Phenomenology and scaling of optimal flapping wing kinematics

Abstract

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.

On the parametrisation of motion kinematics for experimental aerodynamic optimisation

“…The kinematics for the different optimisation studies are evaluated with a robotic flapping wing mechanism immersed in an octagonal water tank (see also [9,8]).…”

“…The flapping frequency of the system is f = 0.25 Hz with a peak-to-peak stroke amplitude of φ = 180°. The flapping frequency and stroke amplitude remain constant throughout this study and lead to a Reynolds number of Re = 2φf cR 2 /ν = 4895 at the radius of the second moment of area R 2 , for a kinematic viscosity of ν 20 °C = 1.00 × 10 −6 m 2 /s for all experiments [8,23].…”

“…The flapping wing apparatus was specifically designed to run unsupervised for large amounts of time and to execute complex motion profiles with high repeatability. Preliminary tests have determined the maximum error between prescribed motion and motor response on the mechanism to be < 0.1°for high acceleration kinematics [8]. The motors are controlled by a motion control board (DMC-4040, Galil Motion Control, USA).…”

“…Small modifications of the parameterisation can lead to a significantly altered solution landscape with a bias towards small but pronounced local optima. Gehrke and Mulleners [8] studied pitching motions with increased complexity, which are able to mimic more closely the motions observed in nature. The stroke motion was fixed and the elevation motion was omitted.…”

“…Here, we will present three different approaches to define and parameterise kinematics for optimisation studies. The different approaches will be presented, compared, and evaluated for the example of the experimental optimisation of the pitching kinematics of a flapping wing, building upon the work presented in Gehrke et al [9] and Gehrke and Mulleners [8]. An optimisation is performed for each kinematic function with 6, 12 and 18 parameters, leading to more than 30 000 individual experiments in total.…”