Origin of localized snakes-and-ladders solutions of plane Couette flow

Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.

Figure 1

Schematic of plane Couette flow rotating around the spanwise ($\widehat{z}$) axis.

Figure 2

Localized traveling wave ${\mathbf{u}}_{\text{TW}}$ (top) and equilibrium ${\mathbf{u}}_{\text{EQ}}$ (bottom) followed to finite rotation $\mathrm{Ro}={10}^{-4}$. (a) and (c) show the velocity in the $y=0$ midplane, with arrows indicating in-plane velocity and the color scale streamwise velocity $u$: dark, medium, light (blue, green, yellow) correspond to $u=[-1,0,1]$. (b) and (d) are the streamwise velocity with in-plane velocity indicated by arrows.

Figure 3

Bifurcation diagram of the localized ${\mathbf{u}}_{\text{TW}}$, ${\mathbf{u}}_{\text{EQ}}$ for fixed $\mathrm{Ro}={10}^{-4}$ in the ($\mathrm{Re},D$) plane with dissipation rate $D=V^{-1}\int {|\mathbf{\nabla }\times \mathbf{u}|}^{2}d{x}^3$ treated as solution measure. Shown also are the rung states connecting ${\mathbf{u}}_{\text{TW}}$ and ${\mathbf{u}}_{\text{EQ}}$ as well as spatially periodic TVF and WVF solutions with spanwise wave number $\zeta =13$ (TVF1,WVF1) and $\zeta =8$ (TVF2,WVF2). Inset (a): The snakes-and-ladders structure observed in the nonrotating system is preserved at finite rotation. The velocity fields for the points marked by the circles are shown in Fig. 2. Inset (b): Both localized ${\mathbf{u}}_{\text{TW}}$ and ${\mathbf{u}}_{\text{EQ}}$ emerge via a tertiary bifurcation from the laminar flow: $\text{LAM}\to \text{TVF}\to \text{WVF}\to ({\mathbf{u}}_{\text{TW}},{\mathbf{u}}_{\text{EQ}})$.

Figure 4

Streamwise-averaged streamwise velocity at the midplane ($y=0$) of critical WVF (top, thick), and the neutral eigenmode (top, thin) of the pitchfork bifurcation creating the localized ${\mathbf{u}}_{\text{EQ}}$ branch. The detuning of the spatial frequency of the eigenmode relative to the base causes phase interference and thereby an amplitude modulation leading to localization (bottom). The eigenmode giving rise to ${\mathbf{u}}_{\text{TW}}$ has an almost identical structure but instead preserving of inversion symmetry it is shift-and-reflect symmetric.

Figure 5

The localized solution before and after the TVF onset, at $\mathrm{Re}=1050$ [(a)–(c)] and $\mathrm{Re}=1000$ [(d)–(f)], respectively. (a),(b),(d),(e) Visualization as in Fig. 2. (c),(f) Filtered spanwise energy density $E(y,z)={\langle {\mathbf{u}}_{3D}^2\rangle }_x/2$ in the range $E=[0\left(\text{light}\right),{10}^{-3}\left(\text{dark}\right)]$. The core of the localized branch resembles streamwise-modulated 3D WVF while the background is formed by 2D TVF. On reducing $\mathrm{Re}$, the TVF background vanishes, leaving the localized core embedded in a laminar background.

Figure 6

Amplitudes of the 3D WVF core, the 2D TVF background, and the periodic WVF and TVF solutions as $\mathrm{Re}$ is reduced from the localizing bifurcation at $\mathrm{Re}=1150$. Amplitudes are quantified by the spanwise maxima of the $xy$-average energy density ${\langle u^2\rangle }_{xy}\left(z\right)/2$ near $z=0$ (WVF core) and $z=\pm L_z/2$ (TVF background), and across the whole spanwise domain for the periodic solutions. The amplitude of the TVF background closely follows TVF and vanishes at $\mathrm{Re}<1035$. The amplitude of the core, however, decouples from WVF and increases as $\mathrm{Re}$ is reduced away from the bifurcation point. The solid circles denote the solutions shown in Fig. 5.