A variational theory of lift

Abstract

In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical 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. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.