Minimum induced power requirements for flapping flight

The Betz criterion for minimum induced loss is used to compute the optimal circulation distribution along the span of flapping wings in fast forward flight. In particular, we consider the case where flapping motion is used to generate both lift (weight support) and thrust. The Betz criterion is used to develop two different numerical models of flapping. In the first model, which applies to small-amplitude harmonic flapping motions, the optimality condition is reduced to a one-dimensional integral equation which we solve numerically. In the second model, which applies to large-amplitude periodic flapping motions, the optimal circulation problem is reduced to solving for the flow over an infinitely long wavy sheet translating through an inviscid fluid at rest at infinity. This three-dimensional flow problem is solved using a vortex-lattice technique. Both methods predict that the induced power required to produce thrust decreases with increasing flapping amplitude and frequency. Using the large-amplitude theory, we find that the induced power required to produce lift increases with flapping amplitude and frequency. Therefore, an optimum flapping amplitude exists when the flapping motion of wings must simultaneously produce lift and thrust.