Effect of chordwise wing flexibility on flapping flight of a butterfly model using immersed-boundary lattice Boltzmann simulations

Wing flexibility is one of the important factors not only for lift and thrust generation and enhancement in flapping flight but also for development of micro-air vehicles with flapping wings. In this study, we construct a flexible wing with chordwise flexibility by connecting two rigid plates with a torsion spring, and investigate the effect of chordwise wing flexibility on the flapping flight of a simple butterfly model by using an immersed boundary-lattice Boltzmann method. First, we investigate the effects of the spring stiffness on the aerodynamic performance when the body of the model is fixed. We find that the time-averaged lift and thrust forces and the required power increase with the spring stiffness. In addition, we find an appropriate range of the spring stiffness where the time-averaged lift and thrust forces are larger than those of the rigid wings. The mechanism of the lift and thrust enhancements is as follows: in the downstroke the flexible wings can generate not only the lift force but also the thrust force due to the deformation of wings; in the upstroke the flexible wings can generate not only the thrust force but also the lift force due to the deformation of wings. Second, we simulate free flights when the body of the model can only move translationally. We find that the model with the flexible wings at an appropriate value of the spring stiffness can fly more effectively than the model with the rigid wings, which is consistent with the results when the body of the model is fixed. Finally, we simulate free flights with pitching rotation. We find that the model gets off balance for any value of the spring stiffness. Therefore, the passive control of the pitching motion by the chordwise wing flexibility cannot be expected for the present butterfly model.

Figure 1

Illustration of (a) a flexible wing and (b) a butterfly model with two flexible wings and a rod-shaped body.

Figure 2

The wing motion during one period for $t/T\ge 1$ when the wings are rigid and $({\theta }_{\mathrm{m}},{\alpha }_{\text{m}},\gamma )=({45}^{\circ },{90}^{\circ },\pi /2)$.

Figure 3

Time variations of the deflection angle ${\theta }_{\mathrm{w}}$ for various values of the nondimensional spring stiffness $N_{\mathrm{K}}$ when the body of the model is fixed, $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 1 in Table 2.

Figure 4

Time variations of (a) lift coefficient $C_{\mathrm{L}}$, (b) thrust coefficient $C_{\mathrm{T}}$, (c) pitching moment coefficient $C_{\mathrm{M}}$, and (d) power coefficient $C_{\mathrm{P}}$ for various values of the nondimensional spring stiffness $N_{\mathrm{K}}$ when the body of the model is fixed, $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 1 in Table 2. Each figure has a small subfigure in the right-hand side. In these subfigures, the left, right, and central bullets indicate the time-averaged values in the downstroke, upstroke, and full cycle, respectively. In the subfigure of panel (a), the dotted line indicates the nondimensional weight force of a Janatella leucodesima equal to 0.605.

Figure 5

Vortex structures viewed from (a-1) the upper side and (a-2) the right side of the butterfly model, and (b) the color maps of the pressure fields on the plane perpendicular to the $z$ axis for various positions relative to the wing root at $t/T=9.23$ when the body of the model is fixed, $N_{\mathrm{K}}=1.0$, $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 1 in Table 2. In (a-1) and (a-2), the model is shown in red, and the isosurface of the $Q$ criterion $[Q=50{(U_{\mathrm{ref}}/L_{\mathrm{ref}})}^2]$ is shown in blue. In (b-1)–(b-5), the wing chord is shown in black, and the color map shows the nondimensional pressure, i.e., $(p-p_0)/\left({\rho }_{\mathrm{f}}{U}_{\mathrm{tip}}^2\right)$, where $p_0$ is the initial value of the pressure.

Figure 6

The ratios of (a) time-averaged lift coefficient $\overline{C_{\mathrm{L}}}$, (b) time-averaged thrust coefficient $\overline{C_{\mathrm{T}}}$, (c) time-averaged power coefficient $\overline{C_{\mathrm{P}}}$, and (d) power-loading coefficient $C_{\mathrm{PL}}$ to the values for the rigid wings $\overline{C_{\mathrm{L}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{T}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{P}}^{\mathrm{rigid}}}$, and $C_{\mathrm{PL}}^{\mathrm{rigid}}$, respectively, as the functions of the nondimensional spring stiffness $N_{\mathrm{K}}$ for various values of the nondimensional hind wing-tip length $\chi$ when the body of the model is fixed, $\text{Re}=500$, and the other governing parameters are given as Set 1 in Table 2.

Figure 7

The ratios of (a) time-averaged lift coefficient $\overline{C_{\mathrm{L}}}$, (b) time-averaged thrust coefficient $\overline{C_{\mathrm{T}}}$, (c) time-averaged power coefficient $\overline{C_{\mathrm{P}}}$, and (d) power-loading coefficient $C_{\mathrm{PL}}$ to the values for the rigid wings $\overline{C_{\mathrm{L}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{T}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{P}}^{\mathrm{rigid}}}$, and $C_{\mathrm{PL}}^{\mathrm{rigid}}$, respectively, as the functions of the nondimensional spring stiffness $N_{\mathrm{K}}$ for various values of the Reynolds number Re when the body of the model is fixed, $\chi =1.0$, and the other governing parameters are given as Set 1 in Table 2.

Figure 8

The ratios of (a) time-averaged lift coefficient $\overline{C_{\mathrm{L}}}$, (b) time-averaged thrust coefficient $\overline{C_{\mathrm{T}}}$, (c) time-averaged power coefficient $\overline{C_{\mathrm{P}}}$, and (d) power-loading coefficient $C_{\mathrm{PL}}$ to the values for the rigid wings $\overline{C_{\mathrm{L}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{T}}^{\mathrm{rigid}}}$, $\overline{C_{\mathrm{P}}^{\mathrm{rigid}}}$, and $C_{\mathrm{PL}}^{\mathrm{rigid}}$, respectively, as the functions of the nondimensional spring stiffness $N_{\mathrm{K}}$ for the two sets of parameters shown in Table 2 when the body of the model is fixed, $\chi =1.0$, and $\text{Re}=500$.

Figure 9

(a) Time variations of the deflection angle ${\theta }_{\mathrm{w}}$ and (b) trajectories of the center of the body for various values of the nondimensional spring stiffness $N_{\mathrm{K}}$ in the free flights without rotation when $\chi =1.0$, $\text{Re}=1000$, and the other governing parameters are given as Set 1 in Table 2.

Figure 10

(a) Time variations of the deflection angle ${\theta }_{\mathrm{w}}$ and (b) trajectories of the center of the body for various values of the nondimensional spring stiffness $N_{\mathrm{K}}$ in the free flights without rotation when $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 2 in Table 2.

Figure 11

(a) Trajectories of the center of the body and (b) time variations of the pitching angle ${\theta }_{\mathrm{p}}$ for the flexible wings with the nondimensional spring stiffness $N_{\mathrm{K}}$ equal to 1.0 and the rigid wings in the free flights with pitching rotation when $\chi =1.0$, $\text{Re}=1000$, and the other governing parameters are given as Set 1 in Table 2.

Figure 12

(a) Trajectories of the center of the body and (b) time variations of the pitching angle ${\theta }_{\mathrm{p}}$ for various values of the nondimensional spring stiffness $N_{\mathrm{K}}$ in the free flights with pitching rotation when $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 2 in Table 2.

Figure 13

(a) Trajectories of the center of the body and (b) time variations of the pitching angle ${\theta }_{\mathrm{p}}$ for various values of the moment of inertia of the body $I_{\mathrm{b}}$ in the free flights with pitching rotation when $N_{\mathrm{K}}=1.0$, $\chi =1.0$, $\text{Re}=500$, and the other governing parameters are given as Set 2 in Table 2.

Figure 14

The coordinate systems $(\xi ,\eta )$ and $({\xi }_{\mathrm{hind}},{\eta }_{\mathrm{hind}})$ fixed to the fore and hind parts, respectively, where $L$ is the wing length and $\chi$ is the nondimensional hind wing-tip length.

Figure 15

Time variations of (a) the lift coefficient $C_{\mathrm{L}}$ and (b) the deflection angle ${\theta }_{\mathrm{w}}$ for the flexible wing with $K^{*}=5$ during the third period when $\text{Re}=200$, $\mathrm{\Phi }={150}^{\circ }$, $\mathrm{\Psi }=0^{\circ }$, and ${\alpha }_{\mathrm{m}}={35}^{\circ }$.