Dynamics of flags over wide ranges of mass and bending stiffness

TL;DR

This paper investigates how flag mass and bending stiffness influence flapping dynamics using simulations and models, revealing specific scaling laws for mode number, frequency, and amplitude across different mass regimes.

Contribution

It provides new insights into the dependence of flapping behavior on physical parameters through combined numerical and linearized modeling approaches.

Findings

Small-mass flags exhibit traveling wave motions at near flow speed.

Flag mode number scales as stiffness to the -1/4 power.

Amplitude scales as the square root of flag mass.

Abstract

There have been many studies of the instability of a flexible plate or flag to flapping motions, and of large-amplitude flapping. Here we use inviscid simulations and a linearized model to study more generally how key quantities -- mode number (or wavenumber), frequency, and amplitude -- depend on the two dimensionless parameters, flag mass and bending stiffness. In the limit of small flag mass, flags perform traveling wave motions that move at nearly the speed of the oncoming flow. The flag mode number scales as the -1/4 power of bending stiffness. The flapping frequency has the same scaling, with an additional slight increase with flag mass in the small-mass regime. The flapping amplitude scales approximately as flag mass to the 1/2 power. For large flag mass, the dominant mode number is low (0 or 1), the flapping frequency tends to zero, and the amplitude saturates in the neighborhood of its upper limit (the flag length). In a linearized model, the fastest growing modes have somewhat different power law scalings for wavenumber and frequency. We discuss how the numerical scalings are consistent with a weakly nonlinear model.