The theory and design of two-airfoil lifting systems

Abstract

By using a general form of conformal mapping from the doubly connected airfoil domain to a canonical ring domain, the problem of two interfering lifting airfoils is reduced to the solution of certain integral equations. It is shown that the solutions of these equations conform to the same general velocity formula in the cases of (a) direct boundary-value problems (given shape), (b) inverse boundary-value problems (given velocities), and (c) simply mixed boundary-value problems (given either shape or velocity on one airfoil and the converse on the other). This velocity formula consists of a singular part in terms of Theta functions and a Laurent series whose coefficients can be adapted to accommodate (a), (b) or (c) without further recourse to integral equations. The theory is used to construct a two-airfoil design computer program which is shown to agree with exact special solutions. As a particular case the velocity for two lifting disjoint circular cylinders is given in a closed form involving only Theta functions.