Resonance of flexible flapping wings at low Reynolds number

Using three-dimensional computer simulations, we examine hovering aerodynamics of flexible planar wings oscillating at resonance. We model flexible wings as tilted elastic plates whose sinusoidal plunging motion is imposed at the plate root. Our simulations reveal that large-amplitude resonance oscillations of elastic wings drastically enhance aerodynamic lift and efficiency of low-Reynolds-number plunging. Driven by a simple sinusoidal stroke, flexible wings at resonance generate a hovering force comparable to that of small insects that employ a very efficient but much more complicated stroke kinematics. Our results indicate the feasibility of using flexible wings driven by a simple harmonic stroke for designing efficient microscale flying machines.

Figure 1

(Color online) (a) Schematic of oscillating wings. The dashed line indicates the wing located beyond the symmetry plane. (b) Lift coefficient $C_L$ and (c) flapping efficiency $\eta$ versus dimensionless frequency $\varphi$ for elastic wings with different values of the added mass parameter $T$. The wing tilt angle is $\theta =40°$ and $\text{Re}=100$. The vertical dashed lines in (b) and (c) show frequencies ${\varphi }_{C_{L_{\text{max}}}}\approx 0.95$ and ${\varphi }_{{\eta }_{\text{max}}}\approx 1.25$, respectively. The inset in (b) shows the maximum wing-tip deflection ${\delta }_{\text{max}}$ as a function of $\varphi$.

Figure 2

(Color online) Snapshots in panels (a) and (b) illustrate the elastic deformations and vorticial flows arising around flexible wings oscillating at ${\varphi }_{C_{L_{\text{max}}}}$ and ${\varphi }_{{\eta }_{\text{max}}}$, respectively. Color surfaces (dark gray) indicate the isovorticial surfaces with the vorticity magnitude $\omega =8f$ in (a) and $\omega =2f$ in (b). The dashed contours in (a) and (b) show the initial position of an undeformed wing. Panels (c) and (d) show wing profiles for, respectively, ${\varphi }_{C_{L_{\text{max}}}}$ and ${\varphi }_{{\eta }_{\text{max}}}$. The profiles for downstroke are shown by the solid green lines, whereas the upstroke profiles are shown by the dashed blue lines. The wing parameters are $\theta =40°$, $T=2/15$, and $\text{Re}=100$.

Figure 3

(Color online) (a) Wing-tip deflection and (b) aerodynamic forces during one period of wing oscillations. The triangle and circle symbols are for $\varphi ={\varphi }_{C_{L_{\text{max}}}}$ and $\varphi ={\varphi }_{{\eta }_{\text{max}}}$, respectively. In (a), the dashed line indicates the wing root position. In (b), the solid and dashed lines show, respectively, the vertical and horizontal components of aerodynamic force on the oscillating flexible wing. The wing parameters are $\theta =40°$, $T=2/15$, and $\text{Re}=100$.

Figure 4

(Color online) Lift-to-weight ratio $L/W$ versus added mass parameter $T$ for a 5 mm flexible wing oscillating in air with frequency $\varphi ={\varphi }_{C_{L_{\text{max}}}}$ and $\text{Re}=100$.