Genesis of Streamwise-Localized Solutions from Globally Periodic Traveling Waves in Pipe Flow

The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can coexist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows, the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate that the origin of the first localized solution is in a modulational symmetry-breaking Hopf bifurcation from a known global traveling wave that has twofold rotational symmetry about the pipe axis. Similar behavior is found for a global wave of threefold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that many global solutions should be expected to lead to more realistic localized counterparts through such bifurcations, which provides a constructive route for their generation.

Figure 1

Localization of the bifurcation from the $N2$ traveling wave (three copies in $z$). Main plot: $E(z)≔{\int }_0^{2\pi }d\theta {\int }_0^{1}sds\frac{1}{2}{\mathbf{u}}^2$ demonstrating localization along the continuation curve. Inset: Continuation in $\alpha =2\pi /L$ against the friction factor, $\mathrm{\Lambda }$ (the axial pressure gradient averaged over the whole pipe and multiplied by $R/\rho U^2$, where $R$ is the pipe radius and $U$ the mean axial flow speed). The branch moves towards smaller domains before turning in a saddle-node bifurcation and localizing. The friction factor’s linear dependence upon $\alpha$ signals localization.

Figure 2

Continuation of the bifurcation from the $N3$ traveling wave (three copies in $z$) in $\alpha$ plotted against the friction factor, $\mathrm{\Lambda }$. The curve initially bifurcates towards smaller domains before turning in a saddle node. After this the solution begins to localize. At $\alpha \approx 0.26$ a secondary bifurcation breaks the ${\mathbf{\Omega }}_{3,T/6}$ symmetry leading to a second localized solution (plotted in red or dashed). Dots (a)–(h) correspond to solutions plotted in Fig. 3 and 3 and 3 to those plotted in Fig. 4. Inset right: Zoomed into region near bifurcation. Inset below: Zoomed into $\alpha <0.15$ region demonstrating linear behavior as domain length tends to infinity ($\alpha \to 0$).

Figure 3

$E(z)$ at dots along the continuation curve (Fig. 2). Inset: Behavior close to the bifurcation as the solution increases in amplitude. Main: Further continuation up to the secondary bifurcation as the solution localizes. Solutions (e)–(h) have been aligned to demonstrate solution evolution along the continuation curve. Solution (e) is plotted in both the main panel and the inset for comparison. Note $E(z)$ has two wavelengths to every wavelength of $N3$ due to the ${\mathrm{\Omega }}_2$ symmetry [see solution (a)].

Figure 4

Energy in streaks ($\theta$-dependent streamwise flow) and rolls ($\theta$-dependent cross-stream flow) as a function of downstream position, $z$, for the two ${\mathbf{R}}_3$-symmetric localized solutions: the ${\mathbf{\Omega }}_{3,T/6}$ symmetric solution, labeled (i); and symmetry-broken solution, labeled (j). Dotted vertical lines indicate slice locations for velocity visualization plotted in Fig. 5. Panel (a): Logarithm of energy, the ${\mathbf{\Omega }}_{3,T/6}$ symmetric solution spans approximately $120R$ including exponential tails while the symmetry-broken solution has length of approximately $180R$. Panel (b): Focused onto the center of the domain, the amplitude of the solutions differ but have very similar roll and streak structure (not plotted).

Figure 5

Flow (deviation from laminar state) at slices (a)–(d) corresponding to the dotted lines, left to right, in Fig. 4 for solution (i). Downstream flow plotted using contours with white for relatively fast flow, and red for slow flow. Arrows indicate cross-stream flow.