On the stability and sound of an unforced flag

A linear theory is proposed for the motion and sound production of a flag in nominally uniform high Reynolds number flow. The flag is modelled as an elastic fluid-loaded filament subject to either clamped-free or fixed-free boundary conditions. Eigenfunctions describing standing waves in the absence of fluid loading are used to expand the loaded flag deflection. The resulting temporal stability problem is analysed and the neutral curve and modes of motion describing the transition to instability are discussed in detail for each boundary condition. It is found that the fixed-free flag is considerably more stable than the clamped-free flag. In the limit of a very heavy flag, instability sets in as a resonant standing wave, while lighter flags exhibit flutter (convective) instability. Acoustic radiation of dipole type is calculated and discussed in the limit where the flag is acoustically compact. At the onset of instability it is found that the acoustic pressure is dominated by the contribution from the wake and that light flags are noisier than heavy.