**1. Introduction**

Much below the linear critical Reynolds number of the parabolic channel flow, transition to turbulence can occur under finite-amplitude perturbations, i.e., via a subcritical transition. Numerous studies have established that turbulence takes the form of discrete turbulent bands that are oblique to the streamwise direction, interspersed with laminar flow, at transitional Reynolds numbers [1–10]. Similar banded turbulent structures have also been observed in other quasi-two-dimensional flows, i.e., systems with one confined dimension and two extended dimensions, such as plane Couette [11–13], Taylor Couette [14,15], annular pipe [16] and Wallefe flows [17]. Therefore, the coexistence of laminar and turbulent states in the form of banded turbulent structures is a common feature of turbulence at transitional Reynolds numbers of a broad variety of shear flows. Recent investigations into these structures have greatly advanced the understanding of the subcritical transition in these flows [10,18]. In the following discussion, for channel flow, the streamwise, wall-normal and spanwise directions are denoted as *x*, *y* and *z*, respectively, time is denoted as *t* and the half-channel-height as *h*. The flow is assumed driven by a constant volume flux and the Reynolds number is defined as *Re* = *Uch <sup>ν</sup>* , where *Uc* is the centerline velocity of the unperturbed parabolic flow and *ν* the kinematic viscosity of the fluid.

The first observation and many numerical studies of turbulent bands in channel flow were performed by numerical simulations in relatively small computational domains, either normal or tilted, in which the structure, kinematics and dynamics of turbulent bands are rather restrained [1,2,4,19,20]. Particularly, narrow tilted domains force turbulent bands to be parallel to the narrow edge, which practically assumes

infinitely long bands in combination with periodic boundary conditions. Nevertheless, this greatly reduces the computational cost and allows studying the kinematics and dynamics of bands over large time scales [9,20] and offers conveniences for studying the mean flow and wavelength of the band pattern [4,10]. In a domain tilted by 24◦, Tuckerman et al. [4] reported that turbulent bands propagate approximately at the bulk speed of the flow, with a slight decreasing trend with the Reynolds number (the speed crosses the bulk speed at *Re* 1100). In a similar approach as Avila et al. [21] for pipe flow and Shi et al. [22] for plane Couette flow, Gomé et al. [20] also showed finite lifetime and splitting nature of bands and determined the on-set of sustained turbulence in channel flow to be at *Re* ≈ 950 by balancing the super-exponential decay and splitting processes, in a domain also tilted by 24◦. The subcritical transition to turbulence in plane Couette flow in tilted domains has been concluded to fall in the universality class of directed percolation [23] and the work of Gomé et al. [20] seems to suggest the same transition scenario in channel flow. However, the imposed tilt angle of the domain seems to affect the statistical results. For example, the simulations in a domain tilted by 45◦ [9] showed very different lifetimes of bands from the results of Gomé et al. [20]. Specifically, the former reported that turbulent bands are sustained at *Re* > 620, whereas the latter suggested that in fact the lifetime stays finite and is below 200 time units at *Re* < 700. The effect of the imposed tilt angle has not been thoroughly investigated. Besides, the usual narrow tilted domain only allows multiple bands to form parallel band pattern, i.e., bands are forced to take the same orientation.

Large domains pose a lesser restriction on turbulent bands. In recent years, a few studies have been dedicated to turbulent bands in large normal domains in experiments [3,9] and simulations [5–8,24,25]. If the domain is large enough, given a proper localized perturbation, turbulence elongates obliquely with respect to the streamwise direction and forms a fully localized band (localized both in its length direction and in its width direction). The existence of the two ends of the band adds further complexity to the flow. Paranjape [9] reported in experiment that at *Re* < 660, a turbulent band shrinks and will decay so that the flow will relaminarize in the end, because the growth at the downstream end (referred to as the head hereafter) is slower than the decay at the upstream end (referred to as the tail hereafter). At higher Reynolds numbers, a turbulent band becomes sustained because the growth at the head outperforms the decay at the tail and will grow in length. Numerical studies [6,7,24] agree with the experiments. Therefore, it has been confirmed that the growth of a band is unidirectional, driven by the head [7–9,24]. Because streaks decay at the tail and are generated at the head, an individual band undergoes a spanwise shift as a whole, aside from being advected in the streamwise direction. Shimizu and Manneville [8] mentioned that the spanwise drift speed is 0.1 and Xiao and Song [24] reported a close value of 0.08. Noticing the periodic streak generation at the head, Kanazawa [7] and Xiao and Song [24], respectively, proposed mechanisms behind the wave generation at the head, which are discussed in more detail below.

In fact, it was found that the length of a band does not grow infinitely. The length 'at equilibrium' of a band at *Re* = 660 was shown to be about 300 *h* and the length seems to increase with *Re* [7]. As the length is sufficiently large, the fast decay of the tail limits the growth, and splitting may occur with a daughter band nucleated. At relatively low Reynolds numbers, the splitting is longitudinal, i.e., the daughter band is parallel to the mother band. As Reynolds number increases (*Re* - 800), transverse splitting (or branching) can also occur, nucleating daughter bands with the opposite tilt direction such that the flow pattern becomes two-sided (the criss-cross pattern) [8,9]. However, the study of the splitting of bands and the underlying mechanism is still rare.

In the presence of multiple bands, given that bands have a spanwise shift speed as a whole and can grow in length, close bands with opposite orientations may collide. Even parallel bands, when located sufficiently close to each other, were shown to interact also [6,8]. The dynamics of individual bands and the interaction between bands determine the pattern that bands can form and therefore, determine the statistical aspect of the transition to turbulence [8]. Using unprecedented large domain and simulating up to very large times (up to O(105) time units), Shimizu and Manneville [8] showed that turbulent

bands can only form one-sided (parallel) pattern at low Reynolds numbers (*Re* 924), breaking the spanwise symmetry, which is restored only at higher Reynolds numbers. Directed percolation was found to reasonably well describe the transition process toward featureless turbulence at further higher Reynolds numbers, as also proposed by Sano and Tamai [26] in experiments. Interestingly, the one-sided pattern of turbulent bands at lowest Reynolds numbers seems to justify the use of tilted domain in which bands are forced to be parallel, although the tilt angle was shown about 40◦–45◦ below *Re* 900 [6–9] rather than 24◦ as used in some studies [4,20].

Although a great advancement in the understanding of turbulent bands has been made in recent studies, many problems even for individual turbulent bands have not been well understood, for example the mechanisms underlying the growth of bands at the head and the decay at the tail, the tilt angle selection and the self-sustaining mechanism of the bulk of turbulent bands. We discuss some of these problems in this paper.
