Bottom-up butterfly model with thorax-pitch control and wing-pitch flexibility*

Despite the diversity in the morphology of butterflies, the wing and body kinematics of butterflies have several common features. In the present study, we construct a bottom-up butterfly model whose morphology and kinematics are simplified while preserving the important features of butterflies. The present bottom-up butterfly model is composed of two trapezoidal wings and a rod-shaped body having a thorax and abdomen, and its wings are flapped downward in the downstroke and backward in the upstroke by changing the geometric angle of attack. The geometric angle of attack is determined by the thorax-pitch angle and wing-pitch angle. The thorax-pitch angle is actively controlled by the abdomen undulation, and the wing-pitch angle is passively determined due to a rotary spring representing the basalar and subalar muscles connecting the wings and thorax. We investigate how effective the abdomen undulation is for the thorax-pitch control and how the wing-pitch flexibility affects aerodynamic-force generation and thorax-pitch control, through numerical simulations using the immersed boundary-lattice Boltzmann method. As a result, the thorax-pitch angle perfectly follows the desired angle by the abdomen undulation. In addition, there is an optimal wing-pitch flexibility that maximizes flying speeds in both forward and upward directions, but the effect of wing-pitch flexibility is not significant on the thorax-pitch control.