Localization in a spanwise-extended model of plane Couette flow

We consider a nine-partial-differential-equation (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localization in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Spatially localized, time-periodic solutions were found instead, which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localized states in very wide domains. This smooth localization behavior, which has also been seen in fully resolved duct flow (S. Okino, Ph.D. thesis, Kyoto University, Kyoto, 2011), indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains.

Figure 1

Laminar flow $\mathbf{u}=\sqrt{2}sin\left(\beta y\right){\mathbf{e}}_{\mathbf{x}}$.

Figure 2

Steady-state solutions found using initial conditions from “turbulence” at $\text{Re}=200$ ($S1$: top) and 300 ($S2$: bottom). The plotted flow is $x$-averaged with contours denoting flow into (red/upper portion of domain) and out of the page (blue/lower portion of domain); and with arrows for cross-stream flow. For both solutions the lower branch solutions are plotted. The third steady-state solution, $S3$, found at $\text{Re}=400$, has a similar appearance (not shown).

Figure 3

Continuation curve (thick black) for the steady state $S2$, first converged at $\text{Re}=300$ (blue diamond) and plotted in Fig. 2. Red dots denote the five $M=3$ modulational bifurcations found for $\text{Re}\le 800$ with dashed lines showing the ensuing solution branches, which either rejoin the original branch or extend to large $\text{Re}$.

Figure 4

Steady solutions (velocity deviation from laminar state) at the bifurcation point from solution $S2$ and at $\text{Re}=800$ for curve, which emerged through a modulational bifurcation of number $M=3$ (dark blue dashed line in Fig. 3). At the bifurcation, the solution is $T_{L_z/3}$ symmetric, which is broken. At $\text{Re}=800$ solution remains global and has similar structure to underlying solution.

Figure 5

Solution (black/dark) and eigenfunction (red/light) for two modes ($A_3$ and $A_6$) at a $M=3$ modulational instability (corresponding to blue dashed curve in Fig. 3 emerging closest to the saddle node). The eigenfunction ${\tilde{A}}_3$ lies in phase with the underlying solution in the center of the domain and out of phase at the edge. For $A_6$ and ${\tilde{A}}_6$, these phases are switched, suggesting that even close to the bifurcation the eigenfunction cannot move all modes simultaneously toward localization.

Figure 6

Continuation curve for solution $S1$ (thick black curve), considering the instabilities with similar structure across different modulation number, $M$. Bifurcations (red dots) lie close together and initially follow similar behavior. Bifurcated solutions fail to localize, irrespective of $M$.

Figure 7

Localized edge periodic orbit, $P1$. Amplitude of the modes as a function of $z$. The $x$-independent modes 1–3 and 9 remain steady (see Fig. 8), while $x$-dependent instability modes 4–8 have simple fluctuations with a single frequency. Time-dependent modes are plotted at four points in the oscillation with time progressing through black (full), red (dashed), blue (dotted), and green (dash-dotted).

Figure 8

The spectrum (maximum amplitude across all modes) for each temporal harmonic, where $a_{nm}^{\left(i\right)}$ is defined in Eq. (20). The temporally constant terms and first harmonic have similar amplitude, whereas the next harmonics have significantly lower amplitude and can be neglected.

Figure 9

Continuation in $\text{Re}$ of the localized edge state $P1$ for $L_z=10\pi$. (a) Time-averaged energy [see Eq. (24)] against $\text{Re}$ for the solution, which undergoes three saddle-node bifurcations before returning to larger Reynolds numbers. The solution at $\text{Re}=400$ on the lower and upper branch are shown in panels (b) and (c). During this process the solution increases in width and amplitude but remains localized.

Figure 10

(a) The energy of periodic orbits $P1$ and $P2$, plotted against spanwise domain length, $L_z$. The localized single edge state ($P1$, red full curve) emerges in a symmetry-breaking bifurcation from solution $P2$ (blue dotted curve, at the black dot). Solution $P2$ can be continued to wide domains, resulting in a symmetric pair of the $P1$ localized state. (b) As above but with the addition of solution $P3$ (green, dashed), the edge state in narrow domains ($L_z<7.5$). Continuation of this solution toward wide domains results in a localized solution pair with symmetry ${\mathbf{T}}_{L_z/2}{\mathbf{R}}_2$. In contrast, the steady states $S1$–$S3$ (pink, black, and orange) found in the previous section remain global throughout continuation in $L_z$.

Figure 11

Visualizations of solution $P1$ at varying domain width corresponding to red dots in Fig. 10. The solution originates at a symmetry breaking bifurcation ${\mathbf{T}}_{L_z/2}{\mathbf{R}}_1$ (panel 1), where the streaks are of equal strength. The slow streak (red) decreases in amplitude as domain length decreases. Beyond the saddle-node (in $L_z$) the slow streak splits into two and the solution localizes. Solution plotted as $x$-averaged (and thus time-averaged) deviations from the laminar state and remains $\mathbf{W}$-symmetric.

Figure 12

Spanwise energy, $\overline{e}\left(z\right)$ [see Eq. (25)] against $z$ for solutions $P1$ (red full line) and $P2$ (all others) as domain width varies. For wide domains (green) $P2$ matches the exponential decay of localized solution $P1$. During continuation to more narrow domains the structure near the energetic maxima of $P2$ is unchanged until a domain of $L_z=11.4$ (orange dot-dashed). Solutions for $L_z\le 14.8$ are plotted in frame (b). At $L_z\approx 7.2$ the bifurcation of $P1$ from $P2$ occurs (brown dashed).

Figure 13

Solution $P3$ during continuation in domain width, $L_z$, corresponding to green dots in Fig. 10). Solution is ${\mathbf{T}}_{L_z/2}{\mathbf{R}}_2$-symmetric and forms the attracting edge state for small domains. As domain width increases ($L_z\approx 7.5$) the solution loses stability (and thus is no longer the edge state). In wide domains ($L_z>20$), the solution localizes and can be viewed as two symmetry related copies of $P1$ separated by $L_z/2$.

Figure 14

Solutions $P2$ and $P3$ ($L_z\approx 39$) plotted in the $(x,z)$ plane at $y=0.9$. Flow in the $y$ direction is plotted with contours, while in-plane flow is denoted with arrows. As described in Eq. (26), solutions are out of phase (both in $x$ and $t$) in one half of the domain (here the left), while in phase in the other half (the right). This subtle difference affects the continuation when localized pairs are brought together through continuation in $L_z$.

Figure 15

Bifurcation diagram depicting downstream localized pipe flow solution (black, unclosed curve) emerging from a downstream periodic solution (blue, closed curve) in a modulational instability. Figure is an adaptation of Fig. 1 (inset) from Chantry et al. (2014), here showing friction factor (pressure gradient down the pipe) against pipe length (in pipe radii).

Figure 16

Dynamics within the edge for two global initial conditions generated from a turbulent trajectory. The first localizes away from the wall and converges toward the low-energy localized periodic orbit found with periodic boundary conditions. The second condition localizes adjacent to the wall to a more complex (both in time and space) and higher-energy solution. The black box encloses a single (half) period, which is further studied in Fig. 17.

Figure 17

Evolution along an edge attracting periodic orbit, $P4$. Top frame shows the energetic evolution along half the period at which point the state is related by symmetry ${\mathbf{R}}_1$ to the initial state. Dots correspond to the 12 $x$-averaged flowfield snapshots (deviation from laminar flow). No-slip conditions are applied at $z=0,16\pi$, with only the left quarter of the domain depicted. Solution initially grows slowly in time while moving slightly toward the wall before breaking down and moving away from the wall. Finally, the structure reforms with inverted streak-roll structure.

Figure 18

Continuation of localized solution $P1$ in $L_z$ for periodic boundary conditions (red) and spanwise-wall boundary conditions (gray dotted). Crosses denote locations of the solutions plotted below. Solutions are indistinguishable until $L_z\approx 10$ when, compared to the periodic solution, the fast streaks (yellow) fail to grow to amplitude as domain width decreases.