Falling plates with leading-edge vortex shedding

TL;DR

The paper develops a robust inviscid vortex-sheet framework for thin plates falling through a fluid, incorporating continual leading-edge vortex shedding through a log-quadrature approach and velocity smoothing. By coupling a rigid-body model to bound and free vortex sheets and enforcing no-penetration and Kelvin’s theorem, the authors reproduce large-amplitude fluttering, tumbling, looping, autorotation, and new bent-plate dynamics, aligning with viscous simulations and experiments. Key contributions include the log-quadrature for near-singular integrals, sub-step fencing to preserve no-penetration accuracy, and the discovery of scaling laws such as $f_{max} \sim R_1^{-1/2}$ in fluttering, along with obtuse fluttering in V-shaped plates. The results provide a computationally efficient yet physically faithful tool for exploring fluid-structure interactions in falling bodies, guiding interpretation of experiments and informing designs for passive aerial systems.

Abstract

We develop a new numerical method for thin plates falling in inviscid fluid that allows for leading-edge vortex shedding. The inclusion of leading-edge shedding restores physical dynamics to vortex-sheet models of falling bodies, and for the first time large-amplitude fluttering and tumbling are observed in inviscid simulations. Leading-edge shedding is achieved by introducing a novel quadrature rule and smoothing procedure for the Birkhoff-Rott equations. The smoothing error is controlled by a novel fencing procedure. We find a transition point between fluttering and tumbling that is consistent with previous viscous simulations and experiments, and other falling motions such as looping, autorotation are also observed as the plate density increases. The dipole street wakes behind the fluttering plates resemble those in experiments. We consider plates bent into V shapes and study the effects of density and bending angle on the qualitative falling dynamics. At small densities, increasing the bending angle stabilizes the falling motion into fluttering, while at large densities, decreasing the bending angle stabilizes the falling motion into autorotation. In the autorotation regime, the magnitude of angular velocity increases as time cubed before it reaches a terminal angular velocity, and in the fluttering regime, the fluttering frequency scales as the $-1/2$ power of $R_1$, the plate density.

Figures (36)

• Figure 1: (a) A flat and (b) V-shaped plate falling under gravity, with center of mass $\zeta_G$ and length $2L$. It has unit tangent and normal vectors $\hat{\textbf{s}}$ and $\hat{\textbf{n}}$ respectively. Off the edges, vortex sheets are being shed with their strength $\gamma_\pm$ colored with a logarithmic scale. They are parameterized by $\zeta_\pm(\Gamma,t)$ where $\Gamma$ is the circulation.
• Figure 2: Side-by-side comparison between (a) the trajectory of the center of mass and (b) the chaotic vortex wake formed by a falling plate with $R_1 = 0.02$ undergoing small-amplitude fluttering. The plate was released with initial angle $\beta(0) = 25^\circ$.
• Figure 3: For $R_1 = 0, 0.01, \ldots, 0.09$, snippets of the center-of-mass trajectories of falling plates for eleven initial angles within $[0,\pi/4]$ during $0\leq t \lessapprox 500$. The darker trajectories correspond to larger initial angles. The plots are shaded according to the qualitative dynamics exhibited by the steady state. Within this $R_1$ range, only fluttering motions occur. Each panel contains an inset that shows the large-scale features of the same trajectories, with a red rectangle indicating the region shown in the main panel.
• Figure 4: The trajectory of the center of mass (a) and the vortex wake (b) formed by a falling plate with $R_1 = 0.03$ undergoing progressive fluttering. The plate was released with initial angle $\beta(0) = 25^\circ$.
• Figure 5: Plate trajectories when $R_1 = 0.03$ for initial angles $4.5^\circ,\ 22.5^\circ,\ 36^\circ, \ 45^\circ$. All trajectories simulated (including those not shown) converge to steady-state periodic fluttering at this value of $R_1$.