Low Reduced Frequency Theory of Airfoils Vibrating in Complex Modes

Abstract

This paper extends a previous study on the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime to the case of complex modes, where the phase of the different points of the same airfoil is not constant. A complex mode-shape can be expressed as the combination of two rigid body modes shifted 90 deg. Therefore, the stability of such modes is more complex to predict than that of the so-called real modes. The aerodynamic work-per-cycle has been deconstructed into its fundamental components, and its dependence on the reduced frequency described. The three distinct contributions to the work-per-cycle of a complex mode have been analyzed theoretically and numerically. First, the mean pressure yields a constant contribution to the work-per-cycle that is independent of the interblade phase angle (IBPA) and the reduced frequency. This term scales with the aerodynamic loading of the airfoil and is null for nonlifting airfoils. Second, the contribution of the two fundamental modes, whose mean value is proportional to the reduced frequency. Finally, the cross works of a fundamental mode over the displacements of the other. The low reduced frequency theory predicts that this term is independent of the frequency and symmetric with the interblade phase angle. The latter has been confirmed using linearized Navier–Stokes simulations, but the former has a constant term, as predicted by the theory, plus a small variation with the frequency that has been associated with acoustic resonances.