Evaluating the Variational Theory of Lift Applied to Conformal Mappings of Airfoils

Abstract

This work assesses a recently proposed variational theory of lift where the circulation generated by a simple closed geometry minimizes the pressure gradients in the flowfield. Unlike the traditional Kutta condition, it is appropriate to apply the variational theory of lift to geometries without sharp trailing edges. This paper demonstrates that, for a Joukowski transformation, in the limit as trailing edge geometry approaches mathematical sharpness, the circulation from the variational theory of lift converges to the Kutta condition circulation. Additionally, in the limit as the shape of a given geometry approaches that of a circular cylinder, circulation from the variational theory of lift converges to zero. Within the confines of the conformal mapping model described in this paper, the variational theory of lift also predicts that reverse airfoils generate negative circulation at positive angles of attack and that all doubly symmetric geometries generate zero circulation independent of angle of attack. These results suggest it may be inappropriate to force flow separation at a single location for geometries without sharp trailing edges. Future work should apply the variational theory of lift to a model which allows for multiple separation points.