A Variational Theory of Lift

In this work, we consider one of the most fundamental questions in fluid mechanics and aerodynamics: lift generation over an airfoil. We exploit a special, less-common, variational principle in analytical mechanics (Hertz' principle of least curvature) to develop a variational analogue of Euler's equations for the dynamics of an ideal fluid. We apply this variational formulation to the classical problem of the flow over an airfoil, which presents, for the first time, an airfoil theory that dispenses with the Kutta condition. The developed variational principle reduces to the Kutta-Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. It is quite a fundamental result that generalizes a 120 years old theory by developing an original framework of the aerodynamics of lifting bodies into a theoretical mechanics formulation, exploiting and reviving forgotten tools from the history of mechanics (Hertz' principle of least curvature).