Optimal flexibility of a flapping appendage in an inviscid fluid

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.