Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow

We have found that in two-dimensional Kolmogorov flow a spatially localized turbulent state (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of the control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmetry of the internal turbulence we introduce two coarse-grained variables characterizing the internal dynamics. These quantities follow the switching process reasonably. This suggests that the twin attracting invariant sets each of which corresponds to a one-way traveling SLT are embedded in the attractor of the moving SLT and the connection of the two sets is too complicated to be represented by a few degrees of freedom but the motion of an SLT is correlated with the internal turbulent dynamics.

Figure 1

Snapshots of vorticity field $\omega (\mathit{x},t)$ of a moving SLT for $n=4$ and $\alpha =0.25$: (a) $\text{Re}=26.75$ and $U_y=0.933$, (b) $\text{Re}=50$ and $U_y=1.46$.

Figure 2

Time evolution of vorticity averaged in $y$-direction $\mathrm{\Omega }(x,t)$ for $\text{Re}=26.75$ and $U_y=0.933$ (a) in the co-moving frame and (b) the laboratory frame.

Figure 3

The same as Fig. 2 except for $\text{Re}=50$ and $U_y=14.6$.

Figure 4

Long time evolution of the center $X\left(t\right)$ of the SLT shown in Fig. 2 for $\text{Re}=26.75$ and $U_y=0.933$. The SLT transits from a moving state to a standing state at $t\sim 120000$.

Figure 5

Time evolutions of the average velocity $\overline{c}$ (blue solid line) and the central position of SLT $X$ (black dotted line). Large black dots denote direction reversals at (a) Re = 26.5 and (b) Re = 50.

Figure 6

$F\left(\mathrm{\Delta }t\right)$ for $(\text{Re},U_y)=(50,1.46)$. Dashed straight line is $F=2\times {10}^{2}exp(-\mathrm{\Delta }t/700)$ for guide.

Figure 7

Phase portrait for $\mathrm{Re}=26.75$ and $U_y=0.933$. Dots indicate the strobe map with interval 10 non-dimensionalized times. Hashed line indicates a reversal around $\mathrm{t}\sim 1000$ in Fig. 5. Closed loops indicate the twin URO. One of the twin URO is used as the initial condition, and the other is just the symmetrical one (also a solution of NS equation).

Figure 8

Time evolutions of $\overline{c}$ (red thin solid line), $\overline{s}$ (black dotted line), and $a=a_p-a_n$ (yellow thick solid line): (a) for Re = 26.75 and $U_y=0.933$ and (b) for Re $=50$ and $U_y=1.4$. Minimum and maximum values are (a) ($-0.125$,0.126) for $a$, ($-0.0268$ 0.0266) for $\overline{c}$, ($-1.36$ 1.79) for $\overline{s}$ and (b) ($-0.0371$ 0.0351) for $a$, ($-0.0174$ 0.0157) for $\overline{c}$, ($-1.50$ 1.36) for $\overline{s}$.

Figure 9

Dispersion of vorticity field $\delta {\mathrm{\Omega }}_n^2\left(x\right)$ defined in Eq. (20) for Re = 26.75 and $U_y=0.933$. The average is taken under the condition $\overline{c}<0$: Solid line for DNS and plus signs for URO. Inset (a) shows a mean vorticity field ${\langle \omega \rangle }_{\overline{c}<0}$ and inset (b) shows dispersion field $\delta {\omega }_n^2(x,y)$. Inset (c) shows $\delta {\mathrm{\Omega }}_n^2\left(x\right)$ for Re = 50 and $U_y=1.46$. Each inset shows only a main part of the field.