From Flutter to Tumble: Inertial Drag and Froude Similarity in Falling Paper

Abstract

In an experiment on thin flat strips falling through a fluid in a vertical cell, two fundamental motions are observed: side-to-side oscillation (flutter) and end-over-end rotation (tumble). At high Reynolds number, the dimensionless similarity variable describing the dynamics is the Froude number $\mathrm{Fr}$, being the ratio of characteristic times for downward motion and pendular oscillations. The transition from flutter to tumble occurs at ${\mathrm{Fr}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.67\ifmmode\pm\else\textpm\fi{}0.05$. We propose a phenomenological model including inertial drag and lift which reproduces this motion, and directly yields the Froude similarity.