Some periodic orbits of chaotic motions for time-periodic forced two-dimensional Navier–Stokes flows

Abstract

In this study, we study the two-dimensional Navier–Stokes flows with time-periodic external forces. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it leading to oscillating/turbulent motions. For the oscillating flows, periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. In addition, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase in energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.