Summary of Chapter 3

In this chapter, small disturbance theory associated with thin airfoils is used to simplify and obtain linearized formulations of the tangency condition and pressure field. The general problem of inviscid, incompressible potential flow past thin airfoils is now fully linear and can be decomposed into symmetric and lifting problems that are handled by a distribution of sources/sinks for the former, and of vortices for the latter.

The symmetric problem does not contribute to lift or moment, however, thick­ness affects the pressure distribution. The lifting problem leads to the fundamental integral equation of thin airfoil theory, which relates the vorticity distribution Г’ to the induced velocity component w along the profile, via the tangency condition. The solution of the fundamental integral equation is worked out using a singular term plus a Fourier series of regular terms for the vorticity along the cut [0, c]. It is shown how the singular term and the Fourier coefficients can be obtained for a given thin cambered plate at incidence. The interpretation of the singular term coefficient as A0 = a — o. adapt points to the existence, for each thin airfoil of an angle of adapta­tion, that depends purely on the airfoil geometry, such that the leading edge satisfies also a Kutta-Joukowski condition. This has a practical interest since separation at a sharp leading edge is likely to occur in real fluid when a = aadapt. The positive role of thick leading edge is also emphasized in this regard.

The forces and moments are obtained for arbitrary airfoils and from these, the center of pressure can be found. In all cases, the drag is zero (d’Alembert paradox). In general, as the incidence varies from large negative to large positive values, the center of pressure will travel from —^ to +ro. At very high or low incidences, the center of pressure moves to the quarter chord, pp = 1. The aerodynamic center for all thin airfoils is found to be at the quarter chord, p – = 4.

Application of the theory to the design of a thin airfoil exemplifies the practical aspects and usefulness of the results derived in this chapter. It is also shown that some particular thickness and camber distributions can be related to the concept of minimum pressure gradient field surrounding the airfoil. This idea has been applied to the design of double element airfoils, which have proved to perform well in practice.

Finally, numerical aspects of the solution of the fundamental integral equation of thin airfoil theory are given, which, properly implemented, contribute to fast and accurate analysis tools for the designer.