Analysis of the sensitivity of periodic flows to subharmonic perturbations using the harmonic transfer function

Many flows in nature and in engineering applications exhibit complicated dynamics arising from the nonlinear interaction between different time scales. For instance, it is well-known that the vortex pairing observed in high-shear flows such as mixing layers and jets is due to the high underlying sensitivity of the flow to subharmonic disturbances. 

Analyzing the sensitivity of these flows to harmonic and subharmonic perturbations may be done by linearizing the governing equations about a time-periodic solution. For example, the recently introduced harmonic resolvent framework allows to study the linear input-output behavior of the flow at integer-multiples of the fundamental frequency of the periodic base flow. 

Here, we show how the harmonic resolvent analysis can be extended to study the linear input-output response of the flow to perturbations that oscillate at a subharmonic of the fundamental frequency. In particular, we compute the singular value decomposition of the harmonic transfer function (Wereley, 1991) evaluated at the subharmonic frequency of interest. The leading right and left singular vectors identify the dominant input-output spatio-temporal flow structures, while the leading singular value measures the overall sensitivity of the flow to forcing at the subharmonic frequency under consideration.

We demonstrate this analysis on a forced incompressible axisymmetric jet under conditions for which vortex pairing is observed.