Power of the wingbeat: modelling the effects of flapping wings in vertebrate flight

Animal flight performance has been studied using models developed for man-made aircraft. For an aeroplane with fixed wings, the energetic cost as a function of flight speed can be expressed in terms of weight, wing span, wing area and body area, where more details are included in proportionality coefficients. Flying animals flap their wings to produce thrust. Adopting the fixed wing flight model implicitly incorporates the effects of wing flapping in the coefficients. However, in practice, these effects have been ignored. In this paper, the effects of reciprocating wing motion on the coefficients of the fixed wing aerodynamic power model for forward flight are explicitly formulated in terms of thrust requirement, wingbeat frequency and stroke-plane angle, for optimized wingbeat amplitudes. The expressions are obtained by simulating flights over a large parameter range using an optimal vortex wake method combined with a low-level blade element method. The results imply that previously assumed acceptable values for the induced power factor might be strongly underestimated. The results also show the dependence of profile power on wing kinematics. The expressions introduced in this paper can be used to significantly improve animal flight models.