Highly deformable flapping membrane wings suppress the leading edge vortex in hover to perform better

Abstract

Airborne insects generate a leading edge vortex when they flap their wings. This coherent vortex is a low-pressure region that enhances the lift of flapping wings compared to fixed wings. Insect wings are thin membranes strengthened by a system of veins that does not allow large wing deformations. Bat wings are thin compliant skin membranes stretched between their limbs, hand, and body that show larger deformations during flapping wing flight. This study examines the role of the leading edge vortex on highly deformable membrane wings that passively change shape under fluid dynamic loading maintaining a positive camber throughout the hover cycle. Our experiments reveal that unsteady wing deformations suppress the formation of a coherent leading edge vortex as flexibility increases. At lift and energy optimal aeroelastic conditions, there is no more leading edge vortex. Instead, vorticity accumulates in a bound shear layer covering the wing's upper surface from the leading to the trailing edge. Despite the absence of a leading edge vortex, the optimal deformable membrane wings demonstrate enhanced lift and energy efficiency compared to their rigid counterparts. It is possible that small bats rely on this mechanism for efficient hovering. We relate the force production on the wings with their deformation through scaling analyses. Additionally, we identify the geometric angles at the leading and trailing edges as observable indicators of the flow state and use them to map out the transitions of the flow topology and their aerodynamic performance for a wide range of aeroelastic conditions.

Keywords: flapping wing flight; fluid–structure interaction; leading edge vortex; membrane wings; unsteady fluid dynamics.

Fig. 1.

(A) Simulated multiexposure photo of the flapping flexible membrane wing. (B and C) Two-dimensional representation of the passively deforming membrane wing for lower and higher flapping frequencies. Here, $c^{\prime }$ represents the shortening of the chord length, and $z_{\max }$ the maximum chord-wise camber due to an increase in the dynamic pressure ($q=0.5\rho U^2$).

Fig. 2.

(A) Stroke-average lift coefficient ${\overline{C}}_L$ contours as a function of the angle of attack amplitude $\widehat{\alpha }$ and the aeroelastic number Ae. (B) Instantaneous vorticity fields and deformed camber lines for $\widehat{\alpha }={55}^{°}$ at $t/T=0.20$, 0.25, and 0.30, for a fully rigid wing (top row) and three values of the aeroelastic number Ae (indicated by the markers in the contour plot in A). (C) Stroke-average power economy $\overline{\eta }$ contours as a function of the angle of attack amplitude $\widehat{\alpha }$ and the aeroelastic number Ae. (D) Instantaneous vorticity fields and deformed camber lines for $\widehat{\alpha }={35}^{°}$ at $t/T=0.20$, 0.25, and 0.30, for a fully rigid wing (top row) and three values of the aeroelastic number Ae (indicated by the markers in the contour plot in C). The cycle-average lift coefficient is defined as ${\overline{C}}_L=\overline{L}/(0.5\rho {\overline{U}}^{2}cR)$, and the power efficiency as the ratio $\overline{\eta }={\overline{C}}_L/{\overline{C}}_P$, with the power coefficient ${\overline{C}}_P=\overline{P}/(0.5\rho {\overline{U}}^{3}cR)$.

Fig. 3.

(A) Maximum wing camber ${\widehat{z}}_{\max }$ during the stroke over aeroelastic number Ae; (B) maximal rotation angles ${\widehat{\gamma }}_{\mathrm{LE}}$ at the leading edge and ${\widehat{\gamma }}_{\mathrm{TE}}$ at the trailing edge during the stroke over aeroelastic number Ae; (C) Maximum wing camber ${\widehat{z}}_{\max }$ over the scaled normal force coefficient $C_{\mathrm{N}}^{\ast }$ and normalized by the angle of attack. The solid lines represent the general trend of the deformation data and are obtained by a least-square fit to the individual measurement data.

Fig. 4.

(A) Stroke-average lift coefficient ${\overline{C}}_{\mathrm{L}}$ and (B) stroke-average power economy $\overline{\eta }$ for the membrane wing over leading ${\widehat{\alpha }}_{\mathrm{LE}}$ and trailing edge angles ${\widehat{\alpha }}_{\mathrm{TE}}$.

Fig. 5.

Vortex phenomenology and its relation to aerodynamic performance over the ${\widehat{\alpha }}_{\mathrm{LE}}$ and ${\widehat{\alpha }}_{\mathrm{TE}}$ space. The positive and negative vorticity is depicted on the wings shortly after mid-stroke ($t/T=0.30$). The red and green contours correspond to the stroke-average lift coefficient ${\overline{C}}_{\mathrm{L}}$ and power economy $\overline{\eta }$ respectively. In the Top Right are the rigid wings with large coherent leading edge vortices, common in hovering flapping wing flight. In the Middle band, indicated by the arrow, we have the region where the membrane flexibility aids to either optimize lift or efficiency by suppressing the leading-edge vortex. This is the optimal region of operation for our flexible membrane wings. If we continue further to the Bottom Left, the lift coefficient and efficiency drop due to overcambering of the membrane.