Unsteady aerodynamics of falling plates

Abstract

We investigate the problem of falling paper by solving the two dimensional Navier-Stokes equations subject to the motion of a free falling body at Reynolds numbers around 10^3, which is typical for a leaf or business card falling in air, and experimentally, by using a quasi two dimensional set up and high speed digital video at sufficient resolution to determine the instantaneous accelerations and thus deduce the fluid forces. We compare the measurements with the direct numerical solutions of the two-dimensional Navier-Stokes equation and, using inviscid theory as a guide, we decompose the fluid forces into contributions due to acceleration, translation, and rotation of the plate. The aerodynamic lift on a tumbling plate is found to be dominated by the product of linear and angular velocities rather than velocity squared as appropriate for an airfoil. This coupling between translation and rotation provides a mechanism for a brief elevation of center of mass near the cusp-like turning points. The Navier-Stokes solutions further provides the missing quantity in the classical theory of lift: the instantaneous circulation, and suggests a revised ODE model for the fluid forces. Experimentally and numerically, we get access to different 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 (side to side oscillations), tumbling (end over end rotation), and apparently chaotic motion. We explore further the transition region between fluttering and tumbling using both direct numerical solutions and the ODE model. In particular, by increasing the non-dimensional moment of inertia in the direct numerical simulations, we observe a wide transition region in which the cards flutter periodically but tumble once between consecutive turning points. In this region, we also observe a divergence of the period of oscillation, with the cards falling vertically for distances of up to 50 times the card width. We analyze the transition between fluttering and tumbling in the ODE model and find a heteroclinic bifurcation which leads to a logarithmic divergence of the period of oscillation at the bifurcation point.