Feigenbaum universality in subcritical Taylor-Couette flow

TL;DR

The paper addresses whether Feigenbaum universality can be observed in a subcritical fluid system, specifically the counter-rotating Taylor-Couette flow, by analyzing a minimal annular domain. It employs direct numerical simulations with a Poincaré section based on torque balance and a PNK solver to resolve a period-doubling cascade up to the seventh bifurcation, yielding near-ideal matches to the universal constants $δ_F$ and $α_F$ and an accumulation point $R_∞$. A novel prediction framework based on a universal map $U_M(ξ)=G(ξ)+MΦ(ξ)$ translates early cascade data into accurate bifurcation diagrams, including predictions beyond the accumulation point, validated by DNS that reveals a stable $P_{12}$ window. The results demonstrate that the Navier–Stokes dynamics near the cascade can be captured by a nearly one-dimensional unimodal map on an appropriately chosen Poincaré section, providing a powerful link between high-dimensional fluid dynamics and classic chaos theory with practical predictive power.

Abstract

Feigenbaum universality is shown to occur in subcritical shear flows. Our testing ground is the counter-rotation regime of the Taylor-Couette flow, where numerical calculations are performed within a small periodic domain. The accurate computation of up to the seventh period doubling bifurcation, assisted by a purposely defined Poincaré section, has enabled us to reproduce the two Feigenbaum universal constants with unprecedented accuracy in a fluid flow problem. We have further devised a method to predict the bifurcation diagram up to the accumulation point of the cascade based on the detailed inspection of just the first few period doubling bifurcations. Remarkably, the method is applicable beyond the accumulation point, with predictions remaining valid, in a statistical sense, for the chaotic dynamics that follows.

Figures (13)

• Figure 1: Taylor-Couette flow. (a) Sketch of the flow configuration. The inner and outer cylinders have radii $r=r_i$ and $r=r_o$, respectively, and rotate with angular velocities $\Omega_i$ and $\Omega_o$. (b) A snapshot of the stripe pattern adopted from figure 2 of WaAyDe22. The colour map shows the radial vorticity at the mid gap $r_m=(r_o+r_i)/2$. The radius ratio and inner and outer cylinder Reynolds numbers, defined in §\ref{['sec:methods']}, are set to $(\eta,R_i,R_o)=(0.883,600,-1200)$.
• Figure 2: (a) Annular-parallelogram computational domain defined by the coordinates of (\ref{['xieta_thetar']}) with wave numbers $(n_1,k_1,n_2,k_2)=(10,2,0,4.5)$ and $\eta=0.883$, adopted from W22. The axial line probe (red dashed vertical line) used in the production of space-time diagrams is located at mid gap $r_m=(r_i+r_o)/2 \approx 8.047$. (b) Three-dimensional flow structure of DRW solution for $(R_i,R_o) = (450,-1200)$. Positive (yellow, $u_{\theta} = 250$) and negative (blue, $u_{\theta} = -100$) isosurfaces of perturbation azimuthal velocity.
• Figure 3: The Poincaré section $\Sigma$ and the periodic orbits P$_1$ (dashed blue) and P$_2$ (solid green). The black squares are DRW solutions. All solutions are computed at $R=395.67$. (a) Projection of the phase space on the $(\tau_o,\tau_i,\kappa)$ coordinates. (b) Inner ($\tau_i$, thick lines) and outer ($\tau_o$, thin) torque time series of P$_1$ and P$_2$. (c) Two-dimensional phase map projection on the $(\tau_o,\tau_i)$ plane. The Poincaré section is shown in transparent grey in panel (a) and as a dashed grey line in panel (c). The circles on the P$_1$ (empty blue) and P$_2$ (filled green) curves correspond to their representation on $\Sigma$.
• Figure 4: Space-time diagrams for (a) DRW, (b) P$_1$ and (c) P$_2$, all computed at $R=395.67$. The roman number labels denote measurements of radial vorticity $\omega_r(z;t)$ along axial probe lines at $(r,\theta)=(r_m,\theta_0)$ fixed to (i) the lab (stationary) reference frame, (ii) a reference frame co-moving with the solution, and (iii) the same co-moving frame but with the temporal mean $\langle\omega_r\rangle_{t}$ subtracted. The azimuthal location, $\theta_0$, is chosen consistently across reference frames and solutions to enable comparison. Colour shading according to $\omega_r\in[-1400,1400]$ or $\omega_r- \langle\omega_r\rangle_{t}\in[-300,300]$, as need be. Dashed vertical lines indicate the natural period of the corresponding solution.
• Figure 5: Bifurcation scenario as recorded on the Poincaré section $\Sigma$. (a) The initial steps of the bifurcation scenario. Shown are DRW (black), P$_1$ (blue) and P$_2$ (green), reported in W22. P$_4$ (green) emerges at the second period doubling bifurcation point PD$_2$. Both stable (solid line) and unstable (dashed) solution branches are shown. (b) Detailed view (close-up of the region bounded by a solid grey box in panel a) of stable solution branches across the period-doubling cascade and beyond. The accumulation point for the period doubling cascade ($R_\infty$) is to be computed in §\ref{['sec:FeiUni']}.