A scaling law for the lift of hovering insects

Abstract

Insect hovering is one of the most fascinating acrobatic flight modes in nature, and its aerodynamics has been intensively studied, mainly through computational approaches. While the numerical analyses have revealed detailed vortical structures around flapping wings and resulting forces for specific hovering conditions, theoretical understanding of a simple unified mechanism enabling the insects to be airborne is still incomplete. Here, we construct a scaling law for the lift of hovering insects through relatively simple scaling arguments of the strength of the leading edge vortex and the momentum induced by the vortical structure. Comparison of our theory with the measurement data of 35 species of insects confirms that the scaling law captures the essential physics of lift generation of hovering insects. Our results offer a simple yet powerful guideline for biologists who seek the evolutionary direction of the shape and kinematics of insect wings, and for engineers who design flapping-based micro air vehicles.

On the autorotation of animal wings

“…Because the LEV is an essential unsteady mechanism for lift production in autorotation [7], we followed a similar reasoning and constructed here a corresponding scaling for animal wings during autorotation. We found that inclusion of the Strouhal number, aspect ratio, and offset of the rotational axis relative to the wing base fits the previously derived law (see Lee et al [12]; figure 6). This similarity in the scaling of lift production in hovering flight and during wing autorotation for small animal fliers deserves further experimental attention, and particularly characterization of the LEV using particle image velocimetry or computational fluid dynamics.…”

“…For the sake of accuracy, the exact fitting (superimposed in figure 5) results to be W ¼ 0:078807St À0:91455 , with a linear regression coefficient R 2 ¼ 0.527, considering seeds data for statistics in order to have a general trend. To identify simple relationships among the dimensionless parameters in equation (4.9), we use a recent scaling law in [12] that predicts that the aerodynamic force F perpendicular to the wing surface behaves as…”

“…when reached a stable descent velocity), so that (see background section of [12]) it seems more appropriate than a potential theory-based scaling for the present rotating wings and seeds where the LEV accounts for the high lift coefficients. As Fcos(u) W (figure 4a), r ¼ fecos(u) and S ¼ ec, wing loading scales as This expression not only agrees with the scaling (4.10) for St, but accounts for the dependence of the non-dimensional wing loading with the rest of the relevant non-dimensional parameters.…”

“…We also tested if freshly killed insects with both wings intact could achieve stable autorotation. Finally, using a recent scaling law for the lift production of hovering insects [12], we derived general relationships for the autorotation of seeds and animal wings.…”

A predictive quasi-steady model of aerodynamic loads on flapping wings

“…However, it is more useful to know the two-dimensional (2D) coefficient (C trans l ) for the wing airfoil that can be used directly in the BEM. Conventionally, the translational velocity at the radius of gyration is taken as the reference to calculate the aerodynamic forces for the entire flapping wings [e.g., Harbig et al, 2014;Lee et al, 2015;Percin & van Oudheusden, 2015]. In this case, the same resultant translational lift can be obtained by BEM with C trans l which takes the value of C trans L , as shown in Appendix A.…”