Mathematically established chaos and forecast of statistics with recurrent patterns in Taylor-Couette flow

TL;DR

The paper studies subcritical chaos in counter-rotating Taylor-Couette flow within a minimal periodic domain and shows that a simple discrete map can accurately approximate the dynamics after a Feigenbaum cascade. By identifying an unstable period-3 orbit and unfolding the return map, the authors construct a one-dimensional map that is chaotic in Li-Yorke and Devaney senses, with a Lyapunov exponent of approximately 0.47 and a topological entropy h_top = ln phi, establishing a dense set of periodic points. They demonstrate a topological conjugacy to the tent map, enabling the reconstruction of turbulence statistics, including the probability density function, solely from unstable periodic solutions and NS data, with striking agreement to direct numerical simulation. The work provides a rigorous, UPO-centered link between deterministic chaos and turbulence statistics, offering a practical alternative to cycle-expansion approaches and a pathway toward applying these ideas to other shear flows, while acknowledging limitations for high Reynolds numbers and NS-system proofs.

Abstract

The transition to chaos in the subcritical regime of counter-rotating Taylor-Couette flow is investigated using a minimal periodic domain capable of sustaining coherent structures. Following a Feigenbaum cascade, the dynamics are found to be remarkably well approximated by a simple discrete map that admits rigorous proof of its chaotic nature. The chaotic set that arises for the map features densely distributed periodic points that are in one-to-one correspondence with unstable periodic orbits (UPOs) of the Navier-Stokes system. This supports the increasingly accepted view that UPOs may serve as the backbone of turbulence and, indeed, we demonstrate that it is possible to reconstruct every statistical property of chaotic fluid flow from UPOs.

Figures (6)

• Figure 1: Taylor-Couette problem. (a) The spiral turbulence regime visualised by isosurfaces of azimuthal vorticity. The small parallelogram-annular domain in orange is the minimal flow unit used throughout this paper, adopted from WaAyDe22. (b) Snapshot of the P$_3$ solution in the minimal flow unit at $R_i=395.7816$. (c) The DNS time signal of inner torque $\tau_i$. Circles denote crossings of the Poincaré section $\Sigma$. The red portion indicates a close visit to $\mathrm{P}_3$.
• Figure 2: Attractor (from DNS data) and the period-three orbit (from PNK). (a) The bifurcation diagram generating the chaotic set. The green dots show torque $\tau$ (on the Poincaré section $\Sigma$) as a function of the inner cylinder Reynolds number $R_i$ for statistically steady states in DNS. The red triangles indicate the period-three orbit (P$_3$) at $R_i=395.7816$, at some distance beyond the cascade's accumulation point $R_\infty$. (b) Phase map projection on $(\tau_i,\tau_o,\kappa)$ of the P$_3$ orbit (red line) and a representation of the chaotic attractor on the Poincaré section $\Sigma$ (green dots) at $R_i=395.7816$.
• Figure 3: Return map analyses based on the Poincaré map on $\Sigma$ at $R_i=395.7816$. (a) Return map of the chaotic attractor (green dots) and of P$_3$ (red triangles). The cusp ($\tau_c$) and minimum ($\tau_m$) points split the map in three distinct branches: B (orange), A1 (black) and A2 (gray). (b) The same return map, now unfolded following \ref{['tautilde']}. The A1 and A2 branches are now labelled as A (black). The inset diagrams indicate branch selection rules.
• Figure 4: Probability density function (PDF) for the spline dynamical system (black), the normalised histogram for DNS data of the chaotic attractor (green), and the prediction from periodic points (gray).
• Figure 5: Complete set of periodic points up to period $n=8$ for (a) the spline map approximation and (b) the Navier-Stokes system. The inset shows a phase map projection of P$_{5a}$ analogous to that of P$_3$ in figure \ref{['fig:P3ChaoticAttractor']}a.