M „ [Si - MQv - Miv2 - M2v3 -...](! +Civ + £2v2 + -•) dv Si + (252 - fSi)v + (353 - 2/52)v2 + and: nSn = / ( n - l ) 5 n _ i + SiCn-i ~ M0£n-2 - M^n-z M n _ 3 £ , - M n _ 2 (C14) n > 2. Tie Selector Function Expansion Coefficients / C.S 158 The flap segment selector T(w) is given in section 4.3: T _i_ j_ ^ 2 _i_ (CIS) as 2 —> oo. Note that this equation is written slightly differently from Equation 4.54 since it is in terms of w2 = § 4- 172, (0 < t-72 < ^ ) . w2 corresponds to x2 o n * n e n a P m 2-plane (/ < x2 < 1). Because of the similarity between T(w) and S(w), obtaining the 7J,'s is straightforward. (C16) 'o = - in — r r r + ~ x " U o o ) 2 ~ G l F~~ 2 If If: = \^x2{x2-s){x2-f){\-x2)xn2 - — G ^ r then Tn and Nn replace Sn and M n , respectively, in Equation C14. ( C 1 7 ) (C18) APPENDIX D Computer Program: The Tandem Airfoil Forces This appendix lists the program used to calculate the tandem thin airfoil theory force predictions presented in Chapter 4 in Figures 10 to 17. The program incorporates all of the 13 steps listed on pages 85 and 86. The programming language is Texas Instrument's "enhanced basic", as described in the user manual for the TI CC-40. In order to run the program in the available 5.7 K of RAM memory, stringent memory conservation measures have been used (the number of lines and variables in the program have been minimized). The table preceeding the program, on the next page, lists the main variables and gives the final values they are assigned by the program. The program accepts four quantities as input: 1) the NACA mean line des-ignation for the slat (i.e., "230" for a 23012 airfoil or " 4 4 " for a 4412 airfoil), 2) the mean line designation for the flap, 3) the flap chord to total chord ratio Rf, and 4) the slot size f-s. Only NACA four and five digit profiles can be processed. The program finishes by displaying £ m a x . The effects of other mean lines (specified with appropriately sized polynomial boundaries) can also be accounted for by the program by manually defining values for the TTV array and " H " , " ! ! " , and " X 4 ' ' variables immediately before line 1 8 5 . This allows, for example, calculation of the effects of slats or flaps attached to the tandem airfoil elements. 1 5 9 Computer Program: The Tandem Airfoil Forces / D 1 6 0 MA = dCmJda/(*/2) AC — xac N = dCe/dri/(2ir) MN = dCmJdV/(*/2) XS = X»ac SA = dCJda/(2n) MSA = dCmJda/(*/2) XF = (*I*c -/)/(! SN dCJdr,/(2v) MSN = dCmJdr,/(*/2) M = dCmJdr, FA = dCtf/da/(2n) MFA = dCmJda/{*/2) MS = dCm,JdV FN = dCtf/dr,/(2v) MFN = dCmJdr,/(*/2) MF = dCmjJdri D = ACt DMO = DM = A C m o c DS A Q . DMSO = DMS = *cmiac DF = ACtf DMFO = m / o DMF = A C m mIae S = s SI = l-a FS = f-S Gl = Gx YI = = uJK* q = Q F = f F l = l - f CT = 3 + 1 - f G2 = G2 Y4 = = U2/K' ER = ^max R = Rf KK = 2K/ir X = = 2Xoo 100 RAD:DIM A(3,2),B(7.4),L(3.2) 105 PRINT "Enter NACA mean l i n e deBgnatna:":PAUSE:PRINT " S l a t : •; 110 ACCEPT SIZE(3)VALIDATE(DIGIT," ")JST:PRINT • ,Flap:";ST;:Q=ST:J=0 116 ACCEPT AT(18)SIZE(-3)VALIDATE(DIGIT. B "),FP:PRINT 120 INPUT "Enter R: ";R."Enter f - s : ";FS:PRINT ST;FP;R;FS; 125 IF Q=230 THEN M=.2025:K=1B.957:GOTO 165 130 IF q=210 THEN M=.068:K=361.4:G0T0 165 135 IF q=220 THEN M=.126:K=51.64:GOTO 165 140 IF q=240 THEN M=.29:K=6.643:G0T0 165 145 IF q=250 THEN M=.391:K=3.23:G0T0 165 ELSE IF H<1 THEN H=l 150 M=INT(q/10):P=(q-10*M)/10:M=M/50 155 L(J,0)-M/P:L(J,1)=-L(J,0)/P:J=J+1:L(J,1)=-M/(1-P)-2:L(J,0)=-L(J,1)*P 160 IF,J=3 THEN X4=P:G0T0 185 ELSE X1=P:J=2:IF q=FP THEN 165 ELSE q=FP:G0T0 125 165 L(J.1)=-K*M:L(J.2)=K/2:D=L(J,1)*M/6:L(J,0)=D*(M-3):L(J+1,0)=D*M:H=2 170 IF 3=2 THEN X4=M:G0T0 186 ELSE X1=M:J=2:IF q=FP THEN 165 ELSE q=FP:GOTO 125 175 BREAK:ER=0:X1=X1/S:X4=(X4-F)/Fl 180 INPUT "Enter f - s : ";FS:PRINT ST;FP;R;FS; 185 F1=R*(1-FS):F=1-F1:S1=FS+F1:S=1-S1:CT=1-FS:X1=S*X1:X4=F1*X4+F:F0R J=0 TO H 190 N=S~J:A(0,J)=L(0,J)/N:A(1,J)=L(1,J)/N:NEXT:F0R 1=2 TO 3:A(I,2)=L(I,2)/Fl-2 195 N=F*A(I.2):D=L(I,l)/Fl-N:A(I,0)=L(I,0)-F*D:A(I,l)=D-N:NEXT:H=H+2 200 K=FS/F/Sl:N=SqR(l-K):D=(l+SqR(N))"2:M=K/(l+N)/D:SA=M"4:P=SqR(l-SA) 205 SN=(1+P)*(l+SqR(P))*2:FA=SA/2/SN:FA=(FA~4*2+1)*FA:q=FA".25:FN=q*FA*2 210 W=FA-4*2:KK=(W+(FN+FA+q)*2+l)-2:DS=-KK*LN(q)/PI:DF=SqR(F*Sl):MSA=KK/DS 216 G2=(K-(8*W+9*FN+4*FA+q)*8/KK*2.5)*DS+2/PI/KK:DMFO=DS*(DS-G2):G2=G2*DF C o m p u t e r P r o g r a m : The Tandem Airfoil Forces / D 161 220 G1=FS*DS*DF/S1:MFA=S1/F*G1:B(4.0)=MFA-G2:G0SUB 600:IF C<0 THEN X=PI-X 225 MFN=W*SIN(4*X):DMS0=W*C0S(4*X):DMF=8*MFN/MSA+DS*T+G1/F/AC:B(3,0)=X/PI-DOT 230 B(3.1)=(G2-G1)/2:B(0,0)=1:T=S:B(0.1)=CT/2:D1=FS/S1:DMO=G1*F1/2:FOR 1=2 TO H 235 K,C=0:Z=I-1:F0R J=l TO Z:C=C+B(0,J)*B(0,I-J):K=K+B(0,J-1)*F~(Z-J):NEXT 240 B(3,I)=(DMO*K+B(3,1)*B(0,Z)+Z*F*B(3,Z))/I:B(I+3,0)=MFA*T+B(I+2.0)*F:T=T*S 245 B(0,I)=((T-1)*D1+1-C)/2:KEXT:Z=X1:DMS=(S*F/S1/F1)-.25:MA=SQR(S1/S/N) 250 MSN=DS*3/DF:N=SqR(N)*DF:FN=2*FA*XI:FA=2*FA*SIN(2*X) 255 K=B(3,1)-DF/KK/PI:J=1:G0SUB 605:Y1=Y4:D1=D4:Z=X4:J=2:C0SUB 605:Y4=-Y4:P=2*PI 260 FOR 1=1 TO 3:F0R J=H TO 1 STEP -1:F0R K=l TO J:B(I,J)=B(I,J)+B(O.K)*B(I.J-K) 265 NEXT:NEXT:NEXT:FOR 1=4 TO 7:N=I-4:F0R J=l TO H:T=J+3:C=0:FOR K=4 TO T 270 C=C+B(K,0)*B(N,T-K):NEXT:B(I,J)=C/J:NEXT:NEXT 275 AC.XS,XF,DM,MS,MF=0:SA=S:SN=FS:C=X1-S:T=X4-S:H=H-2:F0R 1=0 TO H 280 N=A(0.I)-A(l.I):D=A(2,I)-A(3,I):M=A(0.I)-A(3,I)-N*Y1-D*Y4:AC=AC+N*C 285 XS=XS+D*T:XF=XF+M*SN:K=I+2:SA=S*SA:SN=(F-K-SA)/K:C=(XrK-SA)/K:T=(X4*K-SA)/K 290 DM=DM+N*C:MS=MS+D*T:MF=MF+M*SN:NEXT:AC=(AC * D1+XS * D4)*4-DMF0*XF*P 295 XS=DMFO*MF*P-(DM*D1+MS*D4)*4:N=B(3.1)/B(0.1):XI=B(3,0):C=CT*(1-XI-G1/F) 300 SA=(C-B(4,1))/S:SN=(C*N-B(7,1)-DMF0*FS)/S:Z=B(0.1)-2:MA=(1-B(0.2)/Z)*2 305 MN=(N-B(3.2)/Z)*2:MSA=B(4,2):MSN=B(7,2):F0R 1=1 TO H+2 310 B(0.I)=B(0.I)-BC3.I)-BC1,I):B(4.I)=BC4.I)-B(7,I)-B(5,I):B(3.I)=BC3.I)-B(2.I) 315 B(7.I)=B(7.I)-B(6,I):NEXT:D,DS.DM0.DMS0=O:F0R J=0 TO H:T=J+1:M=J+2 320 FOR 1=0 TO 3:K=I+4:D=D-A(I,J)*B(I.T):DS=DS+A(I.J)*B(K.T) 326 DMO=DMO+A(I.J)*B(I,N):DMSO=DMSO+A(I.J)*B(K.M):NEXT:NEXT:D=2*P*D/CT 330 DS=(C*D+AC+P*DS)/S:C=CT*C:T=CT*B(4,0)*2:I=S*S:MSA=(C*MA-T+MSA*4)/I 335 MSN=(C*MN-T*N+(F+S)*FS*DMF0*2+MSN*4)/I:DM0=D/2+DM0*PI/Z 340 DMS0=(C*DM0-T*D/4+XS-P*DMS0)/I:I=CT/Fl:J=S/Fl:FA=I-J*SA:FN=I*N-J*SN:P=PI/2 345 DF=I*D-J*DS:K=I*I:T=J*J:Z=F/F1:MFA=K*MA-T*MSA+Z*FA*4:MFN=K*MN-T*MSN+Z*FN*4 350 DMFO=K*DMO-T*DMSO+Z*DF:AC=-MA*CT/4:XS=-MSA/SA/4:XF=-MFA/FA/4:M=(MN-N*MA)*P 355 MS=(MSN+SN*XS*4)*P:MF=(MFN+FN*XF*4)*P:DM=DMO+D*AC/CT:DMS=DMSO+DS*XS 360 DMF=DMF0+DF*XF:ER=(ER*2*W)"2*8:Z=H+2:IF FKS THEN ER=ER/F1~Z ELSE ER=ER/S"Z 365 PRINT:PRINT "Max err=";ER:PAUSE:GOTO 175 500 I=-l:AC=SqR(S/F/N)+l:XS=l-2/AC:G0T0 510 605 I=0:XF=SqR(Z):MS=SqR(l-Z):AC=DMS*MS+XF:XS=1-2*XF/AC 610 MF=(l-(M*XS)-2)/P:DM=l+SqR(MF):C=XS/M 515 XI=SN*(C-2*2-l-XS-4)/(l+MF)/(DM*P)-2:XI=((XI-2-l)*2*W+l)*XI 520 T=ABS(XI):IF I AND T>1 OR NOT I AND T<1 THEN XI=SGN(XI) 625 T=D*C*SqR(ABS(C-2-l)/P)/DM/2*SA:IF I THEN X=ACS(XI)/2:RETURN 630 MF=XF*MS:DM=MF+N:I=SqR(XI"2-l):C=SGN(C):IF ER<XI THEN ER=XI 635 XS=DMS0*(2*XI-2-l)+FN*XI+l:D4=LN(XI+I)/PI/2:Y4=D4*MSA*C:T=16*W*I*XI*C/KK+T 540 I=(MFN*2*XI+FA)*I*C:MN=Z-S:XI=SqR(MN*(Z-F)):MF=MF*XI/PI 645 B(J.0)=ATN((XS*XI-DM*I)/(XS*DM+XI*I))*2/PI+Y4*DMF 650 B(J,1)=(XI*(DMS*XF-MS)/AC-DF*T)/PI-Y4*K 655 D4=D4-MSN*XI/ABS(MN)*(MS+MA*XF)/AC-DS*T*C:AC=Y4*DM0:F0R 1=2 TO H:XI=I-2 660 XS=0:FOR T=0 TO XI:XS=XS+(MF*Z-T+AC*F~T)*B(O.XI-T):NEXT:XI=I-1 665 B(J,I)=(F*XI*B(J,XI)+B(J.1)*B(0,XI)-XS)/I:NEXT:RETURN 

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