Childhood of turbulent spots in a shear flow

We numerically investigate the temporal aspects of turbulent spot spreading in a plane Couette flow for transitional Reynolds numbers between 300 and 450. We focus our study on the spreading along the streamwise direction and on the shape of turbulent spots. Studying the topology of turbulent spots and the associated large-scale flows, we suggest a decomposition of the streamwise growth rate. On one hand, the quadrupolar large-scale flow heads for the spot along the streamwise direction and act against the growth. The associated growth rate is negative. On the other hand, we also define a positive growth rate associated to inside large-scale flow which enables the convection of the streaks. The sum of these two growth rates is compared to the spot streamwise growth rate and shows good agreement. The resulting shape of the spot is then discussed. A scenario gathering all these elements provides a better understanding of the growth dynamics and of the shape of a turbulent spot in a plane Couette flow. This scenario should be relevant to other shear flows.

Figure 1

Top, left to right: $U_x$ velocity field at $\text{Re}=360$ in the plane at $y=0.77$ (a) and a plane at $y=-0.63$ (b) at $t=150$. Bottom, left to right: $U_x$ velocity field (c) at $\text{Re}=360$ in a slightly off-center plane at $y=0.098$ (same color bar as for the top plots) and in the plane $z=0$ (d) at $t=150$.

Figure 2

Temporal evolution of the turbulent spot at $\text{Re}=380$ in the midplane. Successive snapshots of $U_x$, respectively, correspond to $t=30,t=90,t=150$, and $t=210$ after the initial perturbation.

Figure 3

Temporal evolution of the turbulent fraction Ft (a) for several Re and time-averaged spanwise front velocity $\overline{{\sigma }_{\text{sf}}^z}$ and time-averaged streamwise front velocity $\overline{{\sigma }_{\text{sf}}^x}$ (b) as a function of Re.

Figure 4

Streamwise front velocity ${\sigma }_{\text{sf}}^x$ (a) and spanwise front velocity ${\sigma }_{\text{sf}}^z$ (b) as a function of time for several Re.

Figure 5

$U_x$ at $\text{Re}=360$ at $t=150$ for large scale (left) and small scale (right) in the midplane. The dashed line represents the line $x=0$.

Figure 6

$min\left(U_{\text{LSF}}^x\right){|}_{x>0}$ as a function of time for several Re (a) and $max\left(U_{\text{LSF}}^x\right){|}_{x>0}$ as a function of time for several Re (b).

Figure 7

Sketch of the topology of a turbulent spot and of the associated large-scale flows.

Figure 8

Evolution of the spot streamwise growth rate ${\sigma }_{\text{sf}}^x$ (left) and of $max\left(U_{\text{LSF}}^x\right){|}_{(x>0)}+min\left(U_{\text{LSF}}^x\right){|}_{(x>0)})$ (right) as a function of time for several Re.

Figure 9

Time evolution of the ratio $|min\left(U_{\text{LSF}}^x\right)/max\left(U_{\text{LSF}}^x\right)|$ for several Re.

Figure 10

Temporal evolution of the ratio $l_x/l_z$ for several Re.