**4. Tilt Angle of Turbulent Bands**

The tilt angle of turbulent bands at *Re* < 1000 was reported in experiments by Paranjape [9]. Their measurements showed that the angle stays nearly constant close to 45◦ below *Re* 900 and decreases to approximately 30◦ above *Re* = 950. The decreasing trend was also reported by Shimizu and Manneville [8]. A few numerical studies also reported the tilt angle at some Reynolds numbers; for example, Kanazawa [7] reported 41◦ at *Re* = 660, Tao et al. [6] reported approximately 40◦ at *Re* = 700 and Xiao and Song [24] reported an angle of about 39◦ at *Re* = 750, which are lower than but close to the experimental results of Paranjape [9]. The small difference may be attributed to the periodic boundary condition used in simulations and to the specific methods of quantifying the tilt angle.

However, the mechanism underlying the tilt angle selection is still not well-understood. Prior studies simply measured the tilt angle by considering the entire band as a tilted object based on image processing or in similar manners [6,9]. Differently, here we propose that the tilt angle should be more fundamentally determined by the propagation speed of the head and the advection speed of the streaks inside the bulk. More specifically, the speed of the streaks inside the bulk relative to the head should determine the tilt angle of the band. Based on our measurements shown in Figures 2 and 7, we calculated the tilt angle of the band as

$$\theta = \arctan \frac{|c\_{\rm z,streak} - c\_{\rm z,head}|}{|c\_{\rm x,streak} - c\_{\rm x,head}|}. \tag{1}$$

The result is shown in Figure 8. Our calculations agree well with the experimental result of Paranjape [9] below *Re* 900. However, at *Re* = 1050, our calculation appears to be much higher than their measurement: our calculation gives 37◦ for *Re* = 1050, whereas it was estimated to be around 30◦ in experiments. Nevertheless, our calculation gives the decreasing trend in the tilt angle as *Re* is increased to around *Re* = 1000 and above.

**Figure 8.** The tilt angle of turbulent bands, calculated with Equation (1), at a few Reynolds numbers. The experimental measurements of Paranjape [9] are plotted as the dashed-triangle line for comparison.

The possible reason for the significant difference between our calculation and the experimental measurements at *Re* = 1050 can possibly be understood by inspecting the structure of the band as *Re* increases (see Figure 9). We can see that, at *Re* = 670, the band has a well-defined banded structure, i.e., the width (e.g., the streamwise extension) of the band does not significantly change along the band (see Figure 9a). At *Re* = 950, the tail of the band seems to broaden and the width of the band may not be constant along the length direction any more (see Figure 9b). Further at *Re* = 1050, the band significantly delocalizes: The bulk broadens gradually towards the tail and part of the band turns into an extended turbulent area (see Figure 9c). By image processing the entire band, as in the measurements of Paranjape [9] and Tao et al. [6], the calculated tilt angle at *Re* = 1050 will certainly be smaller than our calculation that is only based on the information of the low-speed streaks and the head. This disagreement will be small at low Reynolds numbers when turbulence is well-banded.

**Figure 9.** Turbulent bands at: *Re* = 670 (**a**); *Re* = 950 (**b**); and *Re* = 1050 (**c**).

The agreement between our calculation and the reported speeds in the literature supports our speculation that the tilt angle of the band is determined jointly by the propagation speed of the head and the advection speed of the streaks inside the bulk. However, what mechanism determines the advection speed of the streaks is still to be investigated. A quantitative study of the large-scale flow may give a hint to the advection of the streaks [9,27,29,30].

It should be noted that the two ends of turbulent bands may not exist in relatively small normal periodic domains or narrow tilted domains, therefore, seemingly our formulation of the tilt angle (Equation (1)) does not apply. In those cases, it is not clear what mechanism determines the tilt angle of turbulent bands. Our speculation is that the tilt angle may be indefinite and is strongly affected by the specific domain selection if the head does not exist. This might explain, for the same Reynolds number, why turbulent bands can exist in tilted domains with very different tilt angles [4,9,20] and why the nonlinear traveling wave solutions that Paranjape et al. [19] obtained can exist in a broad tilt-angle range from 20◦ to 70◦.
