Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum

Many fluid flows, such as bluff body wakes, exhibit stable self-sustained oscillations for a wide range of parameters. Here we study the effect of weak noise on such flows. In the presence of noise, a flow with self-sustained oscillations is characterized not only by its period, but also by the quality factor. This measure gives an estimation of the number of oscillations over which periodicity is maintained. Using a recent theory [P. Gaspard, J. Stat. Phys. 106, 57 (2002)], we report on two observations. First, for weak noise the quality factor can be approximated using a linear Floquet analysis of the deterministic system; its size is inversely proportional to the inner-product between first direct and adjoint Floquet vectors. Second, the quality factor can readily be observed from the spectrum of evolution operators. This has consequences for Koopman/Dynamic mode decomposition analyses, which extract coherent structures associated with different frequencies from numerical or experimental flows. In particular, the presence of noise induces a damping on the eigenvalues, which increases quadratically with the frequency and linearly with the noise amplitude. (C) 2014 AIP Publishing LLC.