Theoretical Solutions for Unsteady Flows Past Oscillating Flexible Airfoils Using Velocity Singularities

A new method of solution is presented for the analysis of unsteady flows past oscillating airfoils. This method is based on the derivation of the singular contributions of the leading edge and ridges (points where the airfoil boundary conditions change) in the solutions of the fluid velocity and pressure coefficient. The method has been validated by comparison with the results obtained by Theodorsen and by Postel and Leppert for rigid airfoil and aileron oscillations in translation and rotation. An excellent agreement was found between the present solutions and these previous results. This method has then been used to obtain solutions for the flexural oscillations of the flexible airfoils, fitted or not with oscillating flexible ailerons, which are of interest for the aeroelastic studies. The aerodynamic stiffness, damping, and virtual (or added) mass contributions in the solutions of the unsteady pressure distribution, lift coefficient, and moment coefficient are specifically determined. An analysis of the relative magnitude of the quasi-steady and vortex shedding contributions in the aerodynamic coefficients is also presented. In all cases studied, this method led to very efficient and simple analytical solutions in closed form.