A note on the thrust of airfoils

Here, we show that the thrust force of oscillating airfoils calculated within the linearised potential flow approach by means of the vortex impulse theory coincides with the one resulting from the integration of the unsteady pressure distribution around the solid obtained by Garrick (1936) when the vertical component of the wake velocity is calculated self-consistently and the analysis retains the contribution of the flux of horizontal momentum induced by the starting vortex. The limitations of the self-consistent linearised potential flow approach for predicting the thrust force of airfoils oscillating periodically with small amplitudes but large values of the reduced frequency are also discussed, as well as the reasons behind the ability of other results in the literature to approximate measurements better than Garrick’s theory. In fact, for those cases in which the airfoil oscillates periodically, the flux of horizontal momentum induced by the starting vortex is negligible and the vortices in the wake are convected parallel to the free-stream velocity, we have deduced an equation for the mean thrust coefficient which differs from previously published results and is in agreement with experimental and numerical results. In addition, for those cases in which the airfoil is suddenly set into motion, we have also deduced an equation that retains the effect of the starting vortex and correctly quantifies the transient thrust force.