**1. Introduction**

Laminar and turbulent flows are two different regimes encountered sometimes at the same parameters for a given geometry. In many flows they are in competition from the point of view of the state space. Shear flows next to solid walls however show this surprisingly robust property that both laminar and turbulent regions coexist spatially on very long time scales, when the laminar state is locally stable. This phenomenon, called 'laminar-turbulent intermittency' is well known in circular pipe flow since the days of O. Reynolds [1] and has lead recently to a burst of interest, a review of which is provided in Reference [2]. Such laminar-turbulent flows have been identified and partly characterized in Taylor-Couette flow [3,4] and in plane Couette flow [4–6]. They also have been identified in other set-ups involving curvature [7–9] or stabilizing effects [10]. The transitional regimes of plane Poiseuille flow, the flow between two fixed parallel plates driven by a fixed pressure gradient, have not received as much attention although this flow is the archetype of wall-bounded turbulent flows. Although this flow is frequently cited as an example of flow developing a linear instability (under the form of Tollmien–Schlichtling waves) [11], coherent structures typical of laminar-turbulent coexistence have been frequently reported in channel flow well below the linear instability threshold and a series of experimental and cutting-edge numerical studies in the 1980s and 1990s have focused on the development of spots [12–16]. Sustained intermittent regimes have not been identified as such before the mid-2000s, when Tsukahara [17] reported large-scale coherent structures from numerics in larger numerical domains. Like their counterpart in Couette flows, these structures display obliqueness with respect to the mean flow direction and a complicated long-time dynamics. The dynamics at onset in particular have remained mysterious [18] and, although this is currently debated, could follow a scenario different from the directed percolation one proposed for Couette flow. [9,19,20]. In recent years, the so-called transitional regime of plane channel flow has attracted renewed attention after new experimental studies. Although the works in Refs [21–23] focused on the minimal transition amplitude

for spot development, other studies [24–29] focused on the sustained intermittent regimes and their statistical quantification.

Experimentally the finite length of the channel sets a limitation to most statistical approaches. Numerical simulation in large domains combined with periodic boundary conditions is a well-established way to overcome such limitations. Surprisingly, despite a large number of numerical studies of transitional channel flow, investigation of spatio-temporal intermittency in large enough domains has not been possible before the availability of massive computational resources. Owing to recent numerical studies [30–32], there is currently a good consensus about a few facts concerning the transitional regime: laminar-turbulent bands with competing orientations emerge progressively as the Reynolds number is reduced below *Re<sup>τ</sup>* « 100, and their mean wavelength increases as the Reynolds number is decreased. At even lower flow rate the bands turn into isolated spots with ballistic dynamics rather than forming a seemingly robust stripe pattern [33–35]. The global centerline Reynolds number for the disappearance of the stripes is close to 660 [18,27]. However, many questions remain open. The most sensible theoretical issues revolve around the (still open) question of the universality class of the transition process (see Reference [18]), the role of the large-scale flows [23,25,36,37] in the sustainment of the stripes, or the mutual way different stripes interact together.

There is also a lack of quantitative data about the patterning regime itself. The present special issue is an opportunity to document the geometric characteristics of the stripe patterns in unconstrained settings. Moreover, there is an ongoing philosophical question about whether traces of spatio-temporal intermittency can be found in the fully turbulent regimes commonly reported at higher Reynolds numbers. In the present paper, using numerical simulation in large domains, we focus on three specific points hitherto undocumented: the angular distribution of turbulent stripes, the statistics of the laminar gaps between them, and high-order statistics of the local and instantaneous bulk velocity, wall shear stress and turbulent kinetic energy. The outline of the paper is as follows: Section 2 introduces the numerical methodology with the relevant definitions. The geometrical statistics of the stripe angles are presented in Section 3.1. The statistics of a few global quantities are presented in Sections 3.2–3.4. A discussion of the results is made in Section 4 with the conclusions and outlooks in Section 5.
