Quasi-Periodic Intermittency in Oscillating Cylinder Flow

Abstract: Fluid dynamics induced by periodically forced flow around a cylinder is analyzed computationally for the case when the forcing frequency is much lower than the von K{\'a}rm{\'a}n vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman Mode Decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric - i.e. not an additive - forcing effect. We find that the dynamics of the flow in this regime are characterized by alternating instances of quiescent and strong oscillatory behavior, and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon -- that we name Quasi-Periodic Intermittency -- using the new normal form model and show that the dynamics are caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomena is also characterized by positive Finite-Time Lyapunov Exponents that, over a long period of time, asymptotically approach zero.