**1. Introduction**

In the past few decades, the understanding of near-wall coherent structures has been greatly improved via the discovery of travelling-wave (TW) solutions [1]. These TW solutions were first obtained by Nagata [2] for plane Couette flow. They are non-trivial invariant solutions to the Navier–Stokes equation and are also sometimes called "exact coherent states (ECS)". Later, Waleffe [3,4]

found ECS solutions for plane channel flow. The spatial structure of these solutions is similar to the commonly observed structure of near-wall turbulence: mean flow with counter-rotating streamwise vortices and alternating low- and high-speed streaks. Most of these ECS solutions are observed to occur in pairs at a saddle-node bifurcation point, arising at a finite value of Reynolds number. The upper branch solution has a higher fluctuation amplitude and higher drag than the lower branch solution [2–5].

One way to investigate the complex turbulent dynamics using TW solutions is to employ "minimal flow units". The minimal flow units or MFU denotes the smallest computational domain where turbulence can persist [6] at a given Reynolds number. Jiménez and Moin [6] observed a cyclic and intermittent behaviour of the fluctuations of all important quantities while employing MFU to study plane channel flow. They also observed a rapid increase in the fluctuations and wall shear stress during the "active" part of the cycle. Later, Hamilton et al. [7] and Jiménez and Pinelli [8] further studied this cycle and observed that during the time when the wall shear stress is near its lowest values the streamwise variation of the flow is also reduced. The presence of intermittency in Newtonian turbulent flow has also been investigated earlier by McComb [9]. Xi and Graham [10] carried out DNS in an MFU for low Reynolds number, *Re<sup>τ</sup>* = *uτh*/*ν* = 85 for both Newtonian and viscoelastic flows. Here, *uτ*, *h* and *ν* are the friction velocity, channel half-height and kinematic viscosity, respectively. They observed that even in the limit of Newtonian flows, there are the moments of "low-drag" or "hibernating" turbulence, which display many similar features to MDR (a phenomenon generally associated with the polymer additives). They coined the nomenclature of a "hibernating" state when the flow was drag-reducing and resembles MDR, and "active" state for the rest of the flow. The major flow characteristics observed during hibernation were only weak streamwise vorticity and three-dimensionality, and lower than average wall shear stress. The frequency of these events increases with increasing viscoelasticity, although the events remain unchanged, i.e., they display similar flow properties as MDR. The connection between the polymeric drag reduction in turbulent flows and transition to turbulence in Newtonian flows has also been discussed earlier by Dubief et al. [11].

Xi and Graham [12] further investigated this phenomenon to provide detailed characteristics of active and hibernating turbulence in Newtonian and viscoelastic flows. They defined hibernation when the area-averaged wall shear stress was below 90% of the mean for a dimensionless time duration of Δ*t* <sup>∗</sup> = Δ*tuτ*/*h* - 3.5, where Δ*t* represents the dimensional time duration. Park and Graham [13] carried out DNS for MFU in a channel flow geometry, close to transition. They obtained five families of ECS solutions, which they denoted as the "P1, P2, P3, P4 and P5" solutions. Out of these five families of solutions, "P4" solution shows the most interesting behaviour. For the upper branch solutions, the velocity profile approaches the classic von Kármán log-law, while for the lower branch solutions the velocity profile approaches the Virk's MDR asymptote. They suggested that most of the time the turbulent trajectories remain at the upper-branch state (or the "active" state) with few excursions to the lower-branch state (or the hibernating state). This result provided a further verification that there are intervals of low-drag in Newtonian flows when the mean velocity profile is close to Virk's MDR profile as previously observed by Xi and Graham [10,12]. The existence of such solutions for Newtonian flows has a potential application in drag reduction, which makes it a practically significant field of research.

One major characteristic of wall-bounded turbulent flows is the so-called bursting process, which is an abrupt breaking of a low-speed streak as it moves away from the wall [14]. Itano and Toh [15] investigated the bursting process for channel flow at *Re<sup>τ</sup>* = 130 by computing TW solutions in a MFU using a shooting method. They observed that the bursting process is linked to the instability of the TW solution. Park et al. [16] studied the connection between the bursting process and the ECS solutions in minimal channel flow for 75 ≤ *Re<sup>τ</sup>* ≤ 115. They focussed on the P4 family of ECS solutions, as identified earlier by Park and Graham [13]. To detect a hibernating event they used the criteria that the area-averaged wall shear stress should go below 90% of the mean wall shear stress and stays there for a duration of Δ*tUcl*,*lam*/*h* > 65, where *Ucl*,*lam* is the laminar centerline velocity. This time-duration corresponds to Δ*t* <sup>∗</sup> > 3 for *Re<sup>τ</sup>* = 85. They defined bursting events based

on an increase in the volume-averaged energy dissipation rate by 50% of its standard deviation for a duration of Δ*tUcl*,*lam*/*h* > 15. They observed that many of the low-drag or hibernating events are followed by strong turbulent bursts. Based on this observation, they divided the turbulent bursts into two categories: weak and strong bursts, and suggested that the strong bursts are the ones which are always preceded by a hibernating event. They also investigated the possible link between the turbulent bursts and the instability of the P4-lower branch solution. Very similar trajectories were observed for the strong bursts and the lower branch of the P4 solution, which provides further evidence that the turbulent bursts are directly related to the instability of the ECS.

Initially, the investigation of these low-drag events was conducted in minimal channels, and therefore the need was to study this phenomenon for fully turbulent flow in extended domains. The relation between the minimal channels and flow in large domains was studied by Jiménez et al. [17] and Flores and Jiménez [18]. They suggested that the flow dynamics in minimal channels have many features that are representative of fully turbulent flows. It has also been seen that some of these solutions are highly localised and display the nontrivial flow only for a small region of an extended domain, whereas the rest of the flow remains laminar [19–21]. Kushwaha et al. [22] carried out an investigation into these low-drag events in an extended domain for channel flow at three Reynolds numbers, *Re<sup>τ</sup>* = 70, 85 and 100. The computational domain, in wall (or inner) units, was *L*<sup>+</sup> *<sup>x</sup>* <sup>≈</sup> <sup>3000</sup> and *<sup>L</sup>*<sup>+</sup> *<sup>z</sup>* ≈ 800 long in the streamwise and spanwise directions, respectively. They carried out a temporal and spatial analysis for extended domains and compared the results between the two. Regions or events of both low- and high-drag events were investigated in large domains, unlike previous MFU studies where the focus was primarily on low-drag events. To study the temporal intermittency, they employed the following criteria to detect low-drag (hibernating) or high-drag (hyperactive) events: the instantaneous wall shear stress (*τw*) should remain below 90% or above 110% of time-averaged value for a time duration of Δ*t* <sup>∗</sup> = Δ*tuτ*/*h* = 3 for low or high drag events, respectively. For studying the velocity characteristics during these low- and high-drag intervals in the flow, a conditional sampling technique was employed. They observed that, although the temporal and spatial analyses are independent of each other, the characteristics of low- and high-drag events obtained using these two methods were very similar. They found that for *Reτ* between 70 and 100, the regions of low-drag in an extended domain show similar conditional mean velocity profiles as obtained from temporal interval of low-drag in minimal channels for *y*<sup>+</sup> = *yuτ*/*ν* < 30, where *y* is the wall-normal distance. This showed that the spatiotemporal intermittency observed in extended channel flow is related to the temporal intermittency in a minimal channel.

Whalley et al. [23,24] carried out an experimental investigation of the low- and high-drag events in a plane channel flow at three Reynolds numbers, *Re<sup>τ</sup>* = 70, 85 and 100. Instantaneous velocity, wall shear stress and flow structure measurements were conducted using laser Doppler velocimetry (LDV), hot-film anemometry (HFA) and stereoscopic particle image velocimetry (SPIV), respectively. They employed the same criteria as Kushwaha et al. [22] to detect the low-drag events, but for the high-drag events, the criteria were slightly relaxed in order to obtain more events, as the high-drag events were found to occur at a lower frequency than the low-drag events. Instantaneous velocity and wall shear stress measurements were made at the same streamwise/spanwise location, enabling conditional sampling of the velocity data to be carried out. The conditionally averaged streamwise velocity and wall shear stress were found to be highly correlated until *<sup>y</sup>*<sup>+</sup> <sup>≈</sup> 40 and a resemblance was observed between the conditionally sampled mean velocity profiles for *y*<sup>+</sup> 40 and the lower branch of the P4 ECS solution as observed earlier in minimal channels [13]. They also observed that the fraction of time spent in hibernation (low-drag) decreases with increasing Reynolds number for 70 < *Re<sup>τ</sup>* < 100.

Recently, Pereira et al. [25] carried out DNS in channel flow of domain size, *Lx* × *Ly* × *Lz* = 8 *πh* × 2 *h* × 1.5 *πh* at *Re<sup>τ</sup>* between 69.26 and 180 for Newtonian flow, and at *Reτ*<sup>0</sup> = 180 for drag-reducing flow (65% drag reduction). The flow was identified as hibernating if the spatially-averaged wall shear stress was lower than 95% of its time-averaged value and no time

criteria were used (unlike previous studies where a minimum time duration was also used to detect a hibernating event, for example, in [16,22,24]). They demonstrated that the transition to turbulence in Newtonian flows shares various common features to the polymer induced drag reduction in turbulent flows.

Until now, these low- and high-drag events are investigated for 70 ≤ *Re<sup>τ</sup>* ≤ 100, and therefore a natural question arises as to what are the characteristics of these events in the so-called fully-turbulent flow regime (often associated with a threshold value of *Re<sup>τ</sup>* ≥ 180 [26]). The Reynolds shear stress characteristics during these events has been studied using the DNS in MFUs [12,13], yet there is no relevant experimental data or numerical data in extended domains available. In this paper, the lowand high-drag intermittencies are investigated using experimental and numerical techniques to answer these fundamental questions. The experiments are conducted in a channel flow facility using wall shear stress and velocity measurements. Recently, Agrawal et al. [27] observed that the flow in the present channel consists only of turbulent events beyond *Re<sup>τ</sup>* ≈ 67 and that significant Reynolds number dependence of the skewness and flatness of wall shear stress fluctuations starts to disappear by *Re<sup>τ</sup>* 73 − 79. Based on these results, in this work, the intermittences associated with the turbulent flow are investigated for *Re<sup>τ</sup>* ≥ 70. An experimental study is made for Reynolds number up to *Re<sup>τ</sup>* = 250, to probe the characteristics of these events for fully-turbulent channel flow. To study the Reynolds shear stress for *Reτ* = 70 and 85, experimental as well numerical techniques are employed.
