New Developments in the Theory of Airfoils of Infinite Span

This paper gives some supplementary developments and revisions of the two dimensional airfoil theory. (1) The chief results are reviewed and the concept of "second axis" of a profile is revised. (2) In discussing stability questions the theory of Hamilton's center is referred to. (3) The method of successive approximations is outlined in its relation to the so-called thinwing theory. (4) The addition of two independent lift-force systems is explained. (5) The general theorem is proved that in any case of several airfoils with mutual interference the lift force follows the same laws in its dependency on the angle of attack as in the case of a single wing. I 1917 and 1920 the author published two papers in which, for the first time, a general and complete solution of the lift problem for an airplane wing of infinite span was given. About the same time Prandtl developed his famous theory of vortex motion behind an airfoil which supplies a most valuable estimation of the influence of the finite ends. Both theories form today the basis of the practical computations in airfoil design and both are the subject matter of numerous papers continuously being published in almost all countries. In that which follows the author presents some supplementary developments and revisions to his former papers, suggested in part by recent publications. 1. T H E CHIEF RESULTS OF THE THEORY The principal results of the theory are the following. To any cross-section of an airfoil which has a sharp trailing edge (B in Fig. 1), there corresponds a point F I G . 1. Lift magnitude, direction, and line of action. F, a straight line 1° and a vector L° parallel to 1°. These elements, depending on five parameters (two coordinates of F, direction and position of /°, magnitude of L°) entirely determine the lift force for every angle of attack a. It is only necessary to draw through F Presented a t an Aerodynamics Session, Eighth Annual Meeting, I.Ae.S., New York, January 25, 1940. a parallel to the velocity V at infinity (at the angle a to the x axis). The lift L is perpendicular to Vy its line of action passes through the point of intersection of the parallel to V with 1° and its magnitude equals the projection of L° (see Fig. 1). It is easy to see that the lines of action so determined envelop a parabola with focus at F and with 1° as vertex tangent. If the direction of the flow at infinity is normal to 1°, then 1° and L° immediately give the line of action and the magnitude of the lift. The elements F, 1°, L° are connected with the crosssection of the profile through the theory of conformal transformation. If z designates the complex number (or the vector) x + iy corresponding to a point x,y of the profile or its surroundings, then a function f(z) = z = x' + iy' relates to the point x,y another point x',y' and this correspondence is called a conformal transformation. If the difference z — z' vanishes for infinite values of z and z\ the transformation leaves infinity unchanged. The decisive theorem is the following: There exists one and only one conformal transformation, leaving infinity unchanged, which transforms the outside of a given closed curve (the profile) into the outside of a circle. This circle, its center M and its radius a (Fig. 2), are uniquely