Active lift inversion process of heaving wing in uniform flow by temporal change of wing kinematics

The transition of the vortex pattern and the lift generated by a heaving wing in a uniform flow was investigated numerically. As a fundamental problem constituting the insects' flight maneuverability, we studied the relationship between a temporal change in the heaving wing motion and the change in the global vortex pattern. At a Strouhal number that generates an asymmetric vortex pattern, we found that temporal angular frequency reduction causes inversion of both the global vortex pattern and the lift sign. The inversion is initiated by the transfer of the leading-edge vortex, which interferes with the vortex pattern generated at the trailing edge. Successful inversion is conditioned on the starting phase and the time interval of the frequency reduction. The details of the process during the transition are discussed.

Figure 1

Configuration of the model. A wing in a two-dimensional uniform flow oscillates perpendicular to the uniform flow. The position of the center of the wing is described by ${\mathit{X}}_w=(0,h\left(t\right))$.

Figure 2

(a) Subregions for the simulation domain. (b) Subregions (divided by thick lines) and the collocation points (represented by the crossing points of thin lines) near the wing.

Figure 3

${\langle C_D\rangle }_{10}$ and ${\langle C_L\rangle }_{10}$ for $\varphi =\pi$. (a) ${\langle C_D\rangle }_{10}$ vs St. (b) ${\langle C_L\rangle }_{10}$ vs St.

Figure 4

Vortex pattern at $t=10T,\varphi =\pi$. The displayed region is $[-5,30]\times [-12.5,12.5]$. The vorticity, $\mathbf{\nabla }\times \mathit{u}$, is shown by filled contours. (a) $\text{St}=0.10$. (b) $\text{St}=0.275$. (c) $\text{St}=0.35$.

Figure 5

(a) Plot of $\left(n-\frac{1}{2},{\langle C_L\rangle }_n\right)$ for different initial phases ($\varphi =\pi /2,\pi$). (b) Snapshot of the wing and vortices for $t=10.5T$ and $\varphi =\pi$. Vorticity is indicated by filled contours; the boundaries are $\mathbf{\nabla }\times \mathit{u}=\pm (4+8k)(k=0,1,...,5)$. (c) Same as (b), but for $t=40.5T$. (d) Same as (b), but for $t=70.5T$ and $\varphi =\pi /2$.

Figure 6

Time sequences of vortex patterns for $\text{St}=0.275$. (a) $t=9.00T$. (b) $t=9.25T$. (c) $t=9.50T$. (d) $t=9.75T$.

Figure 7

$C_L$ vs $t/T$ for $\text{St}=0.275$ ($8T\le t\le 10T$).

Figure 8

Wing kinematics for $\mathrm{\Delta }\omega =\omega /2,t_1=7\frac{1}{3}T,t_2=8\frac{1}{3}T$. (a) $d\mathrm{\Phi }/dt$ (b) $h\left(t\right)/A$.

Figure 9

(a) ${\langle C_D\rangle }_n$ vs $n$ ($n$ is the period) for $t_1/T=7.0$. The open triangle ($▵$), filled triangle ($▴$), open circle ($\circ$), filled circle ($•$), open inverted triangle ($▿$), closed inverted triangle ($▾$), and open square ($\square$) indicate $(t_2-t_1)/T=0.4,0.6,0.8,1.0,1.2,1.4,1.6$, respectively. (b) Same as panel (a), but $t_1/T=7\frac{1}{3}$. (c) ${\langle C_L\rangle }_n$ vs $n$ for $t_1/T=7.0$. (d) Same as panel (c), but $t_1/T=7\frac{1}{3}$.

Figure 10

(a) ${\langle C_L\rangle }_{15}$ vs $(t_2-t_1)/T$ for $t_1/T=7\frac{1}{3}$. (b) Contour of ${\langle C_L\rangle }_{15}$ for sets of $(t_1/T,(t_2-t_1)/T)$, where $t_1/T\in \{7,7\frac{1}{4},7\frac{1}{3},7\frac{1}{2},7\frac{3}{4},7\frac{5}{6}\}$ and $(t_2-t_1)/T\in \{0.4,0.6,0.8,1.0,1.2,1.4,1.6\}$. The region ${\langle C_L\rangle }_{15}<0$ is shaded.

Figure 11

${\langle C_L\rangle }_n$ vs $n$. The open circles indicate the case of no maneuvering ($\mathrm{\Delta }\omega =0$). The filled triangles ($▴$), open squares ($\square$), filled squares ($\blacksquare$), and diamonds ($⧫$) indicate $t_1/T=4\frac{1}{3},7\frac{1}{3},10\frac{1}{3}$ and $13\frac{1}{3}$, respectively.

Figure 12

Vortex patterns during temporal reduction of the local angular frequency $\left(t_1/T=7\frac{1}{3},t_2-t_1=T\right)$. Curves indicate streamlines. (a) $t=6.0T$. (b) $t=6.5T$. (c) $t=7.5T$. (d) $t=7.75T$. (e) $t=8.0T$. (f) $t=8.5T$. (g) $t=9.0T$. (h) $t=14.0T$.

Figure 13

Time expanded image for $t_1/T=7\frac{1}{3},t_2-t_1=T$. This image was generated by stacking the horizontal line distribution of vorticity from top to bottom, for the interval $0\le t\le 20T,-2\le x\le 4$. The legend is the same as that for Fig. 6. (a) Image for the horizontal line $c/4$ above the wing. (b) Image for the horizontal line $c/4$ below the wing.

Figure 14

(a) Reynolds number dependence on the transitions of ${\langle C_D\rangle }_{10}$. (b) Same as panel (a) but for ${\langle C_L\rangle }_{10}$.

Figure 15

${\langle C_L\rangle }_n$ vs $n$ for the cases $\text{Re}=150$, 200, where $t_1/T=7\frac{1}{3}$ and $t_2-t_1=T$.

Figure 16

Time series of the lift on the oscillating wing in a uniform flow. The calculation methods are SEM ($\delta =0.025,0.01$) and IB.