Torsional Instability of Fully Stalled Airfoil (or Supercavitating Hydrofoil)

Abstract

A theoretical study is carried out to determine the conditions under which steady-state pitching oscillations of an airfoil in one degree of freedom are possible in two-dimensional subsonic incompressible stalled flow using Parkin’s [1] coefficients. The results show that such oscillations can occur if the axis of rotation is located at a point that is approximately rearward of airfoil midchord but not too far rearward of the airfoil trailing edge. The practical significance of these results with respect to flutter is briefly examined. Torsional flutter [2] in the case of potential flow can occur only at very low reduced frequencies and under special circumstances, namely, when the rotation point is ahead of quarter chord, K′ > 550 and k < 0.0435. The significant difference between classical and stall flutter (torsional) is that the real part of the pitching moment coefficient [2] is negative in the former case and is positive in the latter case in the region of instability. Therefore, stall flutter can occur at very low values of K′ , unlike classical flutter where K′ has to be greater than 550. The ratio of torsional natural frequency to flutter frequency is greater than 1 in the case of stall flutter, whereas in classical flutter the ratio is less than 1 because of the sign of the real part of the moment coefficient. A single-degree-of-freedom torsional flutter equation was derived, and by equating the real and the imaginary parts of the equation, relationships between the various flutter parameters were obtained. Stability boundaries were obtained for the structural damping coefficient gα = 0, 0.001, 0.005, 0.01, 0.02, 0.04, and 0.1 for various values of K′ , namely, 5, 20, 40, 60, 80, 100, and 1000. The above values practically cover all airplane wings and compressor blades. The significant results are tabulated and some of the important parameters are plotted in the attached figures. By utilizing these figures, a typical airfoil in question could be checked, whether it is flutter-free or not, and if it is in flutter, all the flutter parameters could be determined. The analysis was conducted using Parkin’s coefficients [1], which require that the airfoil be completely stalled during the cycle.