**1. Introduction**

Plane Poiseuille flow (PPF), the flow driven by a pressure gradient between two parallel plates, displays a parabolic velocity profile at its laminar state and becomes linearly unstable when the Reynolds number is larger than the critical value, *Rec* = 5772 [1]. The Reynolds number (*Re*) is defined as 1.5*U*∗ *b h*∗ /ν∗ , where *U*∗ *<sup>b</sup>* is the bulk velocity, *h*<sup>∗</sup> is the half-channel height, and ν<sup>∗</sup> is the kinematic viscosity of the fluid. In practice, PPF may become turbulent at much lower Reynolds numbers than *Rec* due to the subcritical transition, where the finite-amplitude disturbances are necessary and the nonlinear effect cannot be ignored [2–4]. Davies and White [5] measured the friction coefficient of PPF with different aspect ratios of the cross-sections in a wide range of Reynolds numbers. It was shown that the critical Reynolds number of the subcritical transition increases with the ratio between the entrance length and the channel height, and it remains at 667.5 when the entrance length is larger than 108*h*. Patel and Head [6] found experimentally that PPF remained laminar as *Re* < 1035, and intermittent bursts occurred as 1035 < *Re* < 1350. Later experiments by Nishioka and Asai [7] confirmed that the turbulent state could hardly be sustained as *Re* < 1000. Based on flow visualizations, Carlson et al. [8] found that the orifice jet on the wall can trigger turbulent spots when *Re* is about 1000, and when *Re* < 840, the turbulent spots cannot be formed completely and decay eventually. Later experimental,

theoretical and numerical works were mainly focused on the turbulent spots as *Re* > 1000 [9–14]. According to the experiments of Alavyoon, et al. [15], the complete spot cannot be triggered by orifice jet if *Re* < 1100. Recently, turbulent stripes or bands were revealed by numerical simulations for *Re* ≥ 1070 [16,17] and were observed by flow visualizations [18]. It was found experimentally that the turbulent bands would break as *Re* < 1275, and the flow remained stable and laminar at *Re* = 975 [19].

Based on numerical simulations within a tilted long and narrow domain, Tuckerman found turbulent band structures as *Re* > 850 [20]. By applying entrance disturbances and flow visualization techniques, Sano and Tamai [21] obtained the turbulence fraction at a range of Reynolds numbers and defined a threshold of 830 for the transition by fitting the data with the Directed Percolation (DP) model. According to their experimental data, however, the turbulence fractions are not zero as *Re* < 830. Recent numerical simulations revealed that the DP power law is retrieved only when *Re* is above 924, and relaminarization will occur in the long-time limit as *Re* < 700 [22]. Numerical simulations in large domains showed that localized turbulent bands can be obtained when *Re* is reduced to 720 [23]. Further numerical investigations illustrated that the isolated turbulent band, a single banded coherent structure surrounded by a large laminar region, can obliquely extend at moderate Reynolds numbers but will decay eventually as *Re* < 665 [24]. This threshold Reynolds number, in fact, agrees with the experimental observation by Davies and White [5]. It is tested that the periodic turbulent band can sustain as *Re* < 750, though band breaking and band reconnection may occur [25]. Recently, the turbulent bands were observed at *Re* = 750 by flow visualization [26], and the mean growth rate of turbulence fraction was found to become positive at *Re* ≈ 650 [27,28]. Therefore, in the literature, there have been discrepancies on the threshold Reynolds number for sustained turbulence in channel flows.

Besides the turbulence fraction, other statistical parameters are studied as well for the transitional channel flows. Turbulence intensities at the channel center are measured and are found to increase rapidly around *Re* = 1050, reach a peak at *Re* = 1140, and then gradually decrease with increasing *Re* [29]. The intermittent low- and high-drag events are investigated numerically and experimentally [30–32], and it is found that the conditionally averaged Reynolds shear stress is higher than the mean value during the low-drag events [33]. Based on simulations of channel flows with constant pressure gradients, a linear correlation for the wall shear stress is observed between its kurtosis and its skewness squared [34]. It is known that high-order moments of velocity derivatives are important to understand the non-Gaussian behavior of turbulence [35], and the intermittency is a key concept to develop turbulence model for the transitions of incompressible, supersonic, and hypersonic boundary layer flows [36]. However, the study on the relation between the turbulence fraction and the high order moments of velocities in the transitional channel flows is still rudimental.

In this paper, a wind channel with a large width-to-height ratio is used to study the subcritical transition of PPF, and its configuration is introduced in Section 2. In Section 3, it is revealed that the turbulence intensity and the kurtosis of midplane streamwise velocity reach their maxima while the skewness has a negative minimum during the transition. Furthermore, an intermittent structure model is constructed to describe the velocity features of localized turbulent structures and derive theoretically the high-order moments of midplane velocity and the friction coefficient, which are shown to be consistent with the experimental data. In Section 4, conclusions are presented.
