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

This work is an assessment of a recently proposed variational theory of lift where circulation is selected for simply connected cross sections by minimizing the pressure gradients in the flow field. Unlike the traditional Kutta condition, the variational theory of lift is not constrained in its application to geometries with sharp trailing edges. Results in this work show that in the limit as trailing edge geometry approaches mathematical sharpness, the circulation from the variational theory of lift converges to the circulation attained through the Kutta condition. Additionally, in the limit as the profile of a given geometry approaches that of a circular cylinder, the circulation from the variational theory of lift converges to zero from the negative direction. Furthermore, it is shown that the variational theory of lift predicts negative circulation generation for positively inclined sharp leading-edged airfoils and zero circulation for all fore-aft symmetric geometries. This paper does not definitively prove or disprove the variational theory of lift. Instead, it provides detailed instructions for the reader to test the theory in the context of conformal mapping such that the corresponding results can be replicated. The results in this paper demonstrate that further exploration of the variational theory of lift is warranted, and that the limitations inherent in traditional applications of conformal mapping should be considered.