^i/\f/V'TM^3s|f 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1386 
 
 REMARK ON THE THEORY OF LIFTING SURFACES 
 
 By Aldo Muggia 
 
 Translation of *Sulla teoria delle superfici portanti. ' 
 
 Atti della Accademia delle Scienze di Torino, 
 
 VOL 87, 1952-1953. 
 
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 RY 
 117011 
 _w;ILLE, FL 32d! i-/ui i uc<-\ 
 
 Washington 
 January 1956 
 
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 siirface 
 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 \inderstandings, 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 siirface S, which is the projection of the lifting siu-face on the 
 xy-plane. Furthermore, this surface is to be considered Immersed in an 
 incompressible fluid of density p and to have a free-stream pressure and 
 velocity denoted by p and V^^, respectively. This impinging stream 
 
 is assimaed to be directed along and have the same sense as the positive 
 X-axis. In the right-handed Cartesian coordinate system employed here 
 (see fig. 1) it will be assumed that the z-axis is in the direction of 
 the vertically downward pointing vector, as illustrated. 
 
 On the basis of the above-stated hypotheses, it is legitimate to 
 assume that the perturbations to the free-stream uniform velocity V^ 
 that axe produced by the presence of the lifting surface, will be small. 
 Consequently, it follows that the local pressure p will be an harmonic 
 f\inction (a solution to Laplace's equation) of the position-coordinates x, 
 y, z, and, in addition, the overpressure at such a point will be dependent 
 upon the potential cp describing the behavior of the local incremental 
 velocities, through means of the relation 
 
 P - Poo = -PV. ^ 
 
 *"Sulla teoria delle superfici portanti." Atti della Accademia delle 
 Scienze di Torino, vol. 87, 1952-1955. Introduced by Carlo Ferrari, Active 
 National Member, at the Session of 15 May 1955. 
 
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 overpressijre 
 is given by 
 
 P(x,y,z) -P. = i;^/^ 
 
 f(x',y')z dx'dy' 
 r5 
 
 where 
 
 ^f 
 
 f + (y - y'f + 
 
 2 
 
 z 
 
 Upon invoking the stipulation that one must have (p = at an 
 infinite distajice upstream from the surface, it is seen that the sought 
 potential will have the form 
 
 r \ ^ [^ ^ r r f(x',y')dx'dy' 
 
 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 = X]^(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) = -— ^ r dy' / ^ ^ ^^ r f(x,y')<i5 - 
 
 dy' / 2^ f(x,y')dx 
 
 i^«pVoo./_v Jx2(y') r5 Jxi(y-) 
 
 Furthermore, let the following conventions be hypothesized: 
 
 Take S' to be the wake region of the xy-plane; ie., the region 
 which lies downstream of the S region and lying in between the two 
 straight boundaries y = tb. 
 
NACA TM 1586 
 
 Then make the definitions that on the wing region S 
 
 1 r^ 
 
 °^(^'y) = 7:^ / f(x,y)dx 
 
 XI (y) 
 
 while on the waJce region S' 
 
 T pX2(y) 
 m(y) = ^ / f (x,y)dx 
 
 Consequently, it may be seen that the pertiirbation potential is now 
 expressible as 
 
 / V z r r mdx'dy' 
 
 S+S' 
 
 which merely says that the sought value of cp 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 
 y-components given by the following expressions: 
 
 In the S region. 
 
 oy pV„,Jx^(y) dy 
 
 and 
 
 = 5m ^ f(x.y) 
 while in the S' region, 
 
 ' by ' pV„J^^(y) Sy 
 
 ^^ ax pV„ 
 
 ,, . ^ , ^ r^'=" 21 ax 
 
 and 
 
 y = -^ = 
 ^^ Sx 
 
MCA IM 1586 
 
 The above-described 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 
 
 Vz = 
 
 lVrtpV„ 
 
 'S 
 
 ^^^ 1 — -(1 + —^ — rW'^' 
 
 ay' y - y' V X - xV ^ 
 
 Now this vertical component of the induced velocity must be of such a 
 
 ^z Sz 
 magnitude that — = :— holds true, in order that the velocity vector 
 
 Voo dx 
 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 (refs. 1 and 2). In this case one 
 may let 
 
 ^liy) = -| and XgCy) = +1 
 
 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, measiired frcm the angle of attack for zero lift. 
 Thus it follows that the governing integrodifferential equation may be 
 written as 
 
 
 dx 
 
 2 -^ 
 
 >l/2 '^ 
 
 2pVooZ J -1/2 u - X J Js^ y -y \ ^ - ^v 
 
 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 
 
 order of integration will result in the following simplified form for 
 the integrodifferential equation: 
 
 »<^'^» = irC ^ 7^ i^ #) 
 
 + (y - y')' 
 
 where 
 
 1/2 
 
 ^^y') =7k- f f(x',y')dx' 
 P^°°J -1/2 
 
 pi/2 
 
 / 7 (x',y')dx' 
 
 .1/2 
 
 is the circulation fimction for the velocity distribution around the 
 profile that is situated at the spanwise location denoted by y'. 
 
 The above-derived approximate equation equates the vertical velocity 
 
 component a(y)Voo to the induced velocity produced at the point (r->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 bo\ind vortices along the quarter-chord 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 vider generality of this resxilt may be established upon the basis 
 of the following considerations: 
 
 Let it be ass\jmed 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 = XQ(y) line and a distribution of doublets of strength (per unit 
 area) described by means of the m(y) function throughout the S-^ region 
 (and in the S' region as well), where the S-^ region is that area of 
 the wing which lies downstream of the x = XQ(y) dividing line. The 
 
 potential which describes this new redistribution of doublets is express- 
 ible as 
 
 cp*(x,y,z) = - 
 
 z 
 
 'S-L+S' 
 
 m(y' )dx'dy' 
 ^3 
 
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 
 
 cp(x,y,.) - .*(x,y,.) - -^//^^/^^^^^^ r(x,y.)dx. 
 
 pb rX2(y') rX2(y') , 
 
 --^ / dy' / f(x,y')d5 / ^- 
 
 nb pXp(y') pX2(y') 
 
 2 
 
 -b 
 
 ay' / f(x,y')dx / H- 
 
 -'xi(y') 'Jxo(y') r^ 
 
 pb pX2(y') 
 
 / L-^ oW f(x',y') -= ^ - ^' cbc' - 
 
 J-b (y - y')'^ + z"^ Mx, (v) 
 
 '^xpV-J.b (y - y')2 + z2 ^ y,^{y) ^(x - x')^ + (y - y')2 + 
 
 X - xo(y') r^gCy 
 
 r ^ ,n2 , ,>2 2-'xi(y') 
 
 [x - XQ(y')J + (y - y') + z !'■' 
 
 f(x',y')dx' Uy' 
 
 Now let attention be focussed upon that region of space which is 
 composed of all points which have quite small absolute values for the 
 vertical coordinate |z| and which when projected upon the xy-plane 
 fall within the S region; let this portion of space be labeled the 
 2 control volume. For points within Z, therefore, one may replace 
 the cp function with the cp* approximation (and thus it will be per- 
 missible to substitute ^— for ^ on the surface of the wing) pro- 
 
 oz oz / 
 
 vided the quantity standing within the ciirlicue brackets is of small 
 enough size. 
 
 Further, it is to be noted that for wings with sufficiently large 
 
 / 2 2 2" 
 
 spans, the value of y(x -x') +(y-y') +z does not vary to any 
 
 marked degree as one ranges over the values of x' of interest, provided 
 
 the distances y(y - y') + z remain large enough, while on the other 
 
 1/ 2 2 
 hand, if the distances y(y - y') + z are small, then the value 
 
 that X - x' I takes on in the E control zone can be represented to 
 good approximation by use of its average value p ^(y) where the chord 
 distribution function Z (y) is defined as 
 
 Uy) = X2(y) - x^(y) 
 
NACA m 1586 
 
 Thus, in analogy to what was done in the case of the rectangular 
 wing, it will be legitimate to make the approximation that 
 
 |/(X - X')2 + (y - y')2 + z2 ~ ^^ _ xo(y']]2 + (y _ y.)2 ^ 
 
 2 
 z 
 
 provided the point with coordinates (x,y,z) is so situated that it makes 
 the relation 
 
 hold true. 
 
 If this is true, it follows, in consequence, that 
 
 rx2(y') i- -, pX2(y') 
 
 / f(x',y')(x - x')dx' = X - xo(y') / f(x',y')dx' 
 
 from which one obtains the desired definition for XQ(y') as 
 rX2(y') rX2(y') 
 
 r^2\y ) 
 x'f(x',y')dx' / x'7„(x*,y')dx' 
 
 X (y.) ~ ^^i^y') ■■ ^i(y') 
 
 ° rX2(y') px2(y') 
 
 / f(x',y')dx' / 7 (x',y•)dx' 
 ^x-L(y•) ^xi(y>) 
 
 The interpretation of the relationship just deduced is as follows: 
 The proper XQ(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 circiilation for that section. In other words, it is 
 the barycenter of the moments of the vector forces fl taien about the 
 points P(x',y'), where i is the unit vector in the direction of the 
 X-axis and where P is the radius vector out to any arbitrary general 
 point in the S region at which the circulation-function value is 
 7y(x',y'). 
 
 Thus, to close approximation, one may select the Xq abscissae 
 values according to the rule 
 
 xo(y') = xi(y') + i l(y') 
 
NACA TM 1586 
 
 while, when applying the boundary condition, it will be necessarj'-, in 
 addition, to make use of the potential values (or the Induced velocity 
 values) which appertain to the locations (x,y,0) for which it is true 
 that 
 
 x(y) = XQ(y) + I z(y) = x^Cy) + J ^(y) 
 
 This resiilt, which has been deduced by aid of the above-mentioned 
 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 (refs. 3 and 4). 
 
 Translated by R. H. Cramer 
 
 Cornell Aeronautical Laboratory, Inc , 
 
NACA TM 1586 
 
 REFERENCES 
 
 1. Reissner, E,: Note on the Theory of Lifting Surfaces. Proc . Nat. Acad. 
 
 Sci., vol. 35, no. h, Apr. 19i^9. 
 
 2. Lawrence, H. R.: The Lift Distribution on Low Aspect Ratio Wings at 
 
 Subsonic Speeds. Jour. Aero. Sci., vol. I8, no. 10, Oct. 1951, 
 pp. 683-695. 
 
 3. DeYoung, John, and Harper, Charles W. : Theoretical Symmetric Span 
 
 Loading at Subsonic Speeds for Wings Having Arbitrary Plan Form. 
 NACA Rep, 921, 19^8. 
 
 k . DeYoung, John: Theoretical Antisymmetric Span Loading for Wings of 
 Arbitrary Plan Form at Subsonic Speeds. NACA Rep. IO56, 1951. 
 (Supersedes NACA TN 2l40.) 
 
10 
 
 MCA TM 1386 
 
 Figure 1.- Orientation of coordinate axes and definitions of integration 
 
 areas. 
 
NACA TM 1586 
 
 11 
 
 Figure 2.- Location of the bound vortex line and areas of integration. 
 
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