Propulsive Force of a Flexible Flapping Thin Airfoil

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The propulsive force generated by the leading-edge suction of a flapping thin airfoil can be obtained using unsteady aerodynamic notions based on the potential flow theory. Because the potential theory fails to predict the viscous forces, for flapping airfoils, in general there exists a net propulsive force generation. For low Reynolds numbers and for oscillations at low reduced frequencies, however, the viscous forces overcome the propulsive forces to give a net force against the flight direction. In the case of higher Reynolds numbers, flappings with large amplitudes, and high frequencies, the leading-edge suction force generated by the airfoil overcomes the viscous forces. In this study, the critical Reynolds number, the amplitude, and the reduced frequency values that yield the net propulsive force for an oscillating airfoil are predicted with unsteady viscous-inviscid interaction. The airfoil motions are modeled as 1) a thin rigid plate in vertical oscillation, that is, a heaving-plunging motion; 2) a flexibly cambered airfoil whose camber is changing periodically; and 3) the heaving-plunging motion of a flexibly cambered airfoil. The leading-edge suction force for all cases is predicted by means of the well-known Blasius theorem, and the time dependent surface velocity distribution of the airfoil is determined by unsteady aerodynamic considerations. This surface velocity distribution is used as the edge velocity of the unsteady boundary layer to predict the viscous effects. The coefficient of the propulsive force predicted with the present method agrees well with the values given in the literature up to the effective angle of attack at which the dynamic stall takes place.