Unsteady aerodynamics of fluttering and tumbling plates

Abstract

We investigate the aerodynamics of freely falling plates in a quasi-two-dimensional flow at Reynolds number of $10^{3}$, which is typical for a leaf or business card falling in air. We quantify the trajectories experimentally using high-speed digital video at sufficient resolution to determine the instantaneous plate accelerations and thus to deduce the instantaneous fluid forces. We compare the measurements with direct numerical solutions of the two-dimensional Navier–Stokes equation. Using inviscid theory as a guide, we decompose the fluid forces into contributions due to acceleration, translation, and rotation of the plate. For both fluttering and tumbling we find that the fluid circulation is dominated by a rotational term proportional to the angular velocity of the plate, as opposed to the translational velocity for a glider with fixed angle of attack. We find that the torque on a freely falling plate is small, i.e. the torque is one to two orders of magnitude smaller than the torque on a glider with fixed angle of attack. Based on these results we revise the existing ODE models of freely falling plates. We get access to different kinds of dynamics by exploring the phase diagram spanned by the Reynolds number, the dimensionless moment of inertia, and the thickness-to-width ratio. In agreement with previous experiments, we find fluttering, tumbling, and apparently chaotic motion. We further investigate the dependence on initial conditions and find brief transients followed by periodic fluttering described by simple harmonics and tumbling with a pronounced period-two structure. Near the cusp-like turning points, the plates elevate, a feature which would be absent if the lift depended on the translational velocity alone.