On the Theory of the Unsteady Motion of an Airfoil

The paper presents a systematical analysis of the problem of the determination of the unsteady motion about an airfoil moving in an infinite fluid that contains a system of vortices and the determination of the hydrodynamical forces acting on the airfoil. The hydrodynamical problem is reduced to the determination of the function f (xi) which transforms conformally the external region of the airfoil into the interior of a circle. The proposed methods of determining the irrotational motion of a fluid that is produced by any motion of the airfoil are especially simple and effective if the function f (xi) is rational. As an example the flow is determined for the case of an arbitrary motion of an airfoil of the Joukowsky type. The formulas obtained for the determination of the hydrodynamical forces by means of contour integration are similar to those given by S. Chaplygin. These formulas are used to determine the force acting on the airfoil in the cases where the unsteady motion is potential throughout and the circulation about the airfoil is constant and also when the fluid contains a system of vortices. A full discussion is given of the concept of virtual masses together with practical formulas for computing the virtual mass coefficients