Coarse Graining the State Space of a Turbulent Flow Using Periodic Orbits

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier-Stokes equations in a triply periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than ${10}^5$ degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system’s observables as weighted sums over the periodic orbits.

Figure 1

(a) ${\overline{\mathrm{po}}}_{13}$ (green/thick) and a shadowing trajectory (gray/thin) visualized as projections onto the leading three principal components of ${\overline{\mathrm{po}}}_{13}$. (b),(c) The persistence diagrams associated with ${\overline{\mathrm{po}}}_{13}$ (b) and shadowing trajectory segment (c) shown in (a). The data points used for generating the persistence diagrams (b),(c) are marked with dots along the projection curves in (a).

Figure 2

(a) Shadowing distance of a turbulent trajectory from 8 POs. (b) Shadowing decomposition of the turbulent trajectory. (c) State transition graph where the nodes correspond to POs and arrows indicate the possible transitions between them. The self-loops are omitted for clarity and the node sizes are proportional to the probability of observing the respective PO as inferred from the invariant distribution of the corresponding Markov process.

Figure 3

Turbulent trajectories (dots, gray) and POs (loops, colors) visualized (a) on the $ID$ and (b) the $E\dot{E}$ planes along with the long-time averages $⟨D{⟩}_{\infty }=⟨I{⟩}_{\infty }=1.885$, $⟨E{⟩}_{\infty }=10.54$, and $⟨\dot{E}{⟩}_{\infty }=0$ (diamonds, black) and the PO estimates $⟨D{⟩}_{\pi }=⟨I{⟩}_{\pi }=1.874$, $⟨E{⟩}_{\pi }=10.85$, and $⟨\dot{E}{⟩}_{\pi }=0$ (crosses, red). (c) Velocity profiles averaged over POs (solid, colors) along with the long-time average $⟨U(y){⟩}_{\infty }$ (dotted, black) and its PO estimate $⟨U(y){⟩}_{\pi }$ (dashed, red). Only the half domain $y\in [0,\pi ]$ is shown in (c) since the other half corresponds to its mirror image.