Harmonic analysis of unsteady transonic flow

Abstract

Introduction T time averaged mean flow past an airfoil oscillating harmonically at transonic speeds differs from the steady flow obtained at rest. This difference depends on oscillation frequency and amplitude and arises because both mean and disturbance flowfields interact nonlinearly. For small oscillation amplitudes a time linearized description usually suffices: the mean flow is conveniently solved without reference to the unsteady flow and the harmonic flow is solved without explicit reference to the unsteady disturbance amplitude. For larger amplitudes, however, the wave backinteraction induced by the primary disturbance on the mean flow cannot be discounted because the modified mean flow in turn affects the evolution of the harmonic flowfield. This effect may be significant because transonic flows are inherently nonlinear. Linear theory, which generally assumes small unsteady disturbance amplitudes in comparison to the overall thickness, expands the unsteady solution about the known transonic flow corresponding to the stationary airfoil. For oscillatory motions the mean flowfield is therefore defined by a simplified solution rendered independent of disturbance frequency and amplitude: the model implicitly assumes high-frequency oscillations where the mean flow and mean shock position effectively freeze in space. However, for low-frequency motions where large shock excursions are anticipated, the mean flow couples more strongly with the unsteady flow and linear theory may not apply. Thus the practical need arises for a rational harmonic formulation adaptable to unsteady transonic flutter and aeroelastic analyses yielding unstable amplitude as well as frequency boundaries. In this Note, a nonlinear harmonic approach to general unsteady oscillations in transonic flow is developed for those engineering applications where explicit amplitude and frequency information is required. The extent to which nonlinear feedback is significant is addressed, in particular, for the symmetric NACA 64A006 airfoil, unpitched, with a quarter chord oscillating flap executing large amplitude deflections in a subsonic freestream mildly to strongly supercritical. Details of the numerical algorithm are also outlined.