Passive mechanism of pitch recoil in flapping insect wings

The high torsional flexibility of insect wings allows for elastic recoil after the rotation of the wing during stroke reversal. However, the underlying mechanism of this recoil remains unclear because of the dynamic process of transitioning from the wing rotation during stroke reversal to the maintenance of a high angle of attack during the middle of each half-stroke, when the inertial, elastic, and aerodynamic effects all have a significant impact. Therefore, the interaction between the flapping wing and the surrounding air was directly simulated by simultaneously solving the incompressible Navier-Stokes equations, the equation of motion for an elastic body, and the fluid-structure interface conditions using the three-dimensional finite element method. This direct numerical simulation controlling the aerodynamic effect revealed that the recoil is the residual of the free pitch vibration induced by the flapping acceleration during stroke reversal in the transient response very close to critical damping due to the dynamic pressure resistance of the surrounding air. This understanding will enable the control of the leading-edge vortex and lift generation, the reduction of the work performed by flapping wings, and the interpretation of the underlying necessity for the kinematic characteristics of the flapping motion.