An efficient, steady, subsonic collocation method for solving lifting-surface problems

Abstract

In the collocation approach for solving the integral equation of steady subsonic liftingsurface theory, accurate integration of the product of the kernel and pressure functions over the wing surface is required before a stable solution can be achieved. A Gaussian quadrature integration technique is developed which includes correction terms that account for the error incurred while integrating through the singularity and discontinuity of the kernel function. It is shown how inconsistent treatment of the chordwise discontinuity will cause the spanwise integration to diverge with an increasing number of integration chords. An optimum relationship is given between the number of chordwise downwash points and a larger number of chordwise integration points. Also given is an empirically determined optimum ratio of spanwise to chordwise control points based on planform geometry. By comparison with experiment and other theories, the method is shown to be both stable and accurate as well as more efficient than methods based on finite aerodynamic element representations of lifting surfaces.