Free flight simulations of a dragonfly-like flapping wing-body model by the immersed boundary-lattice Boltzmann method

Free flights of the dragonfly-like flapping wing-body model are numerically investigated by using the immersed boundary-lattice Boltzmann method (IB-LBM). First, we simulate free flights of the model without the pitching rotation for various values of the phase lag angle $\phi$ between the forewing and the hindwing motions. We find that the wing-body model goes forward in spite of $\phi$, and the model with $\phi=$0$^{\circ}$ and 90$^{\circ}$ goes upward against gravity. The model with $\phi=180^{\circ}$ goes almost horizontally, and the model with $\phi=270^{\circ}$ goes downward. Secondly, we simulate free flights with the pitching rotation for various values of the phase lag angle $\phi$. It is found that in spite of $\phi$ the wing-body model turns gradually in the nose-up direction and goes back and down as the pitching angle ${\it \Theta}_{\rm c}$ increases. That is, the wing-body model cannot make a stable forward flight without control. Finally, we show a way to control the pitching motion by changing the lead-lag angle $\gamma(t)$. We propose a simple proportional controller of $\gamma(t)$ which makes stable flights within ${\it \Theta}_{\rm c}=\pm 5^{\circ}$ and works well even for a large disturbance.