*2.1. Propagation Speed of the Head*

Firstly, we investigated the advection speed of the head. It has been reported that the head of a turbulent band, which is always located at the downstream end, propagates in both streamwise and spanwise directions [6,8,24]. The spanwise motion can be in either positive or negative spanwise direction and the specific direction is correlated with the orientation of the band (see Figure 1). The head of the upper band moves downward (in negative spanwise direction) while that of the lower band moves upward, given their opposite orientations. Bands with similar orientation as the upper one are referred to as right-going bands, and those with the opposite orientation are referred to as left-going bands. This correlation can be intuitively understood because the head continually generates turbulence by invading laminar flow region on one side. We revisit this point in Section 2.2. Xiao and Song [24] measured the speeds at *Re* = 750 by tracking the head and reported a streamwise speed of *cx* = 0.85 and a spanwise speed of *cz* = 0.1 (absolute value).

**Figure 1.** Turbulent bands with different orientations at *Re* = 750. (**a**,**b**) The streamwise direction is in the positive *x* direction and *z* denotes the spanwise direction. Streamwise velocity fluctuations in the *x*-*z* cut plane at *y* = −0.5 are plotted as the colormap with blue representing low speeds and red representing high speeds compared to the basic flow. The two panels are separated by 320 time units.

To investigate the *Re*-dependence of the speeds and also for calculating the tilt angle of turbulent bands in Section 4, we measured the speeds in the low Reynolds number regime ranging from *Re* = 670, which is nearly the lowest Reynolds number for sustained bands, to *Re* = 1050 at which frequent splitting and branching of bands were reported to occur [8,9]. For this study, the Reynolds numbers, domain sizes and resolutions are listed in Table 1. It has been shown that, at *Re* = 660, a band can continuously grow up to the length of approximately 300 *h* [7]. The length can be much larger at higher Reynolds numbers [7,8]. The domain sizes used in our study are not large enough for the band to reach the length 'at equilibrium', rather we only require the domain size to offer sufficiently long time for the head to reach its characteristic propagation speed. The simulation was stopped when the head and tail were too close to each other and started to interact due to the periodic boundary conditions. Xiao and Song [24] already showed that the speed of the head of turbulent bands at *Re* = 750 is not affected by the domain size by comparing the speeds measured in domains with *Lx* = *Lz* = 120 *h* and *Lx* = *Lz* = 320 *h*.

**Table 1.** The Reynolds number *Re*, domain size *Lx* and *Lz*, number of wall-normal grid point *N* and the ratio between *h* and the grid spacing in *x* and *z* directions, Δ*x* and Δ*z*, respectively.


At each Reynolds number, we generated a fully localized turbulent band directly at low Reynolds numbers using the method proposed by Song and Xiao [25]. After the band has sufficiently developed, the head was tracked over a time window of O(500) time units and the average speed was calculated based on the position and time separation. The results in Figure 2 show that both the streamwise and spanwise speeds stay nearly constant for all Reynolds numbers investigated, at 0.85 and 0.1, respectively. Besides, the speeds were shown to be rather stable, i.e., only fluctuate slightly in time around the respective averaged values for *Re* = 750 [24], which is also the case for other Reynolds numbers in this study.

**Figure 2.** The streamwise (circles) and spanwise (triangles) speed of the head of turbulent bands at various Reynolds numbers. Note that it is the absolute value of the spanwise speed plotted given that the speed can take either positive or negative values. The two solid lines at 0.85 and 0.1 are plotted to guide the eyes. The experimental measurement of the spanwise speed [9] is plotted as the dashed-circle line for comparison.

In experiments, Paranjape [9] showed that the spanwise speed of the head slowly decreases from 0.085 at *Re* 700 to 0.08 when the Reynolds number is increased to *Re* 850 (see the dashed-circle line in Figure 2). Besides, Paranjape [9] reported a streamwise speed of the entire band of about 0.75 between *Re* = 670 and 900, but did not report the streamwise speed of the head. They also reported the speeds between *Re* = 600 and 670, in which regime we could not obtain a sustained turbulent band in our DNS. It can be seen that our spanwise speed is systematically larger than the experimental measurement [9] (see Figure 2). The difference could possibly be attributed to the periodic boundary condition used in our numerical simulations, although Xiao and Song [24] mentioned that the *Lx* = *Lz* = 120 *h* box gives the same speed as that given by the *Lx* = *Lz* = 320 *h* box at *Re* = 750. It may equally be attributed to the side-wall effect in experiments. Simulations in much larger periodic boxes or in a channel with side walls are needed to confirm about this point. Nevertheless, the two sets of speeds are close to each other.
