Searching turbulence for periodic orbits with dynamic mode decomposition

Abstract

We present a new method for generating robust guesses for unstable periodic orbits (UPOs) by post-processing turbulent data using dynamic mode decomposition (DMD). The approach relies on the identification of near-neutral, repeated harmonics in the DMD eigenvalue spectrum from which both an estimate for the period of a nearby UPO and a guess for the velocity field can be constructed. In this way, the signature of a UPO can be identified in a short time series without the need for a near recurrence to occur, which is a considerable drawback to recurrent flow analysis, the current state-of-the-art. We first demonstrate the method by applying it to a known (simple) UPO and find that the period can be reliably extracted even for time windows of length one quarter of the full period. We then turn to a long turbulent trajectory, sliding an observation window through the time series and performing many DMD computations. Our approach yields many more converged periodic orbits (including multiple new solutions) than a standard recurrent flow analysis of the same data. Furthermore, it also yields converged UPOs at points where the recurrent flow analysis flagged a near recurrence but the Newton solver did not converge, suggesting that the new approach can be used alongside the old to generate improved initial guesses. Finally, we discuss some heuristics on what constitutes a "good" time window for the DMD to identify a UPO.

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

“…From a broader perspective, the importance of UPOs in influencing the dynamics of transitional fluid flow has been demonstrated in plane Couette flow [44] and in wall-bounded shear flows [45] via direct numerical simulation of the Navier-Stokes equations. More recently, Dynamic Mode Decomposition and Koopman analysis have been used to provide a method for identifying UPOs in high-dimensional systems [46,47]. The work presented here presents an alternative method for identifying UPOs in high-dimensional fluid flow systems and for understanding their significance in transitional phenomena.…”

Exact coherent structures and shadowing in turbulent Taylor–Couette flow

“…Some of these minima correspond to close passes to s or s, and the corresponding flow fields , time delays and rotation angles represent good initial conditions for the solver. An alternative approach to identifying dynamically relevant initial conditions relies on dynamic mode decomposition (Page & Kerswell 2020). Converged solutions were also numerically continued in using pseudo-arclength continuation (Allgower & Georg 2003); some branches turned around, yielding several additional solutions.…”

Exploiting self-organized criticality in strongly stratified turbulence

“…ECSs have been of particular interest in the context of wall-bounded constant-density shear flows, where a general mechanism supporting such states has been identified (Hamilton, Kim & Waleffe 1995). Numerical computation of ECSs from the governing Navier–Stokes or Boussinesq equations typically requires recurrent flow analyses of expensive DNSs to provide suitable initial conditions for sophisticated Newton-hookstep solvers, although more efficient approaches have been proposed (Page & Kerswell 2020). In contrast, it is clear from the present investigation and related studies (Hall & Sherwin 2010; Beaume et al 2015; Montemuro et al 2020) that asymptotically reduced systems, derived using multiple scales analysis, retain the dominant interactions that sustain such states while self-consistently filtering other dynamics, yielding more efficient algorithms for ECS computations and simultaneously exposing the underlying physical mechanisms.…”