Inviscid analysis of two-dimensional airfoils in unsteady motion using conformal mapping

Abstract

Using a conformal mapping technique, a procedure for calculating the irrotational flow about a two-dimensional airfoil of arbitrary shape in unsteady motion is developed. The two-dimensional form with a sharp trailing edge is transformed into a unit circle by two successive transformations. The latter of the two is a modified version of the Gershgorin integral equation, which yields solutions much more accurate than those obtained with panel methods. The change in circulation around the airfoil due to unsteadiness is modeled by discrete vortices that are shed from the trailing edge and allowed to move freely with the local stream. The strength of these vortices is determined by the Kutta condition that, at each time step, the velocity be zero at the trailing edge in the circle plane. The procedure is efficient since the integral equation is solved only once. Numerical examples are presented for a sinusoidal heaving motion and for an impulsively started airfoil.