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

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 Fr, being the ratio of characteristic times for downward motion and pendular oscillations. The transition from flutter to tumble occurs at Frc0ex0ex=0ex0ex0.67±0.05. We propose a phenomenological model including inertial drag and lift which reproduces this motion, and directly yields the Froude similarity.