*3.1. Angular Statistics of Turbulent Bands*

The self-organization of turbulence into long band–like structures, oriented with an angle with respect to the streamwise direction, is depicted in Figure 2. The (signed) angle is computed using two different methodologies. As in Duguet et al. [36] in the case of pCf, the local *y*–integrated velocity field is found to be parallel to the bands. The same holds for pPf, as is visible in Figure 4a,c for *ReG <sup>τ</sup>* " 60 and 40, respectively. Please note that unlike Couette flow, pPf features advection with a non-zero mean bulk velocity. Hence the local velocity field is here computed by removing this mean advection velocity. A first estimation of the local and instantaneous band angle is therefore computed following Equation (6):

$$\theta\_L(\mathbf{x}, z, \mathbf{t}) = \tan^{-1} \left[ \frac{\int \mathbf{u}\_z' \, d\mathbf{y} - \langle \int \mathbf{u}\_z' \, d\mathbf{y} \rangle}{\int \mathbf{u}\_x' \, d\mathbf{y} - \langle \int \mathbf{u}\_x' \, d\mathbf{y} \rangle} \right] \tag{6}$$

The second estimation is obtained from Fourier analysis and computed from Equation (7), following Reference [44] :

$$\theta\_{\mathbb{F}}(t) = \tan^{-1}(\lambda\_{\mathbb{z}}/\lambda\_{\mathbb{x}}) \tag{7}$$

where *λ* " 2*π*{*k*, with *k* being the leading non-zero wavenumber identified from the power spectra (excluding the *kx* " *kz* " 0 mode). The Fourier spectrum is computed for the quantity *τ*p*x*, *z*, *t*q, but similar results have been observed for other observables such as *Ec f*p*x*, *z*, *t*q and *Ev* " p1{2q ş *u*2 *<sup>y</sup> dy*. The angles can be read directly from the Fourier spectra in polar coordinates, see Figure 4b,d for the same values of *ReG <sup>τ</sup>* " 60 and 40, respectively. The mean angles Ě*θL* and s*θ<sup>F</sup>* are then computed by respectively space-time-averaging and time averaging the data obtained from Equation (6) and (7).

The variation of the mean (signed) angles with *ReG <sup>τ</sup>*, computed using the two methods, is shown in Figure 5a, where the indices 1, 2 stand for the two band orientations. Both methods provide identical results. The variation of the (unsigned) angle of the band denoted by *θ*, computed as *θ* " | Ě*θF*| is shown in Figure 5b. It is found that the mean angle *θ* of the bands remains approximately constant with *θ* " 25˝ ˘ 2.5˝ in the range of values 60 ď *ReG <sup>τ</sup>* ď 90 and increases for lower value of *ReG <sup>τ</sup>* ă 60. In the patterning regime, i.e., for *ReG <sup>τ</sup>* ě 50, the angle of the bands is found to be distributed symmetrically with respect to zero, as a consequence of the natural symmetry *z* Ð ´*z* of the flow. For lower *ReG <sup>τ</sup>* these quasi-regular patterns break down into individual localized structures analogous to individual puffs in cylindrical pipe flow. As the pattern dissolves, one single band orientation ends up dominating the dynamics as shown by Shimizu and Manneville [34] for a similar domain size. The angle *θ* further increases as the regular pattern deteriorates, with *θmax* « 40 at *ReG <sup>τ</sup>* " 39. Previous studies [18,27] have documented that the angle of the bands approach 45˝ close to the onset of transition. The present investigation agrees well with these studies (Figure 5b) while covering a wider range in Reynolds number, highlighting the difference between the puff regime for which *θ* « 40 ´ 45˝, and the patterning regime for which *θ* is almost half this value (see also Figure 2).

(**a**) (**b**)

**Figure 4.** (**a**,**c**) Isocontours of *τ*1 with the local velocity indicated by the normalized velocity vectors, at *ReG <sup>τ</sup>* " 60, 40, respectively; (**b**,**d**) Instantaneous Fourier spectrum in polar coordinates for (**a**,**b**), respectively.

**Figure 5.** (**a**) Variation of the mean (signed) angle of the turbulent bands with *ReG <sup>τ</sup>* , computed from the Fourier spectra (*θ* Ď*F*<sup>1</sup> , *θ* Ď*F*<sup>2</sup> ) and the mean (signed) angle of the local velocity ( Ę*θL*<sup>1</sup> , Ę*θL*<sup>2</sup> ) (**b**) Variation of the mean unsigned band angle *θ* along with the data from Reference [27,30].

Figure 4c shows that across a band, the local large-scale velocity changes orientation [41]. This property is used to sort out the local maxima of *τ* (higher than *τ* ` *σ*p*τ*q) as belonging to one band with a particular inclination. This allows one to define the respective streamwise and spanwise inter-stripe distances *lx* and *lz* between bands of the same orientation. Figure 6a,b displays Ě*lx and* Ě*lz* for orientations 1 and 2, respectively, as a function of *ReG <sup>τ</sup>*. Both increase when decreasing *ReG <sup>τ</sup>*. They vary in parallel in the patterning regime, hence the quasi-constant angle *θ* of the bands. When only one band orientation survives, one observes that the increase in *θ* amounts to the saturation of Ě*lx*1, while Ě*lz*<sup>1</sup> keeps increasing.

**Figure 6.** (**a**,**b**) Space-time-averaged inter-stripe streamwise Ě*lx*1,2 (blue) and spanwise Ě*lz*1,2 (red) distances for bands of orientations 1 and 2, respectively.
