Phenomenology and scaling of optimal flapping wing kinematics

Biological flapping wing fliers operate efficiently and robustly in a wide range of flight conditions and are a great source of inspiration to engineers. The unsteady aerodynamics of flapping wing flight are dominated by large-scale vortical structures that augment the aerodynamic performance but are sensitive to minor changes in the wing actuation. We experimentally optimise the pitch angle kinematics of a flapping wing system in hover to maximise the stroke average lift and hovering efficiency with the help of an evolutionary algorithm and in situ force and torque measurements at the wing root. Additional flow field measurements are conducted to link the vortical flow structures to the aerodynamic performance for the Pareto-optimal kinematics. The optimised pitch angle profiles yielding maximum stroke-average lift coefficients have trapezoidal shapes and high average angles of attack. These kinematics create strong leading-edge vortices early in the cycle which enhance the force production on the wing. The most efficient pitch angle kinematics resemble sinusoidal evolutions and have lower average angles of attack. The leading-edge vortex grows slower and stays close-bound to the wing throughout the majority of the stroke-cycle. This requires less aerodynamic power and increases the hovering efficiency by 93% but sacrifices 43% of the maximum lift in the process. In all cases, a leading-edge vortex is fed by vorticity through the leading edge shear layer which makes the shear layer velocity a good indicator for the growth of the vortex and its impact on the aerodynamic forces. We estimate the shear layer velocity at the leading edge solely from the input kinematics and use it to scale the average and the time-resolved evolution of the circulation and the aerodynamic forces. The experimental data agree well with the shear layer velocity prediction, making it a promising metric to quantify and predict the aerodynamic performance of the flapping wing hovering motion.