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The Motion of a Flapping Flag: Chaos or Noise?

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Samuel E. Mermall

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2013

Observing a flag flapping in a passing breeze is so commonplace that the rich dynamical behavior it exhibits is often overlooked. This experiment sought to observe and categorize the motion of a flag in a fluid flow to be either that of a chaotic system or a noisy system. A jet fan plugged into a variac autotransformer allowed precise wind speed control, which was used to produce laminar airflow. Time series were collected for the speed of air leaving the trailing end of the flag and the drag force experienced by the flag. Several minutes of time series data were analyzed numerically using delay coordinate embedding, Fourier analysis, and three dimensional trajectory reconstruction. This data indicates that the motion of a flapping flag is dominated by weakly colored noise at lower wind speeds and white noise at higher wind speeds.

The three dimensional trajectory reconstructions were done using a stereoscopic triangulation system. It was designed to use a single high speed camera and a series of mirrors configured so that the camera can see the flag from two different vantage points. The apparent shift of a point on the flag between the two perspectives, or parallax, provides the means to triangulate the point's position in three dimensional space. The point's position was tracked in two dimensions over a couple of seconds from both perspectives. This position data was read and interpreted using Wolfram *Mathematica*, which produced three dimensional trajectory plots of this motion over time. These plots show how the flag's fluttering patterns change qualitatively with increases in wind speed.