Analysis of the dynamics of subharmonic flow structures via the harmonic resolvent: Application to vortex pairing in an axisymmetric jet

In this paper we extend the harmonic resolvent analysis to study the dynamics of subharmonic perturbations about a periodically time-varying base flow. In doing so, we recover an input-output operator that is closely related to the harmonic transfer function introduced in Wereley [Ph.D. thesis, MIT (1991)], and we also elucidate the nature of the cross-frequency interactions between subharmonic flow structures in the proximity of the base flow. We first demonstrate the use of this method on the Rössler system, under conditions for which the dynamics are sensitive to period-doubling perturbations. We then apply it to a forced incompressible axisymmetric jet, and we study how the jet's sensitivity to subharmonic perturbations varies as a function of the Reynolds number. This analysis suggests that as the Reynolds number is increased, the cross-frequency interactions between subharmonic structures with period 2T and the T-periodic base flow become increasingly important. Remarkably, we also demonstrate that the well-known nonlinear vortex pairing phenomenon is driven by the spatiotemporal structures contained within the first right singular vector (or input mode) of the harmonic resolvent evaluated at the 1/2-subharmonic of the fundamental frequency. In particular, we show that if the nonlinear flow is forced with an input that is orthogonal to the first right singular vector of the harmonic resolvent, then no pairing will occur.