Refining Kutta’s Flow over a Flat Plate: Necessary Conditions for Lift

Abstract

In this paper, we present a variational theory of lift that, unlike Kutta’s theory, is derived from first principles in mechanics: Hertz’s principle of least curvature. In this theory, the unique value of circulation is determined by minimizing the Appellian of the flowfield. Interestingly, it recovers the Kutta condition in the special case of an airfoil with a sharp trailing edge. In this paper, we apply such a theory to the classical problem of the flow over a flat plate. The resulting ideal flow does not match Kutta’s solution in this case; it results in a nonlifting solution for any uncambered, fore-aft symmetric shape, confirming experimental findings in superfluids. This result provides necessary conditions for lift generation in an ideal fluid. For a real fluid over a flat plate with a sharp leading edge, viscosity plays an important role, leading to a flow separation at the leading edge, even at small angles of attack. This separation bubble creates asymmetry in the outer inviscid flowfield (outside the bubble), which enables lift. This problem is discussed in the light of the developed variational theory of lift and some historical details about the development of Kutta’s theory.