Interaction between the fore- and hind-wings in hovering flight of modelled dragonfly*

In the present study, we investigate the interaction between the fore- and hind-wings in hovering flight of modelled dragonfly using 3D numerical simulation. The three-dimensional wing shape is based on that of {\it Aeschna juncea} (Norberg 1972) and numerically realized using an immersed boundary method (Kim et al. 2001). The wing flapping motion is modelled using a sinusoidal function and the stroke plane angle is 60$^{\circ}$. We consider 12 different phase differences between the fore- and hind-wings ($\phi=0^{\circ} \sim ~$330$^{\circ}$). The Reynolds number is 1,000 based on the maximum translational velocity and mean chord length. In counter stroke ($\phi=180^{\circ}$), the wing-tip vortices from both wings are connected, generating an entangled wing-tip vortex (e-WTV). A strong downward motion induced by this vortex decreases the vertical force in the following stroke (Kweon \& Choi 2008). In parallel stroke ($\phi=0^{\circ}$), both wings meet e-WTV during the upstroke and thus the decrease of vertical force is small. At $\phi=270^{\circ}$, although e-WTV is generated on a relatively narrow region, the hind-wing moves downward along with e-WTV, resulting in a significant reduction of vertical force on the hind-wing. Therefore, the sum of vertical forces on both wings is maximum with parallel stroke and minimum at $\phi=270^{\circ}$. The power required has a similar trend to the vertical force and thus the efficiency does not show a large variation with the phase difference.