Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence

Considering a wall turbulence as a chaotic dynamical system, we study regeneration cycles in a minimal wall turbulence from the viewpoint of orbital instability by employing the covariant Lyapunov analysis developed by [F. Ginelli et al. Phys. Rev. Lett. 99, 130601 (2007)]. We divide the regeneration cycle into two phases and characterize them with the local Lyapunov exponents and the covariant Lyapunov vectors of the Navier-Stokes turbulence. In particular, we show numerically that phase (i) is dominated by instabilities related to the sinuous mode and the streamwise vorticity, and there is no instability in phase (ii). Furthermore, we discuss a mechanism of the regeneration cycle, making use of an energy budget analysis.

Figure 1

Illustration of the plane Couette flow system. $x,y,z$ directions are referred to as streamwise, spanwise, wall-normal directions, respectively.

Figure 2

Lyapunov spectrum with the different averaging time $T$.

Figure 3

Closeup of the Fig. 2 focusing on the zero Lyapunov exponents (5th–8th Lyapunov exponents).

Figure 4

Lyapunov spectrum with different truncation numbers. Red filled circles are the $8\times 8\times 32$ case and black open circles are the $10\times 10\times 36$ case.

Figure 5

Snapshots of the streamwise velocity field $u_x(\mathit{x},t)$ (contour lines) and the streamwise vorticity field ${\omega }_x(\mathit{x},t)$ (tone levels) at (a) $t=2730$, (b) $t=2750$, (c) $t=2760$, (d) $t=2770$, (e) $t=2790$, (f) $t=2820$. The upper figure of each snapshot is a cross-sectional view taken along $z=0$ and the lower one is a cross-sectional view taken along $x=1.5$ [indicated by a small arrow in (a)].

Figure 6

Time series of model rms velocities $\sqrt{{\langle |\widehat{\mathit{u}}{\left(\mathit{k}\right)|}^2\rangle }_z}$ for (a) multiple regeneration cycles $\left(2500\le t\le 3500\right)$ and (b) single regeneration cycle $\left(2730\le t\le 2830\right)$. Solid lines (red): $\mathit{k}=(0,1)$ (streak mode), dotted lines (blue): $\mathit{k}=(1,0)$ (meandering mode), dashed dotted lines (green): $\mathit{k}=(1,1)$, thin solid lines (pink): $\mathit{k}=(2,0)$, and dashed double-dotted lines (light blue): $\mathit{k}=(1,2)$.

Figure 7

Time series of “horizontal” rms of streamwise vorticity $\sqrt{{\langle {\omega }_x^2\rangle }_H}$ of the single regeneration cycle at the midplane $(z=0)$. Blue circle in the figure represents the value of $\sqrt{{\langle {\omega }_x^2\rangle }_H}$ at $t=2750$ for reference in a later section.

Figure 8

Lyapunov spectrum ${\lambda }_j(j=1,2,3,\cdots ,30)$ of the minimal Couette turbulence. The minimal Couette turbulence possesses four positive Lyapunov exponents $[{\lambda }_j>0(j=1,2,3,4)]$ and three zero Lyapunov exponents $[{\lambda }_j=0(j=5,6,7)]$. The maximum Lyapunov exponent is ${\lambda }_1=0.021$.

Figure 9

Summation of the Lyapunov exponents ${\sum }_{j=0}^i{\lambda }_j$ of the minimal Couette turbulence. The Lyapunov dimension is $D_L=14.8$ and Kolmogorov-Sinai entropy is $h_{\text{KS}}=0.048$.

Figure 10

Upper panel: Time series of local Lyapunov exponents ${\tilde{\lambda }}_j\left(t\right)$ for (solid line; red) $j=1$, (dotted line; green) $j=2$, (dashed dotted; blue) $j=3$, and (dashed double-dotted; pink) $j=4$. Lower panel: The “horizontal” rms of streamwise vorticity $\sqrt{\langle {\omega }_x^2\rangle H}$ at the midplane $(z=0)$.

Figure 11

Finite-time growth rate $\mathrm{\Lambda }(t_0,\tau )$, (a) $t_0=2730$, (b) $t_0=2760$, for ${\tilde{\lambda }}_j\left(t\right)$ for (solid line; red) $j=1$, (dotted line; green) $j=2$, (dashed dotted; blue) $j=3$, and (dashed double-dotted; pink) $j=4$. The black dot horizontal line denotes ${\mathrm{\Lambda }}_j(t_0,\tau )\equiv 1$ (i.e., neutral).

Figure 12

Lyapunov modes at the initial stage of the phase (i) $\left(t=2730\right)$. Streamwise vorticities of the Lyapunov modes $\delta {{\omega }_x}_j$ are shown as color tone for (a) $j=1$, (b) $j=2$, (c) $j=3$, (d) $j=4$, and streamwise velocities of the base flow are shown as counter lines.

Figure 13

Cross-sectional views of the most unstable Lyapunov mode at the initial stage of the phase (i) $\left(t=2730\right)$ at (a) $x=L_x/4$, (b) $x=L_x/2$, (c) $x=3L_x/4$, (d) $x=L_x$. Streamwise vorticities of the Lyapunov mode $\delta {{\omega }_x}_1$ are shown as color tone and streamwise velocities of the base flow are shown as counter lines.

Figure 14

Lyapunov modes at the final stage of the phase (i) $\left(t=2760\right)$. Streamwise vorticity of the Lyapunov mode $\delta {{\omega }_x}_j$ is shown as color tone for (a) $j=1$, (b) $j=2$, (c) $j=3$, (d) $j=4$, and streamwise velocity of the base flow is shown as counter lines.

Figure 15

Budget analysis of the evolution equation of the “modal energy” (10) in the case of the streak mode (i.e., $\mathit{k}={\mathit{k}}_s)$. The red (solid) line is the time derivative term (left-hand side of the evolution equation), the green (dashed) line is the mean flow interaction term, the blue (dot) line is other nonlinear terms, and the pink (dashed-dot) line is the viscous term.

Figure 16

(a) Wall-normal profile of the mean flow interaction term of the streak mode $g({\mathit{k}}_s,z)$ (red line with closed circles) and the meandering mode $g({\mathit{k}}_m,z)$ (blue line with open circles) at $t=2800$. (b) The cross-sectional view of the streamwise velocity field consisting of the streak mode only, $u_x^{{\mathit{k}}_s}(y,z)$ (color tone), and that consisting of all modes (solid lines) at $t=2820$.

Figure 17

Budget analysis of the evolution equation of the streak mode “energy” Eq. (11). The red (solid) line, the green (dashed) line, and the pink (dashed-dot) line are the same as in Fig. 15. The blue (dot) line is the other nonlinear terms and the navy (dashed double-dotted) line is the nonlinear interaction terms with the meandering mode in Eq. (11).

Figure 18

Budget analysis of the evolution equation of the streak mode “energy” Eq. (13) for $2730\le t\le 3030$ including nearly three regeneration cycles. The red (solid) line is the time derivative term [left-hand side of Eq. (13)], the blue (dot) line is the interaction term between the streak mode and the meandering mode ${\langle h({\mathit{k}}_s,z)\rangle }_z$, and the green (dashed) line is the rest term ${\langle r({\mathit{k}}_s,z)\rangle }_z$.

Figure 19

Conceptual diagram of the energy flows in the regeneration cycle for phases (i) and (ii). In the diagrams, “Mean flows” denotes $\mathit{k}=(0,0)$ mode, “Streaks” denotes streak mode $\left({\mathit{k}}_s=(0,1)\right)$, and “$e^{i\alpha x}$ mode” denotes meandering mode $[{\mathit{k}}_m=(1,0)]$.