Exact coherent structures as building blocks of turbulence on large domains

Abstract

Exact unstable solutions of the Navier-Stokes equations are thought to underpin the dynamics of turbulence, but are usually computed in minimal computational domains. Here, we extend this dynamical systems approach to spatially extended turbulent flows featuring multiple interacting 'substructures', and show how new simple invariant solutions can be constructed by spatial tiling of exact solutions from small-box calculations. Candidate solutions are found via gradient-based optimization of a scalar loss function which targets autorecurrence in spatially-masked regions of the flow. We apply these ideas to a vertically-extended Kolmogorov flow, where we first identify large numbers of relative periodic orbits (RPOs) which are combinations of high-dissipation, small-box solutions with laminar patches. We then show that vertically-stacked combinations of pairs of distinct small-box RPOs can form robust guesses for dynamically-relevant two-tori in the larger domain. Finally, we show how our optimization procedure can identify 'turbulent' trajectories which locally shadow a small-box RPO for multiple periods in a subdomain. These small-box combinations are possible as the flow spends prolonged periods in a regime where it can be effectively considered as a pair of weakly-coupled small-box systems, due to shielding effects associated with higher-dissipation flow structures.

Figures (4)

• Figure 1: Cross-stream coupling in the tall-box Kolmogorov flow. (a) Cross-coupling terms $C_{12}$ and $C_{21}$ (see equation \ref{['eqn:coupling_defn']}) for a turbulent $2\pi\times4\pi$ trajectory of total length $T = 10^5$ (contours; levels evenly spaced logarithmically from 0.4 to 228.0). Also shown are the coupling coefficients for the RPO-laminar tiles (green curves/points) and the RPO-RPO tiles (blue curves). (b, c, d) Examples of weakly coupled, two-way coupled and one-way coupled vorticity fields respectively, corresponding to the magenta symbols in (a). Background filled contours are the explicit coupling terms ${\bf u}_1\cdot\boldsymbol \nabla\omega$ (left) and ${\bf u}_2\cdot\boldsymbol \nabla\omega$ (right) with contours in ranges (a) $[-24.2,24.2]$, (b) $[-17.5,17.5]$, and (c) $[-25.0,25.0]$. Line contours show the vorticity, evenly spaced $-11.4<\omega<11.4$, in each half domain which is used to compute the induced velocities $\{\mathbf u_i\}$.
• Figure 2: Exact coherent states in the tall $2\pi \times 4\pi$ box. (a) Snapshots of out-of-plane vorticity for RPOs formed from a combination of a small box ($2\pi\times 2\pi$) RPO with the laminar solution. The first panel includes the 'starting' pair of solutions alongside the converged state. (b) Snapshots of out-of-plane vorticity for trajectories shadowing two-tori in the $2\pi \times 4\pi$ box. Left panels show snapshots from the pair of RPOs used to initialize the optimization, right panel is a snapshot from the final state. (c) Masked near-recurrence for the solutions in (b). Colors match the symbols on the panels in (b); dashed lines are top solution, solid lines the bottom. Symbols in (c) identify integer multiples of the fundamental period within the masked subdomains (blue solution shown in SI).
• Figure 3: Production/dissipation ($I$ and $D$ respectively) in the $2\pi \times 4\pi$ domain. Gray histogram is a pdf of a long ($4000$ advective time units) turbulent orbit, which runs logarithmically from 0.4 to 275.8. Also shown are the candidate two-tori (blue) and some example 'turbulent' trajectories which shadow an RPO in a masked region of the domain (red). The dashed/faded components of these curves span the first half of the optimization period $nT$ for tiles with turbulence and $\min(n_1T_1,n_2T_2)$ for RPO-RPO tiles.
• Figure 4: RPO shadowing in a subdomain. (a) Out-of-plane vorticity above two periods of the RPO used to construct the solution (contours run $-11.4 \leq \omega \leq 11.4$). (b) Masked autorecurrence $R(t,\tau)$ in the top (left panel) and bottom (right panel) halves of the domain.