Falling cards

In this study we consider the unsteady separated flow of an inviscid fluid (density $\rho_{f}$) around a falling flat plate (thickness $T$, half-chord $L$, width $W$, and density $\rho_{s}$) of small thickness and high aspect ratio ($T \ll L \ll W$). The motion of the plate, which is initially released from rest, is unknown in advance and is determined as part of the solution. The flow solution is assumed two-dimensional and to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. Throughout its motion, the plate continually sheds vorticity from each of its two sharp edges and the unsteady Kutta condition, which states the fluid velocity must be bounded everywhere, is applied at each edge. The coupled equations of motion for the plate and its trailing vortex wake are derived (the unsteady aerodynamic loads on the plate are included) and are shown to depend only on the modified Froude number $\Fr = T\rho_{s}/L\rho_{f}$. Crucially, the unsteady aerodynamic loads are shown to depend on not only the usual acceleration reactions, which lead to the effect known as added mass, but also on novel unsteady vortical loads, which arise due to relative motion between the plate and its wake. Exact expressions for these loads are derived. An asymptotic solution to the full system of governing equations is developed for small times $t > 0$ and the initial motion of the plate is shown to depend only on the gravitational field strength and the acceleration reaction of the fluid; effects due to the unsteady shedding of vorticity remain of higher order at small times. At larger times, a desingularized numerical treatment of the full problem is proposed and implemented. Several example solutions are presented for a range of modified Froude numbers $\Fr$ and small initial inclinations $\theta_{0} <\pi/32$. All of the cases considered were found to be unstable to oscillations of growing amplitude. The non-dimensional frequency of the oscillations is shown to scale in direct proportion with the inverse square root of the modified Froude number $1/\sqrt{\Fr}$. Importantly, the novel unsteady vortical loads are shown to dominate the evolution of the plate's trajectory in at least one example. Throughout the study, the possibility of including a general time-dependent external force (in place of gravity) is retained.