The quasi-steady / unsteady transition for two dimensional flapping wings

We use 2D DNS at Re=1000 to investigate the unsteady characteristics of flows which characterize the heaving and the pitching motion component contributing to the forward flight of insects. This reveals the existence of two critical reduced frequencies above which the added mass reaction and the rotational forces start to compete with the quasi steady circulatory forces which dominate the lower frequency regimes. We also show that an increase in the frequencies of these movement leads to a reduction of the leading edge vortex (LEV) size and to promotion of the wake capture mechanism. A frequency diagram is proposed where the various critical frequencies are listed and were three regimes are emphasized : a quasi steady regime, a transitional regime in which unsteady effects interact with quasi steady forces, and a pure unsteady regime in which unsteady forces are dominant. A comparison with living species show that these three different regimes correspond respectively to birds, large insects and small insects. Then, we study a realistic motion where heaving, pitching and sliding motions are combined. Using a kinematics adapted to a quasi steady regime, we show that when frequency is increased, the lift and the thrust are first leaded by the rotational force in the transitional regime. Reduction of the LEV size and wing wake interactions then distort the instantaneous efforts during the upstroke, whereas the contribution of the added mass reaction still remains unclear.