Complexity of the laminar-turbulent boundary in pipe flow

Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry reduction, which we formulate here for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling-wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.

Figure 1

Two traveling-wave solutions of pipe flow $(\mathrm{S}1$ and $\mathrm{S}1\mathrm{N}$, to be described later in the article) and their continuum of symmetry copies obtained from the axial translations and the azimuthal rotations, visualized as red and blue wire frames projected from the state space. Projection bases are the same as in Figs. 4 and 5. Apparent intersections of the curves are the artifacts of finite-dimensional projection from the infinite-dimensional state space. In the symmetry-reduced state space, each torus will be represented by a single point.

Figure 2

Phase velocities (25) ${\dot{\varphi }}_z$ (blue) and ${\dot{\varphi }}_{\theta }$ (green) measured in a typical turbulence simulation. The inset shows a zoom into the time interval $\tau \in [103.3,103.5]$, where a sharp increase in ${\dot{\varphi }}_z$ is apparent. Dots in the inset correspond to adjacent time steps of the simulation, showing that this episode is in fact time resolved.

Figure 3

(a) Isosurfaces of streamwise velocity at $u=0.26$ (red) and $u=-0.33$ (blue) and streamwise vorticity at ${\omega }_z=\pm 0.40$ (green and purple, respectively) for the traveling wave $\mathrm{S}1$ in three dimensions. Also shown are the color-coded streamwise velocity and cross-stream velocity (arrows) averaged over (b) the half pipe length $z\in (0,L/2]$ for $\mathrm{S}1$ and the unstable eigenvectors (c) $V_1^{\mathrm{S}1}$, (d) ${\widehat{V}}_1^{\mathrm{S}1}$, (e) $V_2^{\mathrm{S}1}$, and (f) ${\widehat{V}}_2^{\mathrm{S}1}$. The flow direction is into the page.

Figure 4

(a) First approximation to the unstable manifold of $\mathrm{S}1$, as forward-integrated trajectories with initial conditions (30) visualized as projections onto the local coordinate frame (31). Here $L$ denotes the laminar solution. The inset shows the same 12 trajectories on the unstable manifold for the time interval $\tau \in [0,2.5]$, illustrating the initial almost uniform expansion. Isosurfaces of streamwise velocity and vorticity at (b) $u=0.36$ (red), $u=-0.50$ (blue), and ${\omega }_z=\pm 0.99$ (green and purple, respectively); (c) $u=0.50$ (red), $u=-0.62$ (blue), and ${\omega }_z=\pm 1.5$ (green and purple, respectively); and (d) $u=0.66$ (red), $u=-0.71$ (blue), and ${\omega }_z=\pm 9.0$ (green and purple, respectively). The flow direction is into the page.

Figure 5

(a) Unstable manifold of $\mathrm{S}1$ approximated by bisecting between trajectories that relaminarize and those that develop into turbulence. (b) Time series of kinetic energy for trajectories that stay on the edge for the longest times.

Figure 6

(a) Self-recurrence measured over an edge trajectory (drawn in blue in Fig. 5) on the unstable manifold of $\mathrm{S}1$. Also shown are the color-coded (fast and slow in red and blue, respectively) streamwise velocity and cross-sectional velocity (arrows) averaged over the half pipe length $\left(z\in [0,L/2]\right)$ at (b) initial time $\tau =0$, (c) local minimum $\tau =245$, (d) local minimum $\tau =320$, and (e) final time $\tau =345$. The flow direction is into the page.

Figure 7

(a) Isosurfaces of streamwise velocity at $u=0.27$ and $-0.38$ (red and blue, respectively) and streamwise vorticity at ${\omega }_z=\pm 0.96$ (green and purple, respectively) of $\mathrm{S}1\mathrm{N}$. (b) Color-coded (fast and slow in red and blue, respectively) streamwise velocity and cross-sectional velocity (arrows) averaged over the half pipe length $\left(z\in [0,L/2]\right)$ visualization of $\mathrm{S}1\mathrm{N}$. (c) Leading linear stability eigenvalues of $\mathrm{S}1\mathrm{N}$ on the complex plane computed in shift-and-reflect invariant subspace (black circles) and full state space (red crosses).

Figure 8

Two-dimensional surfaces associated with the three leading unstable eigenvectors of $\mathrm{S}1\mathrm{N}$ approximated as a set of trajectories with initial conditions given by (32) for (a) $k=1$, (b) $k=3$, and (c) $k=5$.

Figure 9

(a) Time evolution of kinetic energy for trajectories starting from $a_{\mathrm{S}1\mathrm{N}}\pm {10}^{-4}{V}_{11}^{\mathrm{S}1\mathrm{N}}$ (green and purple, respectively). Also shown are the color-coded streamwise velocity and cross-sectional velocity (arrows) averaged over the half pipe length $\left(z\in [0,L/2]\right)$ at times (b) and (d) $\tau =100$ and (c) and (e) $\tau =200$ for (b) and (c) the transition to turbulence [purple in (a)] and (d) and (e) laminarization [green in (a)].

Figure 10

(a) Time evolution of turbulent kinetic energy for edge tracking trajectories obtained through bisection. (b) Distance of bisection trajectories from $\mathrm{S}1$ (blue) and $\mathrm{S}1\mathrm{N}$ (orange). Minima of distances at $\tau =110[min\parallel {\tilde{a}}_{\mathrm{S}1\mathrm{N}}-\tilde{a}\left(\tau \right)\parallel ]$ and at $\tau =165[min\parallel {\tilde{a}}_{\mathrm{S}1}-\tilde{a}\left(\tau \right)\parallel ]$ are marked black.

Figure 11

Color-coded streamwise velocity and cross-sectional velocity (arrows) averaged over the half pipe length $\left(z\in [0,L/2]\right)$ of (a) $\mathrm{S}1\mathrm{N}$, (b) closest approach of the trajectory on the edge to $\mathrm{S}1\mathrm{N}$ at $\tau =110$, (c) $\mathrm{S}1$, and (d) closest approach of the trajectory on the edge to $\mathrm{S}1$ at $\tau =165$. Each figure uses a fixed color scale $(min,max)=(-0.35,0.35)$.