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\u2014s 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 \u2014* 1 (so that s \u2014\u2022 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 \/\u2014s = 0.1333 (figures (b), an interesting midpoint between f\u2014s = 0 and f\u2014s = 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; \u2022--<>--\u2022 exact-numerical theory J 1 \" 1* I 1 1 1 L*. I 1 1 ( !_\u00bb. | | | | \u2022 I \u2022 \u2022 \u2022 I \u2022 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\u00b0 o\u00b0 a\u00b0 o\u00b0 a\u00b0 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> k> \/^ooi A i \u00bb *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\/? \u2014 i sin\/? \/ \u00ab 2 * 2 - ( l + s+\/)* + \/ 1-! 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) = \u00bbsin\/? 2 l+s+\/ 6 j 2 >'+2 6s f F' 1 - il-\u00bb)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: 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 \u00b0 = T*Vpf' X c - X 0 = T*h- (5-23) Then, with k given by: ' .2 = \/-\u00ab = (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\u00b0 ~ 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 = > \/ ( * \/ - * < > ) ( * \u00ab - * . ) \u2022 (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 \u2014> 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' \u2014* 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' \u2014\u00bb 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 \u2014 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 ( = \u2014Ax), 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\u2014s 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 \u2014 : \u2014 ; 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\u2014s 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\u2014s. However: jgt 2 K'~Hf=s) a3\/-a-*\u00b0 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\u2014s as the lncj term does in the g-series expansion solution of z(u), Eq. 5.27. E'\/K' - \u00bb O a s \/ \u2014 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 \u2014\u2022 0 with overlap. They are not completely general since they assume x2 > 0 and z 22 > a typical slotted flap arrangement. As f\u2014s \u2014\u2022 0: *(0 -* 1 ~ l \u00b0 S ^ X 2 2 + x22(tcoaP - isinpy\/t^t-l)) (5.29) Kz) ~* ^\u2014- + \u2014(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 * *\u00b0 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-*-]\u2014 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\/? \u2022t + Z0 + -j- + -jjt + '-' > as*->oo (5.32) where: and: Z0 = x0 + \u00bb ( 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 \u00a3{z), the generalized staggered slit version of the boundary condition selector, are: 1) its real part is zero on all airfoil element surfaces. 2) \u00a3(z) \u2014* 1 as z \u2014\u2022 oo 3) the poles in \u00a3(z) must be at the airfoil and flap trailing edges so that w'(z)\u00a3(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 \u00a3(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, \u00a3(z) is unique: \u00a3 { z ) = Vt(t-s)(t-f)(t-l) + (t-tn)(t-t22) 1 T22~tll \\ \/<22(<22-*)(<22-\/) ( t22-l) \\All(*ll ~ *)(*!! ~ 1) t \u2014 t22 '\"\"^11 . (5.36) This function is obtained through a linear superposition of the -^ -(w-Wn) and -r^(w\u2014w22) 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 \u2014 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 \u00a3(z) finite at t = t^ and t = t22. Alternatively, when h < 0, tj = t^t^ft^ ,kt22 and \u00a3{z) is finite at txl & t22. Staggered Two-Element Thin Airtoil Theory \/ 5.2 H3 To calculate the forces, the large z expansion of \u00a3(z) is needed. If: + + ^ + + , a s * - o o (5.37) defines the A\u201e's, then the z(t) expansion, Eq. 5.32, can be used to obtain the \u00a3 n ' s : n f> p \u00a3{z) ~l + -\u00b1 + \u00b1l + Ji + ... , as z-> oo (5.38) Z Z Z> and: Cl==sw\u00a3l>> i2 = ZoCl + -^T\u00a3t>> 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 \u2014 (5.41) * l l + * 2 2 - ( * l + *2)(*ll+<22) + g 2 cos\/? \u2022-\u00bb < 2 2 \\ \/ < 2 2 ( ^ 2 - s ) ( < 2 2 - \/ ) ( * 2 2 - l ) ~ hi\\A11 - \/ ) ( ' 11 ~ 1) sin\/3 <22 - zw'(z) dz ( ' i + c 2 ) 2 tol 4(*-\u00a3Ju\/(*K(*)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 - ^ \u00a3 ( 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 \u00b0 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 \u2014\u2022co f+S + sf - ( l + S 4 + \/ ) 2 + < U + *22 - ( (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 \u2014\u2022 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 \u00bb 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- \u2014-^-(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 \u2022 ~ \/o H 1\u2014~ H , as 2r \u2014\u2022 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) \u00ab\u2022 V 2 2 \/ c 0 8 ^ G (1 + a + \/ _ 4 f 2 ) 2 ^ 1 t 2 ( i + \/ - a ) 2 - \/ ( l + \/ - a ) , 4 T = 8 + \u00ab 2 l i ' Ti and 72 are obtained by replacing \u00a3 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 \/? \u2014\u2022 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 \u00bb m\u00ab (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-\u00a3(*;*i\u00bb<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 \u2014~{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 - _ - _ . - ! ( \u201e ) + hi-s t{t-f)(t-l) t-tn\\ t-3 , V'II('II-\u00ab)(\u00abII-\/)('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 \u00b1 \u00a3 l A o C l - * \u00a3^[A{u)-i]d*-\u00a3\u00a3%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 \u2014 Xyy, then, upon integrating by parts, becomes: \u201e c i + c 2 A \u201e 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 = \u2014\u2014 dz H 2\\\/t(t \u2014 s)(t\u2014f)(t \u2014 l) cos\/? \u2014tsin (5.70) which is purely imaginary on the slit surfaces and reduces to Equation 4.34 when \/? \u2014> 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\u2014xn)~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 \u00a3(z) oc 1 \u2014 [>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 \u00a3 [x{A -1 - #) + ih(A - 1)A] ^ dz Ci Ci airfoil - i \u00a3[x{A+l-$) - ihA{l-A)}^}dz . flap Staggered Two-Element Thin Airfoil Theory \/ 5.3 121 The real part of the integral on the LHS of this equation can be evaluated (by residues) in terms of the overall lift and moment (the real part of the second term in the integrand integrates to zero). The real part of the first integral on the RHS gives the airfoil moment, while that of the next integral is zero. The last tv\/o integrals on the RHS can be evaluated numerically, with all the problems that entails, but it would be more convenient to convert them to a form that is more easily solved by residues. Until now, this was accomplished by integrating by parts so that the resulting integrands consisted of a boundary condition multiplied by a function whose complex form was purely imaginary on both slit surfaces \u2014 a characteristic which greatly simplified the formulation of the complex version of the boundary condition. However, the A A products in the above integrands negate the effectiveness of this process. The problem can be seen in a different way, as suggested by the forms of the mapping function, Eq. 5.15, and the A(OJ) and A(OJ) functions. If: H Z = x0 + sin\/3 and, writing Z as Z(t;tn), if: then one can show that: i r n * ,-n:n fl>\/*('-')(\u00ab-7)('-l) + >\/*ii(\u00abn-\u00ab)(*n-\/)(*n-l) i cos p \u2014 \\ sin p t-hi (5.74) 2 = Z(t;t22), (5.75) z - Z = ih{A-\\) - (5.76) z - Z = -ihA . Z and 2 are purely real on the surfaces of the slits. The integrands of the last two integrals of Equation 5.73 can now be rewritten: 2 7 9 x{A-l-$) + 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 \u2014\u2022 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\u00b0 V c\\ ) 2 1 airfoil flap The Z f - ^ integrands can be handled by converting to one complex integral: ax ij-\u00bbyzzV(*) ZZw'{z)dz\\. (5.79) The difficulty is with the remaining x \u2014 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\u00b0 or 30\u00b0, 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\u2014for 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 \u201e Regions. AIAA Journal, Vol. 14, No. 8, p. 100G-11 \u2022 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 \u2014 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) \u00bb y cos ct - z sin a + \u2014 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 \u2014 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\u00b0. 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\u00a3y products (\u00a3y 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) = + iipt then: 0(\u00ab) = ze~ia + \u00b1 J 7(5) ln(* - zs) dS (A4) s where zs is on surface 5 in the complex plane z \u2014 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(\u00ab) \u00ab ze~ia + JL \u00a3 Tj j Hz - z() d\u00a3 . (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 \u00a3 7y[(*-*i>-''> ln(*-*iy) - ) \u00ab - ' ' > l n ( * - \u00ab * , ) - (,\u2022]. (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 = \u00a3\\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 \u2014\u00bb oo: real terms + Thus: (All) (A12) (A13) ( c i + ' - H w ) 2 ^ ' ^ V - J \u2014 \u2022 ( 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 + - \u2022+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&,)]\u00a3\u00a3+E\u00a3+\u00a3\u00a3+\u00a3\u00a3 \u2022 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\u00a3~\u00b0(J as z \u2014\u2022 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\u00bb j G{z) dz . (A20) This expression can now be evaluated conventionally: *2, 2fft 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 \u2022 \u2022 \u2022 \u2022 -1% 2.18 2.16 20 _1 L_ 40 60 80 100 a) Overall lift vs. Nf.. 0.08 \u2022 p r e s e n t t h e o r y \u2022 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 \u2022 i T \u2022 *\u2022 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\u00b0. Exact-Numerical Potential Flow Computations \/ A.S 137 2.44 0% \"1 \u2022 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 \u2022i 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\u00b0. Exact-Numerical Potential Flow Computations \/ A.3 138 -0.26 -1% +1% \u2022 \" - - - -\u2022 \u2022 w w z 1 1 \u2022 -0.24 \u20220.22 -0.20 \u2022 present theory \u2022 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% \u20221% 0.66 \u2022 m -I 1 1 \u2022 \u2022 - \u2022 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\u00b0. Exact-Numerical Potential Flow Computations \/ A.3 139 \u2022 present theory \u2022 CPj assumed constant over all of segment \/ -0.92 -0.90 -1% \u20220.88 +1% -* \"I 1\" \u00bb--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\u00b0. Exact-Numerical Potential Flow Computations \/ A.3 140 \u20221.02 -m -1.00 10 -0.98 \u20221% +1% \u20220.96 \u2022 present theory. \u2022 Cp j assumed constant over all of segment \/. 1 1 1 * 20 40 60 80 100 Nk a) Airfoil moment vs. iVj.. -0.58 \u20220.56 - \u2022 -m 20 -0.54 +1% -0.52 - \u2022 \u2022 \u2022 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\u00b0. 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 \u00b14K\u00b1 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\u00ab sin u \u2014 2q* sin 3w + 2q~* sin 5w i \\ 1 9 21 V2[u,q) = 2q* cos a; + 2q* cos 3a; + 2g < cos5a; H ^3(w\u00bb 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: , \u201e 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)\u00ab\/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~' - *\u00bb(o.g ) - *4(Q,q) _ 02(0,g4) m i m \" - l + v\/it7 U^) + U0tq) \u00ab\/3(0,?4)' 1 ' This infinite series for e0 in powers of q can be inverted to give: q = c0 + 2e50 + lSe90 + 150e\u00a33 + O (lOOOeJ7) . (Bll) Thus, q is calculated using Equation B l l , K is obtained as above, and: K' = -\u2014lnq. (B12) Elliptic Functions \/ B.l 144 For typically small values of f\u2014s, these are all very rapidly converging series. As f \u2014 s \u2014* 0, k \u2014> 0, k' \u2014> 1, and q \u2014\u00bb 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): \u00b0^ = OT; ^ =yfS; \u2122=^\u00a3R- (B13) Also: d d d , \u2014 sn A = cn A dn A ; \u2014 cn A = \u2014 sn A dn A ; \u2014 dn A = \u2014k 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 \u2014[to] \u2022 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 \u2014 + 2E' 2K' 1 TT sn2A d dX \u2014 + _ TC 2E' 2K'dn2X 7r cn2 A d dX '2ui \u2014 + 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-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: \u00a3 = ^ (B21) cos2w = tf[l - g 4 ( 2 - 4 \u00a3 2 ) + g 8 ( 3 - 2 0 \u00a3 2 + 3 2 \u00a3 4 ) - g 1 2 ( 6 - 7 6 \u00a3 2 + 2 7 2 \u00a3 4 - 3 2 0 \u00a3 6 ) +0(g 1 6)]. (B22) Note that u can be real or imaginary. For u real, it can be shown that |\u00a3| < 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 V F + 1 since dn fi^ \u2014 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 \u2014(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 ^ \u2022 (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\u2014s is small enough, they cannot be used over as large a range of f\u2014s 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 = \u2014 determines q2 1 - k' k2 = JJ determines g 4 1 + kx etc. In fact: Now, if e02 is defined: ko \u2014 *4 (l + A ^ l + v\/* 7) 4 ( 2 e \u00b0 ) 4 ' 2, _ l - V ^ = 0 2 ( O , g 1 6 ) then, from Eqs. BIO and B l l : (B27) so that: 94 = e02+2\u20acg2 + 15eg2 + 150e05 + .- . 1\/4 1 , 1^4 117^8 , n(\u201eM\\] 1 + 2 % - l 2 - \u00a3 0 2 +0{q (B28) where: 1\/4 2jo \u00b02 ~ r . . . . . \u2014 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\u00a3 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,.,\u201e = 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 \u00a3 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, ? = \u00a3 2 [ l - 2 g 1 6 + 5g32 + 0(g4 8)] cos4UJ = (2{l + 4g16[el-l] + 4 g 3 2 [ 3 - l l \u00a3 2 + 8 \u00a3 2 4 ] + O (lOO^g 4 8)}. (B32) An investigation of the characteristics of \u00a3 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 |\u00a32| < 1. However, from Equation B20 it can be seen that cw has a pole at u = | \u00b1 27rr, so that in the rectangle (\u2014^ < 17 < T^) it Elliptic Functions \/ B.3 150 will take on its maximum magnitudes along u = x\u00b1 although this is still some distance from the pole. At w values of | \u00b1 ^ , and \u00a7 \u00b1 y-, |ej = 1 so that \u00a3 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 \u00a3 2 and Equation B32 for cos4w. Since ew is odd in x about u \u2014 J, one can use its sign to determine which value to choose from the arccosine operation that gives u>: \u20acw > 0 for 0 < Au < n for u real. (B33) \u20acw < 0 for TC < Au) < lit To calculate u>i = t'71 and w2 = | +i*72, corresponding to xx (0 < x\u00b1 < s) and x2 (\/ < x2 < 1), one makes use of the fact that: s ( l - x t ) dnA>' = V ( i ^ ; f o r \u00b0 ^ 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 < \u00bb7y < ^ : cosh2 47y + yjcosh2 47y \u2014 1 7y = ^ ln (B36) 0* When evaluating the \u2022^ \u2022(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: \u2122 S \u00b1 V , _ ( & ) (B37) * J \" ' IT 1 + I \u00a33 dn(A2, fc2) When w is real, as when E 3 is being evaluated, and when 0 < x 5; \u00a7> 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 = \u00b1n\/(g2-i. sn(A2, k2) is odd in 7 for x \u2014 0 a n d X = \u00a7\u2022 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: \u00bb 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), \u00a3j = 1 + 2g4 cos 4Xoo cosh 47y + 2g16 cos 8Xoo cosh 87y + 0 ( ^ . g 3 6 ) Ji = 2g4 sin 4Xoo sinh 4 7 y + 2ql\u00ab sin 8Xoo sinh 8 7 y + O ^ f .g36) R2 = + Vxy(l-Xy) 72 = \u00b1 | (xy - s) (zy\u2014\/) | 5 , + for airfoil upper surface \u2014 for flap upper surface (B40) then: i m t ? 4 ( X o o - \" y ) = 2 a r ( R j - i J A * 1 1 \u00ab ? 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 \u00a3(z) A N D F(z) \u00a3(z) is first used in section 4.1: ^ 2 ) = y ( z _ a ) ( z _ 1 ) ~ i + T + ^ + ^ - + \" If v = 1\/z, then: = 1 + 2CiV + ( 2 \u00a3 2 + C] )v2 + ( 2 \u00a3 3 + 2\u00a3ii :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 - ( \/ - \u00ab ) 2 \u00a3 2 = l - ( \/ - s ) ( l + s ) - \u00a3 2 2 \u00a3 3 = 1 - (\/-s)(l+s+a 2) - 2 t i t 2 (Cl) , _ n - l 2 \u00a3 \u00bb = 1 - 7 - ^ ( 1 - 0 - \u00a3 4 \u00bb 4 . - m , \u00bb > 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 \u2014* oo. Of course, F[z) = l\/C{z) which leads to a simple recurrence relation for the Fn's in terms of the \u00a3n'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\u201e's is desirable: 2F1 = \/-(l+a) n - l 2 ^ = [\/2-\/(l+s) + s ] \/ B _ 2 - \u00a3 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. \u201e, , , 2UJ K'A\\. . . A >f2 Jlw =1 - T T T 1 \" ~ 4 + \u2014 + + JT A 1\/j z zl as 2 \u2014\u2022 oo. Using Equations B7 and since = Xoo \u00b1 y It is convenient to define: w K #, 4 .o (Xoo) .o (Xoo) and using Equations B15, 2.5, and 2.9: \u00a3 4 = \u00a33 + 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 \u2014 s) JT Ai + 2X2v + 3A3v2 + (1 -av)(l - v) 1-fv or: 1-s G^l+sv+aV+sV-r-- \u2022 \u2022) - G2 2 L \/ (C4) [l + \u00a3 i V + Z2v2 + \u00a33v3 + \u2022\u2022\u2022] = 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 = ( \/ + \u00ab ) ( \u00ab - l ) \/ l n _ i - af{n-2)An-3 + + 5 \u00bb > 3. The flap selector is introduced in section 2.2 and is used throughout the thesis: , , , 2u K'$'2. . _ Ti 72 as z \u2014\u2022 oo. In much the same way as the A\u201eB 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\u00a3n-3 + \/ 2 \u00a3 n _ 4 + \u2022 \u2022 \u2022 + 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 ) \u00bb *J , C 7 ) \/. 5i 52 ~ 5 0 H T - -5 - -1 , as 2 \u2014> 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 * \u00ab 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 \u2014 1\/z as is done here, leads to a doubly periodic function which can be represented by algebraic functions of z: dS(uj) dX \u201e \u201e \u201e . o ,\\ ' \u2014 ~ 5i + 252v + 3S3tT + \u2022 \u2022 \u2022 , 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 ^ \u2022 ( ( j - u \/ J - ^ - ( w + w j TT t?'4 2sn A t cn A t dn Ax = \" 2 2 T ^ + s n ' A \u2014 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 \u2014 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 \u2014 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 \u2014 3 TT Z\u2014f 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=^-\u20147== 2 if ' G 2 -K (C12) Now, if: (C13) and since, as z \u2014\u2022 oo: 1 ~ , *, (i + +i:2v2 + \u00a33t;3 H \u2014 ) 1 - sv){l - fv){l - v) l - f v K 1 then: (1 _ fv)^M \u201e [Si - MQv - Miv2 - M2v3 -...](! +Civ + \u00a32v2 + -\u2022) dv Si + (252 - fSi)v + (353 - 2\/52)v2 + and: nSn = \/ ( n - l ) 5 n _ i + SiCn-i ~ M0\u00a3n-2 - M^n-z M n _ 3 \u00a3 , - 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 \u2014> oo. Note that this equation is written slightly differently from Equation 4.54 since it is in terms of w2 = \u00a7 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 \u2014 r r r + ~ x \" U o o ) 2 ~ G l F~~ 2 If If: = \\^x2{x2-s){x2-f){\\-x2)xn2 - \u2014 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 \u00a3 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 \u2014 xac N = dCe\/dri\/(2ir) MN = dCmJdV\/(*\/2) XS = X\u00bbac 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 : \u2022; 110 ACCEPT SIZE(3)VALIDATE(DIGIT,\" \")JST:PRINT \u2022 ,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=(Xr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l: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