Linearized propulsion theory of flapping airfoils revisited

A vortical impulse theory is used to compute the thrust force of a plunging and pitching airfoil in forward flight at high Reynolds numbers within the framework of linear potential flow theory. The result is significantly different from the classical one of Garrick, which considered only two effects, the leading-edge suction and the projection in the flight direction of the pressure force on the airfoil. By taking into account the complete vorticity distribution on the airfoil and the wake the mean thrust coefficient contains, in addition to the pressure force projection term, a new term that generalizes the leading-edge suction term in Garrick's theory. This term depends on Theodorsen function C(k) and on a new complex function C-1(k) of the reduced frequency k. The main qualitative difference with Garrick's theory is that the propulsive efficiency, or ratio of the mean thrust power and the mean input power required to drive the airfoil, tends to zero as the reduced frequency increases to infinity (as k(-1)), in contrast to Garrick's propulsive efficiency that tends to a constant (1/2). Consequently, for pure pitching and combined pitching and plunging motions, the maximum of the propulsive efficiency is not reached as k -> infinity like in Garrick's theory, but at a finite value of the reduced frequency that depends on the remaining nondimensional parameters. The present analytical results are in good agreement, for small amplitude oscillations, with numerical results from unsteady panel methods, and with experimental data and numerical results from the Navier-Stokes equations, except for small reduced frequencies where viscous effects are obviously important.