Dynamic lift enhancement mechanism of dragonfly wing model by vortex-corrugation interaction

The wing structure of several insects, including dragonflies, is not smooth, but corrugated; its vertical cross section consists of a connected series of line segments. Some previous studies have reported that corrugated wings exhibit better aerodynamic performance than flat wings at low Reynolds numbers $[\text{Re}\simeq \mathit{O}\left({10}^3\right)]$. However, the mechanism remains unclear because of the complex wing structure and flow characteristics. A corrugated structure modifies the formation and behavior of aerodynamic flow features arising during unsteady wing motion, such as the leading-edge vortex (LEV). These modifications can be key to lift enhancement in many insects, though the details of these benefits remain imperfectly understood. In this study, we analyzed the flow around a two-dimensional corrugated wing model that started impulsively by direct numerical simulations. We focused on the period between the initial generation of LEVs and subsequent interactions before detachment. For the flat wing, it is known that a secondary vortex with a sign opposite to that of the LEV, the $\lambda$ vortex, develops and erupts to discourage lift enhancement. For corrugated wings, such an eruption of the $\lambda$ vortex can be suppressed by the corrugation structure, which enhances the lift. The detailed mechanism and its dependence on the angle of attack are also discussed.

Figure 1

Wing model descriptions. (a) Vertices $(x_k,y_k)$, adjacent displacement vectors ${\mathit{L}}_k$, and angles ${\theta }_k$. (b) Line segments and thickness $w$ of the wing model. (c) Flat wing model ($\alpha =0$). (d) Corrugated wing model ($\alpha =1$). (e) Spectral elements around corrugated wing model ($\varphi ={35}^{\circ }$).

Figure 2

Convergence of calculations for corrugated wing ($\alpha =1$): (a) $C_L(1,{20}^{\circ };t^{*})$ and (b) $C_L(1,{35}^{\circ };t^{*})$.

Figure 3

Time series of the lift coefficient, $C_L(\alpha ,\varphi ;t^{*})$ (${20}^{\circ }<\varphi <{40}^{\circ }$). (a) Flat wing ($\alpha =0$). (b) Corrugated wing ($\alpha =1$).

Figure 4

Normalized vorticity fields for (a) a flat wing ($\alpha =0$) for $\varphi ={35}^{\circ }$ at $t^{*}=0.50$, (b) a corrugated wing ($\alpha =1$) for $\varphi ={35}^{\circ }$ at $t^{*}=0.50$, (c) a flat wing ($\alpha =0$) for $\varphi ={35}^{\circ }$ at $t^{*}=3.25$, and (d) a corrugated wing ($\alpha =1$) for $\varphi ={35}^{\circ }$ at $t^{*}=3.25$.

Figure 5

(a) $C_{L,\text{max}}$ during $0.5<t^{*}<3.25$ for the flat wing (blue, $\alpha =0$) and corrugated wing (red, $\alpha =1$). (b) $C_{L,\text{mean}}$ during $0.5<t^{*}<3.25$ for the flat wing (blue, $\alpha =0$) and corrugated wing (red, $\alpha =1$). (c) ${\mathrm{\Delta }}_{\text{max}}$ and ${\mathrm{\Delta }}_{\text{mean}}$ during $0.5<t^{*}<3.25$. Vertical axis is in a percentage form. The dashed lines indicate $0\%$ and $5\%$. (d) $C_{D,\text{mean}}$ during $0.5<t^{*}<3.25$ for the flat wing (blue, $\alpha =0$) and corrugated wing (red, $\alpha =1$).

Figure 6

Spatiotemporal distributions of $C_p$. ${\xi }^{*}=0.15$. $s^{*}$ axis and $t^{*}$ axis are the horizontal and vertical axes, respectively.

Figure 7

(a) Time series of lift coefficient (blue, flat wing ($\alpha =0$); red, corrugated wing ($\alpha =1$); $\varphi ={35}^{\circ }$). (b) Snapshots of the pressure field around the wings at $t^{*}=t_{\text{max}}^{*}$.

Figure 8

Snapshots of the normalized vorticity fields ($\varphi ={35}^{\circ }$) for (a) the flat plate wing ($t^{*}=1.00$, $t^{*}=1.70$, and $t^{*}=t_{\text{max}}$) and (b) the corrugated wing ($t^{*}=1.00$, $t^{*}=1.70$, and $t^{*}=t_{\text{max}}$).

Figure 9

Typical snapshots of the normalized vorticity fields around the corrugated wing ($\alpha =1,\varphi ={20}^{\circ }$): (a) $t^{*}=2.15$, (b) $t^{*}=2.45$, and (c) $t^{*}=2.81=t_{max}^{*}$.

Figure 10

Spatiotemporal distributions of the vorticity field. ${\xi }^{*}=0.15$. $s^{*}$ axis and $t^{*}$ axis are the horizontal and vertical axes, respectively.

Figure 11

Extraction of $\lambda$ vortex eruptions. (a) The corrugated wing ($\alpha =1$, $\varphi ={20}^{\circ }$). (b) The corrugated wing ($\alpha =1$, $\varphi ={35}^{\circ }$). (c) Results of linear regression for the distribution of the $\lambda$ vortex. Phase diagram of slope and mean squared error.

Figure 12

Mean pressure fields around (a) the flat wing ($\alpha =0$, $\varphi ={35}^{\circ }$) and (b) the corrugated wing ($\alpha =1$, $\varphi ={35}^{\circ }$), and the definition of the region around the corrugated wing model for understanding the flow characteristics. Mean normalized vorticity fields around (c) the flat wing ($\alpha =0$, $\varphi ={35}^{\circ }$) and (d) the corrugated wing ($\alpha =1$, $\varphi ={35}^{\circ }$).

Figure 13

Mean pressure fields around (a) the flat wing ($\alpha =0$, $\varphi ={20}^{\circ }$) and (b) the corrugated wing ($\alpha =1$, $\varphi ={20}^{\circ }$), and the definition of the region around the corrugated wing model for understanding the flow characteristics. Mean normalized vorticity fields around (c) the flat wing ($\alpha =0$, $\varphi ={20}^{\circ }$) and (d) the corrugated wing ($\alpha =1$, $\varphi ={20}^{\circ }$).

Figure 14

Typical snapshots of the flow field near the leading edge of the corrugated wing ($\alpha =1$, $\varphi ={35}^{\circ }$). Pressure fields at (a) $t^{*}=1.80$ and (b) $t^{*}=2.15$. Vorticity fields at (c) $t^{*}=1.80$ and (d) $t^{*}=2.15$.