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r  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1586 REMARK ON THE THEORY OF LIFTING SURFACES* By Aldo Muggia SUMMARY First of all, a brief synopsis of the Weissinger method, as it applies to a rectangular wing, is set forth, in order to show how lifting surface theory is applied in this simple case and to show that his idealization of the vortex system is justifiable in this particular instance. By building on this framework and merely adding a few approximations and unrestrictive understandings, it is demonstrated how the same sort of vortex system can be devised, and can find sanction, for the treatment of the lift problem presented by any thin wing of arbitrary plan form. 1. To begin with, let attention be directed to the aerodynamics of a lifting surface (which is a suitable idealization of an actual wing) having the simple physical property that it departs but slightly from the flat surface S, which is the projection of the lifting surface on the xyplane. Furthermore, this surface is to be considered immersed in an incompressible fluid of density p and to have a freestream pressure and velocity denoted by pm and V,, respectively. This impinging stream is assumed to be directed along and have the same sense as the positive xaxis. In the righthanded Cartesian coordinate system employed here (see fig. 1) it will be assumed that the zaxis is in the direction of the vertically downward pointing vector, as illustrated. On the basis of the abovestated hypotheses, it is legitimate to assume that the perturbations to the freestream uniform velocity V, that are produced by the presence of the lifting surface, will be small. Consequently, it follows that the local pressure p will be an harmonic function (a solution to Laplace's equation) of the positioncoordinates x, y, z, and, in addition, the overpressure at such a point will be dependent upon the potential (p describing the behavior of the local incremental velocities, through means of the relation C)T p p = pV  *"Sulla teoria delle superfici portanti." Atti della Accademia delle Scienze di Torino, vol. 87, 195219553. Introduced by Carlo Ferrari, Active National Member, at the Session of 15 May 19535. NACA TM 1586 The pressure jump occasioned by passage from the underside to the topside of the surface will be a certain unknown function of the points of S; call it f(x,y). The value of this function must become zero along the edge of the surface S, and thus one may write that the overpressure is given by p(xyz) p 1 f(x',y')z dx'dy' 4vyS 1 where r = ((x x')2 + (y y')2 + z2 Upon invoking the stipulation that one must have Cq = 0 at an infinite distance upstream from the surface, it is seen that the sought potential will have the form f(x',y')dcx'dy' Ifs r3 qp(x,y,z) = /4npV f dx ^ilpVoo m < Now let it be assumed, for convenience' sake, that the equation denoting the leading edge of the lifting surface is to be taken as x = xl(y) and that for the trailing edge as x = x2(y), while, likewise, as is customary, the semispan is to be denoted by b. Then, upon carrying through a few obvious transformations it is possible to rewrite the expression giving the perturbation potential as p(x,y,z) = 42 Jbd' z d 4 n P V. f S2 (y') x1(y') d 7 %rx I(Y r x(' b npV, dy' f oC+m dx x2 Y') 2x(y') r' x (y') f(xy')di Furthermore, let the following conventions be hypothesized: Take S' to be the wake region of the xyplane; ie., the region which lies downstream of the S region and lying in between the two straight boundaries y = tb. NACA TM 1586 5 Then make the definitions that on the wing region S m(x,y) = lf f(x,y)dx xPVJ (y) while on the wake region S' p, x2(y) m(y) = 1 f(x,y)dx PV. x(y) Consequently, it may be seen that the perturbation potential is now expressible as T (., Y; z m dx'dy' p(x,y,z) =  which merely says that the sought value of qp is the potential which describes the behavior of a distribution of doublets of strength m per unit area in the S + S' region. This system of doublets may be replaced by a system of vortices, spread over the same S + S' region, by having recourse to the equiva lency principle between doublets and vortices. The local strength of this equivalent vortex distribution, per unit distance, will have x and ycomponents given by the following expressions: In the S region, x = = L y pPV xl(y) a and = m =f(xy) Yy ax PV' while in the S' region, =, k x2(y) 7x =y P V" ) fa and 7 0 Y x NACA TM 1386 The abovedescribed system of vortices will induce, at any arbitrary general point of the S region, a certain velocity, the vertical component of which will be given by the integral v= =41 1 f +L + dxdy+ S4xpVo f s by' y yI x x' Now this vertical component of the induced velocity must be of such a v g magnitude that = holds true, in order that the velocity vector V"0 ax representing the total flow at the point in question shall be tangent to the wing surface. Thus one now has obtained an integrodifferential equation to work with for the determination of the unknown function, f(x,y). 2. The exact solution of this equation will, however, entail rather formidable difficulties, and for this reason it is best to fall back upon a much simpler approximate procedure for attaining the desired result. To this end, let the situation in regard to the simple rectan gular wing be examined first of all refss. 1 and 2). In this case one may let xi(y) = 2 and x2(y) +' 2 2 The angle of attack for any profile section situated at a spanwise dis tance y from the center line of the wing may be denoted by a(y), where this symbol is meant to denote the true aerodynamic angle of attack of the profile in question, measured from the angle of attack for zero lift. Thus it follows that the governing integrodifferential equation may be written as 1/2 1+ x1/2 + x a(y)V. = 0 dx= vz dx Z Ja/2 Ix x x J /2 1 x r2/ x F2 1/2 + = 1 2 CX 1 (1 + r dx'dy' a 2pVo 1 11/2 (2 x  Now make the approximation that the value of Ix x'l, that enters into the expression for r, is to be replaced by its average, which is simply 1/2 in this case. Making this substitution, and changing the NACA TM 1586 5 order of integration will result in the following simplified form for the integrodifferential equation: c.(y)V. = 1 b d .AL[ d (12 +26y a2t b dy' y/' Q Y y' 2 where S1/2 1/2 r(y') = f(x',y')x' = 2 ly7 (x',y')dx' is the circulation function for the velocity distribution around the profile that is situated at the spanwise location denoted by y'. The abovederived approximate equation equates the vertical velocity component a(y)Vg to the induced velocity produced at the point (,y,o) by action of a special system of vortices, which may be considered as derived from the actual distribution of vortices by concentration of all the bound vortices along the quarterchord line. 5. The line of reasoning followed in section 2 is only valid for the case of rectangular wings, but the result obtained is known to hold true even for other cases, as has already been pointed out by Weissinger. The wider generality of this result may be established upon the basis of the following considerations: Let it be assumed that the bound vortices are concentrated along a line denoted by x = xo(y), as illustrated in figure 2. The precise way in which this line is to be selected will be explained in full later on. It may be remarked here, however, that the acceptance of this represen tation for the bound vortices is equivalent to replacement of the actual distribution of doublets in the S region by means of a special kind of doublet system, consisting of null doublets everywhere upstream of the x = xg(y) line and a distribution of doublets of strength (per unit area) described by means of the m(y) function throughout the S1 region (and in the S' region as well), where the Si region is that area of the wing which lies downstream of the x = xO(y) dividing line. The potential which describes this new redistribution of doublets is express ible as qp*(x,y,z) =  m(y)dx'dy' 6 NACA TM 1386 and thus the error in this approximate expression for the potential of the flow is given by the difference in the two potentials, i.e., this difference is fx.y' zI Z.Iq1xy, 4 f / C4 r3 K frx(,y' dx + No//' let2^'1 ttentio e s u nJ'.',, J x ly'l veta odxay')t ae(7 Xhic e friIY dx' + iJIIL.V. X YJ I 4 .t o. io I F. f m 1.1c t Y 2 + w X, In fx' Sr .*X' I' ly z [ X ,' I [( y') J Now let attention be focused upon that region of space which is composed of all points which have quite small absolute values for the vertical coordinate zi and which when projected upon the xyplane fall within the S region; let this portion of space be labeled the S control volume. For points within E, therefore, one may replace the cp function with the cp* approximation land thus it will be per missible to substitute C for 2 on the surface of the wing) pro kz 6z / vided the quantity standing within the curlicue brackets is of small enough size. Further, it is to be noted that for wings with sufficiently large spans, the value of j(x x')2 + (y y')2 + does not vary to any marked degree as one ranges over the values of x' of interest, provided 2 2 the distances V(y y') + z remain large enough, while on the other hand, if the distances V(y y')2 + z are small, then the value that x x' takes on in the E control zone can be represented to good approximation by use of its average value I I(y) where the chord 2 distribution function I(y) is defined as 1(y) = x2y) x1(Y) NACA TM 1386 Thus, in analogy to what was done in the case of the rectangular wing, it will be legitimate to make the approximation that V(x x)2 + (y ')2 + z2 V xo 2 (y j2+ )2 z2 provided the point with coordinates (x,y,z) is so situated that it makes the relation ix xO(y) = 1(y) hold true. If this is true, it follows, in consequence, that X2(y') x2 (y1) f7y x2(y') (x',y')(x x')dx' x xO(y'] f(x',y')dx x (y') (y') from which one obtains the desired definition for x0(y') as 1 x'f(x',y')dx' Xx1(y') yX2(y') 1x2 x2f(x',y')dx' 7 (x',y')dx' xl(y') xl(y) The interpretation of the relationship just deduced is as follows: The proper xo(y') abscissa coordinate to choose at each profile section through the wing is the one which corresponds to the location of the barycenter of the circulation for that section. In other words, it is the barycenter of the moments of the vector forces fi taken about the points P(x',y'), where i is the unit vector in the direction of the xaxis and where P is the radius vector out to any arbitrary general point in the S region at which the circulationfunction value is 7y(x',y'). Thus, to close approximation, one may select the x0 abscissae values according to the rule xo(y') x (y') + (y') '4 8 NACA TM 1586 while, when applying the boundary condition, it will be necessary, in addition, to make use of the potential values (or the induced velocity values) which appertain to the locations (x,y,O) for which it is true that x(y) = x0(y) + 1. 1(y) = x1(y) + l(y) This result, which has been deduced by aid of the abovementioned list of specific observations and series of approximations, may be arrived at by examination of the general equations applying to lifting surfaces. This result is important, for example, in those cases where one wishes to obtain the distribution of circulation (and thus of the lift) which exists out along the span of swept wings refss. 5 and 4). Translated by R. H. Cramer Cornell Aeronautical Laboratory, Inc. NACA TM 1386 9 REFERENCES 1. Reissner, E.: Note on the Theory of Lifting Surfaces. Proc. Nat. Acad. Sci., vol. 55, no. 4, Apr. 1949. 2. Lawrence, H. R.: The Lift Distribution on Low Aspect Ratio Wings at Subsonic Speeds. Jour. Aero. Sci., vol. 18, no. 10, Oct. 1951, pp. 683695. 3. DeYoung, John, and Harper, Charles W.: Theoretical Symmetric Span Loading at Subsonic Speeds for Wings Having Arbitrary Plan Form. NACA Rep. 921, 1948. 4. DeYoung, John: Theoretical Antisymmetric Span Loading for Wings of Arbitrary Plan Form at Subsonic Speeds. NACA Rep. 1056, 1951. (Supersedes NACA TN 2140.) NACA TM 1586 / X Voo Figure 1. Orientation of coordinate axes and definitions of integration areas. NACA TM 1386 11 b x x (y) X X2 (y) x= x0 V) x YSj +bI Figure 2. Location of the bound vortex line and areas of integration. NACA Langley Field. 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