Falling Paper
Description of Research
A piece of paper or a leaf flutters and tumbles down in a seemingly unpredictable manner, but it might give us some insight about flapping flight. A stumbling block in constructing a simple theory of freely falling objects is the quantitative description of the fluid force. Computationally and experimentally we have studied the dynamics of thin rigid objects falling freely in a fluid. A surprise came from the analysis of the fluid force. Our computations revealed that the circulation around a tumbling plate is proportional to the angular velocity, not to the translational velocity as in the case of a steadily translating plate.
(a) experimental setup; (b) picture of release mechanism
In a table top experiment we have further investigated this type of dynamics and explored the two-dimensional trajectories of rigid metal plates falling in a water tank. Examples of the observed trajectories include periodic fluttering (side to side oscillations) and periodic tumbling (rotation and sideways drift). The measured trajectories give us the translational and the rotational plate acceleration, and since the plate dynamics is governed by gravity and fluid forces alone this permits us to extract the instantaneous fluid forces directly. Our model of the fluid forces with translational and rotational lift compares favorably with our experimental results for both fluttering and tumbling.
(a) fluttering trajectory; (b) tumbling trajectory
The vorticity plots shown below (with positive vorticity in red and negative vorticity in blue)are obtained by solving the two dimensional Navier-Stokes equations subject to the motion of free falling ellipses at Reynolds numbers around 10³. The Navier-Stokes equations are written in the vorticity-stream function formulation and solved efficiently via Fast Fourier Transform on a conformal grid.
The vortices in the wake of the tumbling ellipse (b) are either shed when the ellipse makes a 180° turn (the pair in the top left corner and the pair closest the ellipse) or generated when its wake becomes unstable (all other pairs). For computational reasons the flow around the fluttering ellipse (a) is at a lower Reynolds number (Re=400). In this case the wake instability has no time to develop and all the vortex pairs are generated at the turning point of its trajectory.
Vorticity diagrams: (a) fluttering; (b) tumbling
See movies of these calculations: fluttering (1.69Mb) and tumbling (2.11Mb). (Requires Quicktime or compatible player.)
In addition we have investigated the transition between periodic fluttering and periodic tumbling using the force model and direct numerical simulations. The dynamics depends on three non-dimensional numbers, the thickness to width ratio, the non-dimensional moment of inertia, and the Reynolds number. At intermediate Reynolds number we observe a transition between fluttering and tumbling with increasing non-dimensional moment of inertia, thereby confirming previous results which show that the value of the non-dimensional moment of inertia is crucial for predicting the effect of the aerodynamic coupling between the plate and the fluid. In the force model we find that the transition is a heteroclinic bifurcation which leads to a characteristic logarithmic divergence of the period of oscillation at the bifurcation point.