Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

Figure 1

Schematic of symmetry reduction by the method of slices. The blue point is the template ${\widehat{x}}^{\prime }$. Group orbits are marked by dotted curves, so that all pink points are equivalent to $\widehat{x}$ up to a shift. The relative periodic orbit (green) in the $d$-dimensional full state space $\mathcal{M}$ closes into a periodic orbit (blue) in the slice $\widehat{\mathcal{M}}=\mathcal{M}/G$, a $(d-1)$-dimensional slab transverse to the template group tangent $t_{}^{\prime }$. A typical group orbit crosses the slice hyperplane transversally, with a non-orthogonal group tangent $t=t\left(\widehat{x}\right)$. A slice hyperplane is almost never a global slice; it is valid up to the slice border, a $(d-2)$-dimensional hypersurface (red) of points ${\widehat{x}}^{*}$ whose group orbits graze the slice, i.e., points whose tangents $t^{*}=t\left({\widehat{x}}^{*}\right)$ lie in $\widehat{\mathcal{M}}$. Beyond the slice border (dashed ‘chunk'), group orbits do not cross the slice hyperplane locally.

Figure 2

(Top left) Schematic of the first Fourier mode slice, with $a_1,a_2$ defined in the text. In this projection the slice border is a zero-measure ‘point’ at the origin. (Bottom) For a generic ergodic trajectory the phase velocity $c=\dot{\ell }\left(t\right)$ appears to encounter singularities whenever it approaches the slice border, which, however, is never reached [10]. Closer inspection reveals a rapid but continuous change in the shift (top right) by $\approx L/2$ in the $\ell \left(t\right)-\overline{c}t$ Galilean frame, moving (for our parameter values) at $\overline{c}=1.1092$. These apparent jumps are well resolved: each ‘+’ corresponds to 10 time integration steps.

Figure 3

Projection of 32 relative periodic orbits and traveling waves using symmetry-invariant coordinates, $I/I_{\mathrm{lam}},D/D_{\mathrm{lam}}$, where $I_{\mathrm{lam}}=D_{\mathrm{lam}}$ are energy rates for the laminar flow. All discovered traveling waves are included: (B) ${\text{TW}}_{2.04}$, (C) ${\text{TW}}_{2.03}$, (D) ${\text{TW}}_{1.97}$, (G) ${\text{TW}}_{1.93}$, (H) ${\text{TW}}_{1.89}$, (F) ${\text{TW}}_{1.85}$, and (J) ${\text{TW}}_{1.78}$, except for ${\text{TW}}_{N4L/1.38},{\text{TW}}_{N4U/3.28}$, and ${\text{TW}}_{1.57}$, which lie far outside the ergodic cloud (grey dots).

Figure 4

Projection of the symmetry-reduced infinite-dimensional state space onto the first 3 PCA principal axes, computed from the L2-norm average over the natural measure (the gray ‘cloud') in the slice. 32 relative periodic orbits, and a subset of the 7 shortest relative periodic orbits, together with traveling waves (A) ${\text{TW}}_{N4U/3.28}$, (B) ${\text{TW}}_{2.04}$, (C) ${\text{TW}}_{2.03}$, (D) ${\text{TW}}_{1.97}$, (E) ${\text{TW}}_{1.98}$, (F) ${\text{TW}}_{1.85}$. While ${\text{TW}}_{1.93}$ appears to lie in the very center of the $(I,D)$ projection Fig. 3, it is revealed in this state space projection to lie far from the ergodic cloud, outside the box plotted, as are (E) ${\text{TW}}_{N4L/1.38}$ and (F) ${\text{TW}}_{1.85}$. Due to a ‘rotate-and-reflect’ symmetry, each solution appears twice, with the exception of (A) ${\text{TW}}_{N4U/3.28}$ (and the far-away ${\text{TW}}_{N4L/1.38}$), which belong to the ‘rotate-and-reflect’ invariant subspace. Our relative periodic orbits capture the regions of high natural measure very well. The symmetry-invariant subspace has a strong repulsive influence, separating the natural measure into two weakly communicating regions. The inset shows the ergodic cloud from another perspective.