Application of Gauss' Principle to the Classical Airfoil Problem

Gauss’s principle of least constraint is applied to the classical problem of lift of a two-dimensional airfoil in ideal, incompressible flow and examines the role of the empirical Kutta condition in classical airfoil theory. Gauss’s principle demonstrates that the fluid pressure field of a lifting airfoil comprises both a constraint pressure and an impressed pressure of arbitrary strength. Gauss’s principle also proves that a flow solution that minimizes the difference between the acceleration and the impressed pressure gradient is a valid solution of Euler’s equation. Furthermore, Gauss’s principle proves that a first-principles closure condition cannot exist thus reaffirming the need for the empirical Kutta condition to complete classical airfoil theory. This investigation also disproves a new variational theory of lift derived from Hertz’s principle of minimum curvature that claims to replace the empirical Kutta condition with a first-principles based closure condition. This new theory predicts airfoil lift for conventional airfoils consistent with classical theory, however, for other airfoils, the predictions are highly implausible; e.g., zero lift for the flat plate, and negative lift for an airfoil in reverse flow. The present investigation explains why the new theory is invalid.