Time behavior of the secondary flow between time-periodically corotating cylinders: a two-frequency forcing case

We consider the oscillatory flow between time-periodically corotating cylinders, in the case of a two-frequency forcing. The angular velocity Omega(t) of the cylinders is the sum of a low-frequency omega(1) oscillation plus a harmonic frequency omega(2) oscillation at a lower amplitude, Omega(t)=Omega(1) cos(omega(1)t)+Omega(2) cos(omega(2)t). The temporal behavior of the secondary flow is characterized by ultrasound Doppler velocimetry. For a single-frequency forcing at omega(1), above a critical amplitude Omega(1), within one cycle, the secondary flow measurements exhibit a spikelike behavior with several successive growths, dampings, and periods of quietness. The effect of the superimposed omega(2) frequency is the following, although if alone it would be stable: to sharpen the spikes, restabilizing the flow for some intervals that exhibit secondary flow for the single forcing at omega(1); and to induce further secondary flow spikes during what is a quiescent stage in the single omega(1) frequency forcing case. Numerical calculations for the finite-gap quasisteady linear stability analysis are presented and provide a good prediction of the times of growth and damping of the secondary flow observed experimentally during a flow period.