^{1,a)}, Kei Senda

^{1}, Makoto Iima

^{2}and Norio Hirai

^{3}

### Abstract

Forward flights of a bilaterally symmetrically flapping butterfly modeled as a four-link rigid-body system consisting of a thorax, an abdomen, and left and right wings are numerically simulated. The joint motions of the butterflies are adopted from experimental observations. Three kinds of the simulations, distinguished by ways to determine the position and attitude of the thorax, are carried out: a tethered simulation, a prescribed simulation, and free-flight simulations. The upward and streamwise forces as well as the wake structures in the tethered simulation, where the thorax of the butterfly is fixed, reasonably agree with those in the corresponding tethered experiment. In the prescribed simulation, where the thoracic trajectories as well as the joint angles are given by those observed in a free-flight experiment, it is confirmed that the butterfly can produce enough forces to achieve the flapping flights. Moreover, coherent vortical structures in the wake and those on the wings are identified. The generation of the aerodynamic forces due to the vortical structures are also clarified. In the free-flight simulation, where only the joint angles are given as periodic functions of time, it is found that the free flight is longitudinally unstable because the butterfly cannot maintain the attitude in a proper range. Focusing on the abdominal mass, which largely varies owing to feeding and metabolizing, we have shown that the abdominal motion plays an important role in periodic flights. The necessity of control of the thoracic attitude for periodic flights and maneuverability is also discussed.

Numerical computation in this work was carried out at the Yukawa Institute Computer Facility. This work was partially supported by Grants-in-Aid for Scientific Research, MEXT.

I. INTRODUCTION

II. NUMERICAL SCHEME

A. Flow field

B. Butterfly

III. NUMERICAL RESULTS

A. Tethered simulation

B. Prescribed simulation

1. Forces and torques during flap

2. Vortical structures

IV. DISCUSSION: FREE-FLIGHT SIMULATION

V. CONCLUSION

### Key Topics

- Aerodynamics
- 61.0
- Rotating flows
- 43.0
- Torque
- 35.0
- Drag reduction
- 17.0
- Viscosity
- 10.0

##### F15D

## Figures

Definitions of coordinates and the Euler angles. The positive directions of the thoracic and joint angles θt, θa, β, η, and θ are expressed by the round arrows. (a) First, the coordinate of the left wing corresponds to that of the thorax on the right side. (d) After the three rotations (1: about the axis by β + π, 2: about the axis by −η, 3: about the axis by −θ) the coordinate corresponds to . (e) The coordinate of the thorax corresponds to that of the abdomen after the rotation about the axis by −θa (4). The positive directions of the Euler angles of the rotations ϕ ij are determined by the right-hand rule.

Definitions of coordinates and the Euler angles. The positive directions of the thoracic and joint angles θt, θa, β, η, and θ are expressed by the round arrows. (a) First, the coordinate of the left wing corresponds to that of the thorax on the right side. (d) After the three rotations (1: about the axis by β + π, 2: about the axis by −η, 3: about the axis by −θ) the coordinate corresponds to . (e) The coordinate of the thorax corresponds to that of the abdomen after the rotation about the axis by −θa (4). The positive directions of the Euler angles of the rotations ϕ ij are determined by the right-hand rule.

Observed joint angles of the abdomen and the wings of a tethered butterfly. The abdominal pitching angle, the approximate flapping angle, the approximate lead-lag angle, the approximate feathering angle are denoted by θa, β, η, and θ, respectively.

Observed joint angles of the abdomen and the wings of a tethered butterfly. The abdominal pitching angle, the approximate flapping angle, the approximate lead-lag angle, the approximate feathering angle are denoted by θa, β, η, and θ, respectively.

Observed thoracic coordinates during a free flight. The means of the vertical and horizontal coordinates during the flapping period are set to 0. The pitching angle is scaled by the right-hand axis.

Observed thoracic coordinates during a free flight. The means of the vertical and horizontal coordinates during the flapping period are set to 0. The pitching angle is scaled by the right-hand axis.

Upward and streamwise forces and longitudinal torque acting on the thorax of the tethered butterflies in the numerical simulation and the experiment. (a) Upward force, (b) streamwise force, and (c) longitudinal torque. The aerodynamic forces and pitching moment in the simulation are also drawn as the lift, drag, and pitching moment, respectively.

Upward and streamwise forces and longitudinal torque acting on the thorax of the tethered butterflies in the numerical simulation and the experiment. (a) Upward force, (b) streamwise force, and (c) longitudinal torque. The aerodynamic forces and pitching moment in the simulation are also drawn as the lift, drag, and pitching moment, respectively.

Smoke structures in the tethered experiment. The smoke is released on the vertical segment of z = 4y tip/5 on the inflow boundary. (a) Topview and (b) sideview.

Smoke structures in the tethered experiment. The smoke is released on the vertical segment of z = 4y tip/5 on the inflow boundary. (a) Topview and (b) sideview.

Passive tracers in the tethered simulation. (a) Topview and (b) sideview.

Passive tracers in the tethered simulation. (a) Topview and (b) sideview.

Lift, drag, and pitching moment due to aerodynamic force in the prescribed simulation. The pitching moment is scaled by the right-hand axis.

Lift, drag, and pitching moment due to aerodynamic force in the prescribed simulation. The pitching moment is scaled by the right-hand axis.

Upward and streamwise forces and longitudinal torque due to extra control applied to the thorax in the prescribed simulation. The longitudinal torque are scaled by the right-hand axis.

Upward and streamwise forces and longitudinal torque due to extra control applied to the thorax in the prescribed simulation. The longitudinal torque are scaled by the right-hand axis.

Torques applied to the joints of the abdomen and of the left wing τwlβ, τwlη and τwlθ in the prescribed simulation.

Torques applied to the joints of the abdomen and of the left wing τwlβ, τwlη and τwlθ in the prescribed simulation.

Power due to aerodynamic force P air, extra control P 1, and joint control forces P 2 in the prescribed simulation. The power due to the joint control is the sum of those due to the abdominal and wing joints, i.e., P 2 = P a + P wl + P wr.

Power due to aerodynamic force P air, extra control P 1, and joint control forces P 2 in the prescribed simulation. The power due to the joint control is the sum of those due to the abdominal and wing joints, i.e., P 2 = P a + P wl + P wr.

Vortical structures visualized by the Q-criterion, i.e., the second invariant of velocity gradient tensor in the prescribed simulation. The isosurface of Q = 1 × 105 s−2 is drawn. (a) Topview and (b) sideview.

Vortical structures visualized by the Q-criterion, i.e., the second invariant of velocity gradient tensor in the prescribed simulation. The isosurface of Q = 1 × 105 s−2 is drawn. (a) Topview and (b) sideview.

Schematic illustration of the three-dimensional vortical structures. The directions of the coherent vortices are expressed by the arrows.

Schematic illustration of the three-dimensional vortical structures. The directions of the coherent vortices are expressed by the arrows.

Near-field structures viewed from above and behind in the prescribed simulation. The forces and the isosurface of Q = 2.5 × 105 s−2 are drawn. The streamlines starting from two segments are drawn at t = T/4. The spanwise vorticities ω z on the right wing and the streamwise vorticities ω x in the downstream are also drawn on the corresponding cross-section at t = T/4. (a) Forces and Q-criterion at t = T/4, (b) streamlines and vorticities at t = T/4, (c) forces and Q-criterion at t = 5T/8, and (d) forces and Q-criterion at t = 7T/8. The magnitudes and the direction of the forces are, respectively, expressed by density (density of blue) and by the displacement from the wing. The forces and the streamlines are shown on the left side while the isosurface of Q and the vorticities are shown on the right side.

Near-field structures viewed from above and behind in the prescribed simulation. The forces and the isosurface of Q = 2.5 × 105 s−2 are drawn. The streamlines starting from two segments are drawn at t = T/4. The spanwise vorticities ω z on the right wing and the streamwise vorticities ω x in the downstream are also drawn on the corresponding cross-section at t = T/4. (a) Forces and Q-criterion at t = T/4, (b) streamlines and vorticities at t = T/4, (c) forces and Q-criterion at t = 5T/8, and (d) forces and Q-criterion at t = 7T/8. The magnitudes and the direction of the forces are, respectively, expressed by density (density of blue) and by the displacement from the wing. The forces and the streamlines are shown on the left side while the isosurface of Q and the vorticities are shown on the right side.

Lift, drag, and pitching moment due to aerodynamic force during the first free-flight flapping period.

Lift, drag, and pitching moment due to aerodynamic force during the first free-flight flapping period.

Thoracic coordinates during the first free-flight flapping period.

Thoracic coordinates during the first free-flight flapping period.

Lift, drag and pitching moment due to aerodynamic force during the three free-flight flapping periods with the smaller abdominal mass 3m a/4.

Lift, drag and pitching moment due to aerodynamic force during the three free-flight flapping periods with the smaller abdominal mass 3m a/4.

Thoracic coordinates during the three free-flight flapping periods with the smaller abdominal mass 3m a/4.

Thoracic coordinates during the three free-flight flapping periods with the smaller abdominal mass 3m a/4.

Upward and streamwise forces and longitudinal torque in four tests for computational validation.

Upward and streamwise forces and longitudinal torque in four tests for computational validation.

## Tables

Physical properties of butterflies (chestnut tiger).

Physical properties of butterflies (chestnut tiger).

Numerical conditions of three simulations.

Numerical conditions of three simulations.

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