Kutta–Joukowski Theorem for Unsteady Linear Aerodynamics

This paper presents the extension of the Kutta–Joukowski theorem to unsteady linear aerodynamics. Starting from the formulation developed by Theodorsen for the solution of the velocity potential for circulatory flows around thin, rectilinear airfoils, the frequency response function between bound circulation and circulatory lift is derived. To provide a formulation suitable for time-domain applications, two finite-state approximations of the Kutta–Joukowski frequency response function, with different degrees of complexity and accuracy, are also proposed. The validity of the derived unsteady Kutta–Joukowski theorem is verified by correlation with numerical predictions of the circulatory lift given by a validated boundary-element-method solver for potential flows. Furthermore, the proposed finite-state approximations of the unsteady Kutta–Joukowski theorem are applied to problems concerning the circulatory lift response to damped oscillatory airfoil motion and gust perturbation. The higher-accuracy finite-state model provides predictions that correspond perfectly to those obtained by the convolution integrals of the step responses given by the theories of Wagner and Küssner. The results given by the simpler finite-state model derived from the linear approximation of the frequency response function are satisfactory for low-frequency problems, and are compared with those provided by a widely used approximate unsteady version of the Kutta–Joukowski theorem available in the literature