Unsteady Propulsion of a Two-Dimensional Flapping Thin Airfoil in a Pulsating Stream

Abstract

The cruising velocity of animals, or robotic vehicles, that use flapping wings or fins to propel themselves is not constant but oscillates around a mean value with an amplitude usually much smaller than the mean, and a frequency that typically doubles the flapping frequency. Quantifying the effect that these velocity fluctuations may have on the propulsion of a flapping and oscillating airfoil is of great relevance to properly modeling the self-propelled performance of these animals or robotic vehicles. This is the objective of the present work, where the force and moment that an oscillating stream exerts on a two-dimensional pitching and heaving airfoil are obtained analytically using the vortical impulse theory in the linear potential flow limit. The thrust force of the flapping airfoil in a pulsating stream in this limit is obtained here for the first time. The lift force and moment derived here contain new terms in relation to the pioneering work by Greenberg (1947), which are shown quantitatively unimportant. The theoretical results obtained here are compared with existing computational data for flapping foils immersed in a stream with velocity oscillating sinusoidally about a mean value.