"Energy-minimizing kinematics in hovering insect flight" by Gordon J. Berman & Z. Jane Wang
J. Fluid Mech. 582 (2007), 153--168
Summary: "We investigate aspects of hovering insect flight by finding the optimal wing kinematics which minimize power consumption while still providing enough lift to maintain a time-averaged constant altitude over one flapping period. In particular, we study the flight of three insects whose masses vary by approximately three orders of magnitude: fruitfly (Drosophila melanogaster), bumblebee (Bombus terrestris), and hawkmoth (Manduca sexta). Here, we model an insect wing as a rigid body with three rotational degrees of freedom. The aerodynamic forces are modelled via a quasi-steady model of a thin plate interacting with the surrounding fluid. The advantage of this model, as opposed to the more computationally costly method of direct numerical simulation via computational fluid dynamics, is that it allows us to perform optimization procedures and detailed sensitivity analyses which require many cost function evaluations. The optimal solutions are found via a hybrid optimization algorithm combining aspects of a genetic algorithm and a gradient-based optimizer. We find that the results of this optimization yield kinematics which are qualitatively and quantitatively similar to previously observed data. We also perform sensitivity analyses on parameters of the optimal kinematics to gain insight into the values of the observed optima. Additionally, we find that all of the optimal kinematics found here maintain the same leading edge throughout the stroke, as is the case for nearly all insect wing motions. We show that this type of stroke takes advantage of a passive wing rotation in which aerodynamic forces help to reverse the wing pitch, similar to the turning of a free-falling leaf."