Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics

TL;DR

This study investigates how the dynamical relevance of relative periodic orbits (RPOs) in two-dimensional Kolmogorov flow evolves with increasing Reynolds number by arclength-continuing a large RPO library from $Re=40$ and $Re=100$ up to $Re$ values near $1100$. It reveals three dissipation-based classes: Class 1 with $D \sim Re$ that connects to unforced Euler solutions, Class 3 with $D \sim 1/Re$ consistent with forced Euler dynamics, and Class 2 remaining near the turbulent attractor; most RPOs become dynamically irrelevant at high $Re$. The authors further relate many RPOs to exact relative periodic solutions of a doubly periodic point-vortex system via gradient-based optimisation, supporting the Euler connection for class 1 states while highlighting limitations for others. They also attempt to reconstruct turbulent statistics from a fixed weighted RPO expansion using KL-divergence minimisation, finding robust predictions for some observables (e.g., $D$ and $E$) only over a limited $Re$ range due to incomplete libraries and dynamical irrelevance at high $Re$. Overall, the work clarifies the limits of RPO-based turbulence descriptions in 2D and motivates developing longer-period solutions and higher-dimensional extensions.

Abstract

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly-periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers $Re \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the $Re$-values where they were obtained, it is unclear how their dynamical importance evolves with increasing $Re$. We perform arclength continuation on this library of solutions to show that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either well above or below those associated with the turbulent attractor at high $Re$. The scaling of the high dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the 'unimodal' solutions considered in previous work (Kim & Okamoto, Nonlinearity 28, 2015). On the other hand, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The apparent dynamical irrelevance is associated with poor statistical reconstructions away from the $Re$ values where the RPOs were originally converged. Motivated by the connection to solutions of the Euler equation, we show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (1) matches the dynamics of the point vortices to the turbulent vortex cores and (2) insists the point vortex evolution is itself time-periodic.

Figures (19)

• Figure 1: Application of the vortex extraction criterion discussed in the text to two example snapshots at $Re=100$. Contours are of the out-of-plane vorticity, white lines identify regions identified by the threshold (\ref{['eq:vort_extraction_cond']}), and red lines (which here enclose areas so small they appear as points in the visualisation) highlight which of these regions were discarded due to the bound imposed on the region area.
• Figure 2: Continuation of periodic orbits and their overlap with the turbulent dissipation PDF. (Top) Histograms of the number of monotonic-in-$\Rey$ subsections $N_{upo}$ of all solution branches (grey) and monotonic-in-$\Rey$ subsections fully within the dynamically important region (red) as function of $\Rey$. Both histograms are computed using 50 bins, spaced logarithmically over the range $\Rey \in [20, 1000]$. (Middle) The time-averaged dissipation rate $\Rey \, D$ against $\Rey$ of the arclength continuation of the initial library of UPOs starting at both $\Rey = 40$ and $\Rey = 100$. The contour plot shows the reference PDFs of dissipation rate, with the $1^{st}$ and $99^{th}$ percentiles indicated by the dashed black lines. The red/blue/green branches indicate those with terminal UPOs above/within/below this dynamically important region. Circles denote the terminal solution along each branch. Filled circles denote the branches which were terminated as 50 states were converged, while empty circles denote the branches which could not be continued further. (Bottom) The time averaged, scaled dissipation rate $\overline{ \Rey \, D }$ of a long-time simulation is shown as a function of $\Rey$ (black), as well as the scaling law $\Rey^{1/2}$ (blue).
• Figure 3: The time-averaged, scaled dissipation rate $\Rey \, D$ against $\Rey$ for three representative branches in class (1). Three UPOs along each branch are sampled (indicated by crosses) and visualised on the right of the figure. Vorticity field snapshots for each sampled UPO along each branch are shown in a row, increasing in arclength from left to right. Colour maps are held constant per row to highlight the development of the vorticity gradients with $\Rey$. Note the different colour bars for each subfigures, highlighting the strengthening vortical structures.
• Figure 4: The scalings with $Re$ of the periods $T_{\text{RPO}}$ (left), maximum vorticity values $\max |\omega|$ (middle) and maximum velocity values $\max |\bm u|$ (right) for a subset of branches in class 1.
• Figure 5: The time-averaged, scaled dissipation rate $\Rey \, D$ against $\Rey$ for three representative branches in class (2). Three RPOs along each branch are sampled (indicated by crosses) and visualised on the right of the figure. Vorticity field snapshots for each sampled RPO along each branch are shown in a row, increasing in arclength from left to right.