On the unsteady wake-induced lift on a slotted airfoil, part II: The influence of displacement thickness fluctuations

Abstract

In the preceding companion paper [1] a theoretical model for determining the influence of a slot in a thin airfoil on the unsteady lift/radiated sound caused by vortices shed into the wake was presented. The unsteady motion produces additional vorticity at the upstream edge of the slot, and it was shown that, at sufficiently low reduced frequencies based on the width of the slot, this vorticity can prevent penetration by the flow, so that the airfoil behaves as if the slot were absent. At higher frequencies, however, both the lift and the sound power were predicted to be significantly reduced relative to their respective levels for the unslotted airfoil. The analysis is extended in this paper to include the effects of displacement thickness fluctuations of the boundary layers on the “flap” downstream of the slot. These fluctuations arise as a result of the periodic ejection of vorticity from the slot. It is concluded that the earlier predictions of a reduction in the lift/sound pressure level are enhanced by the displacement thickness effects.