**4. Discussion**

The present simulations of the transitional regime of pPf confirm and extend previously documented knowledge, such as the constancy of *Cf* in the patterning regime and the variation of the band orientations close to the transition point.

The statistical analysis of the distribution of laminar gaps reveals that the distributions are exponentially tailed over the entire parameter range 39 ď *ReG <sup>τ</sup>* ď 100, demonstrating that even the value *ReG <sup>τ</sup>* " 39 remains away from any sort of critical regime, which would be marked by algebraic distributions. This is consistent with the existing estimation of the location of the transitional critical point *Recl* « 660 [18,27], which translates to *ReG <sup>τ</sup>* « 36. The entire patterning regime should thus be seen as bona fide spatio-temporal intermittency, with the critical behavior and transition point being relegated to values of *ReG <sup>τ</sup>* ă 39. Exploring the statistics of the flow closer to the critical point would require even larger domains and longer observation times. Such an investigation is outside the scope of the current study.

The orientation of the bands in the patterning regime for 60 ď *ReG <sup>τ</sup>* ď 90 (1800 ď *Recl* ď 4050) is essentially constant, with an angle *θ* " 25˝ ˘ 2.5˝. This validates the choice of *θ* " 24˝ as a suitable value in the numerical approach of Tuckerman et al. [5,31,32], where slender computational domains are tilted at a chosen value of the angle. However, this angle of 24˝ no longer fits the mean orientation of the independent turbulent bands in the lower range *ReG <sup>τ</sup>* ď 60 (*Recl* ď 1800), where the orientation of the bands increases by a factor close to two, with *θ* « 40˝ for *ReG <sup>τ</sup>* " 39.

We confirm the observation of a constant *Cf* <sup>Ğ</sup> in the patterning regime, which also implies *Reτ* " Ğ*Reb*, as reflected in Figure 10a. This constant value of *Cf* in the transitional regimes further enforces the long lasting analogy with first order phase transitions [49], for which the thermodynamic parameter conjugated to the order parameter remains constant while the system evolves from one homogeneous phase to the other, when a suitable control parameter is varied. At the mean-field level, a trademark of phase coexistence, is then the presence of a bimodal distribution of the order parameter in the coexistence regime. Capturing this bi-modality is however known as being a challenge, even in simulations of standard equilibrium systems: first, not all protocols allow for observing the phase coexistence; second, the order parameter must be coarse-grained on appropriate length-scales as

compared to the correlation lengths such that non-mean-field effect do not dominate [50]. More than often, the bi-modality of the order parameter distribution is replaced by a mere concavity and a large kurtosis. If the two phases have very different fluctuations, as is the case here, one also expects a strong skewness of the distribution. Our observations extend the analogy, already reported at the level of the mean observable, to their fluctuations. However, a lot remain to be done to further exploit this analogy, in particular by making more precise what the relevant order and control parameters are. Let us stress that whether the analogy with a first order transition is valid or not, it does not preclude the dynamics at the spinodals from obeying a critical scenario, such as directed percolation close to the laminar phase spinodal [51] and a modulated instability of the turbulent flow close to the turbulent one [4].

Finally, the statistical moments showcased here demonstrate a correlation between the skewness and the kurtosis of both *Re<sup>τ</sup>* and *Ec f* . Such a correlation, observed in both the transitional regime and higher Reynolds number turbulence but originally developed for the latter only [48], suggests a universal turbulent character, beyond the mere distinction transitional/featureless.
