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.

On the uncertainty quantification of the unsteady aerodynamics of 2D free falling plates

“…Here, we can see how, as the mesh becomes finer, the velocities as well as the descent angle increase whereas the angular velocity decreases. The results are in good agreement between those found experimentally by [21]. The numerically study carried out by [26] uses [21] as reference and some bigger discrepancy (if compared with ours) exists for v x and the descent angle while ω and v y prediction are slightly closer to the experiment.…”

“…The results are in good agreement between those found experimentally by [21]. The numerically study carried out by [26] uses [21] as reference and some bigger discrepancy (if compared with ours) exists for v x and the descent angle while ω and v y prediction are slightly closer to the experiment. In the following will be presented in detailed the results obtained with the finest mesh.…”

“…In 1998, Belmonte et al [20] studied the transition from fluttering to tumbling in a quasi-two-dimensional experiment in which flat strips were dropped in a narrow container filled with fluid (water, glycerol/water or petroleum ether) and mechanical constraints ensured the twodimensional plate motion. Further improvements were done by Andersen et al [21] who eliminated the mechanical constrained and used a release mechanism for dropping the aluminium plates. By using a high-speed digital video camera they determined the instantaneous acceleration and thus the instantaneous forces acting on the plate.…”

“…For the resolution a conformal mapping was used and, to avoid singularities, an elliptical cross section plate was modelled. Andersen et al [21], using the same mathematical formulation, qualitatively compared the numerical solution, for tumbling and fluttering cases, with the experimental one measured in the same paper and ascribed the discrepancies to the differences in geometry between the rectangular cross-section in experiment and the elliptical one in numerical simulation. In [25], the same authors conduct a series of numerical simulations in air and solved numerically the free fall governing differential equations.…”

“…This particular choice is due to the moderately high angular velocity of the tumbling plate: to minimize numerical errors, the internal rotating circle should not slide on the external rectangle more than one fourth of cell each time step. According to [21,26], most of results are represented in function of dimensionless parameters. The meaning of these numbers is reported below:…”