*3.2. Advection Speed of the Streaks inside the Bulk*

Next, we quantitatively studied the advection speed of the streaks. The advection speed can be estimated by tracking an individual low-speed streak, as shown in Figure 6. Alternatively, it is possible to measure the speed of an array of streaks as a whole. We adopted the latter approach. We used the velocity data on the cut plane of *y* = −0.5, which well cuts through the streaks and offers a nearly optimal visualization of the flow pattern (see Figure 5). Nevertheless, a cut plane close to the wall, which would cut through high-speed streaks that are located close to the wall (see Figure 5), is equally applicable. We used the Structural Similarity Index Measure (SSIM) method [28] from image processing, which accesses the similarity between two images based on luminance, contrast and structure of the images. The method is detailed in Section 6.

The advection speed of low-speed streaks for a few Reynolds numbers are shown in Figure 7. The results show that, in the low Reynolds number regime between 670 and 1050, the streamwise advection speed slowly decreases from 0.68 to 0.63, whereas the spanwise speed seems to stay nearly constant at around 0.07. Note that the streamwise speed is very close to the bulk speed of the flow, which is 0.67. In fact, in Figure 6, the parallelogram was moved at the speeds we measured in this way and very well tracked the streak over hundreds of time units. Paranjape [9] reported that the phase speeds of the exact nonlinear traveling wave solution they obtained at *Re* = 720 are *cx* = 0.77 and *cz* = 0.06, which are close to our results, suggesting a strong connection between their traveling wave solution and turbulent bands.

**Figure 7.** Advection speed of the low-speed streaks at a few Reynolds numbers. The spanwise speeds are *cz* = 0.068, 0.076, 0.069, 0.071 and 0.068, and the streamwise speeds are *cx* = 0.68, 0.66, 0.65, 0.65 and 0.63, for *Re* = 670, 750, 850, 950 and 1050, respectively.
