OSCILLATING AIRFOILS IN TWO-DIMENSIONAL. INCOMPRESSIBLE FLOW

13.1 THE PROBLEM SPECIFIED

The problem of an oscillating thin airfoil in a two-dimensional incom­pressible flow will be considered in this chapter. The mean motion of the airfoil is a rectilinear translation of speed U with respect to the fluid at infinity. To this mean motion a simple-harmonic oscillation of infinitesimal amplitude is superposed. Since an arbitrary motion of an airfoil can be analyzed into harmonic components by means of a Fourier analysis, and, conversely, by a synthesis of the simple-harmonic com­ponents, any general motion can be established, the analysis of harmonic oscillations actually forms the basis of a general airfoil theory in unsteady motion (of small amplitude).

The airfoil to be considered is a planar system. The treatment will be limited to a linearized theory.

Within the framework of a linearized theory, solutions may be super­posed to generate another solution. The solution of an oscillating airfoil with finite but small thickness and camber at a given mean angle of attack can be obtained by a superposition of an unsteady solution for an oscill­ating airfoil of zero thickness and zero camber at zero mean angle of attack, and a steady-state solution for an airfoil of the given thickness and eamber at the given mean angle of attack. The steady-state solution can be found in many textbooks on aerodynamics. Therefore, in discussing the aerodynamics of an oscillating airfoil, it is sufficient to consider an airfoil of zero thickness and zero camber at zero mean angle of attack.

The two-dimensional nonstationary airfoil theory was first formulated by Birnbaum and Wagner. A short historical review of the earlier works of Kiissner, Glauert, and Theodorsen is given in § 5.8. More recent developments were made by Cicala, Kiissner, Schwarz, von Karman, Sears, Dietze, W. P. Jones, Biot, and others. See bibliography.