Intrinsic Stability of a Body Hovering in an Oscillating Airflow

We explore the stability of flapping flight in a model system that consists of a pyramid-shaped object hovering in a vertically oscillating airflow. Such a flyer not only generates sufficient aerodynamic force to keep aloft but also robustly maintains balance during free flight. Flow visualization reveals that both weight support and orientational stability result from the periodic shedding of vortices. We explain these findings with a model of the flight dynamics, predict increasing stability for higher center of mass, and verify this counterintuitive fact by comparing top- and bottom-heavy flyers.

Figure 1

Stable hovering of a pyramid in an oscillating airflow. (a) Snapshots from high-speed video of free flight of a paper pyramid (mass 0.22 g and height $L=3.2\mathrm{cm}$). The flow oscillates up and down with peak-to-peak amplitude $A=1.9\mathrm{cm}$ and frequency $f=20\mathrm{Hz}$ (period $T=0.05\mathrm{s}$). (b) Typical traces of the body tilt angle $\theta$, which is measured from the vertical. (c) An aerodynamic potential is reconstructed from the tilt dynamics of 16 movies (gray points), and data for $\theta \ge 0$ are reflected about the vertical to reveal a stable potential well (dashed line).

Figure 2

Vortex ejection for a $\Lambda$-shaped body in an oscillating flow. A water-filled pan is rocked back and forth, and flow structures are visualized with shadowgraphs. (a) Snapshots of the flow around an upright $\Lambda$. A vortex curls around each side of the body as the flow moves upwards ($t=0$), a counterrotating vortex then forms on the downward flow, and the pair are shed downward ($t=T/2$). (b) Schematic of the flow field for an upright body. (c),(d) When tilted rightward, the left side emits a pair outward and nearly perpendicular to the body axis ($t=0$), and the right side of the body emits a weaker vortex pair downward ($t=T/2$).

Figure 3

A point-force model of stability. Time-averaged aerodynamic forces (dark arrows) on a triangular body are assumed to act at the sites of vortex emission. (a) As the body tilts, the force vector on the left side maintains constant magnitude, and its angle is 3 times the tilt angle $\theta$. (b) The force on the right points upward and decreases in strength as $\theta$ increases to 30 deg. Similarly, it is directed downward and grows in strength for $30<\theta <60$. (c) Potential for the nominal case in which the center of mass is midway between the apex and base, as well as for top-heavy (dashed) and bottom-heavy (dotted) arrangements.

Figure 4

Center of mass (c.m.) location and flight stability. Weights are added to the pyramid to make top- and bottom-heavy bodies. The pyramid height is 1.7 cm in both cases, and the c.m. is located at the apex and 1.5 cm below the base, respectively. The objects are released into the flow with tilt $\theta =30\mathrm{deg}$. (a),(b) A top-heavy pyramid recovers the upright orientation, while a bottom-heavy body tips over. (c) Body tilt dynamics for top- and bottom-heavy pyramids.