**5. Time Spent in Low- and High-Drag Events**

Here we study the effect of three different scalings, i.e., inner scaling, mixed scaling and outer scaling for the time-duration criteria to detect a conditional event. Outer scaling is simply Δ*tUb*/*h*. Inner scaling (Δ*t* <sup>+</sup> = Δ*t u*2 *τ <sup>ν</sup>* ) and the mixed scaling (Δ*t* <sup>∗</sup> = Δ*t <sup>u</sup><sup>τ</sup> <sup>h</sup>* ) are related by the following relation.

$$
\Delta t^+ = \mathcal{R}e\_7 \Delta t^\*.\tag{4}
$$

From Equation (4), it can be observed that with increasing Reynolds numbers, the Δ*t* <sup>+</sup> value increases for the same Δ*t* ∗ value. Whalley et al. [24] studied the fraction of time spent in low- and high-drag events with changing Reynolds numbers where the time-duration criterion was kept constant in mixed scaling. They observed that with increasing Reynolds number between 70 ≤ *Re<sup>τ</sup>* ≤ 100, the fraction of time spent in low-drag events decreases by approximately 500% while increasing the *Re<sup>τ</sup>* from 70 to 100. The effect of other scalings has not been considered previously.

The fraction of time spent in the conditional events is investigated for *Re<sup>τ</sup>* = 70, 85, 120, 180 and 250 using all three scalings. For *Re<sup>τ</sup>* = 70, Δ*tuτ*/*h* = 3 corresponds to about *tu*<sup>2</sup> *<sup>τ</sup>*/*ν* = 200 and *tUb*/*h* = 42. Based on this information, three values are chosen for each scaling to study the effect of Reynolds number on the fraction of time spent in the conditional events. For the mixed scaling, Δ*tuτ*/*h* = 1, 2 and 3, for outer scaling, *tUb*/*h* = 15, 30 and 45, and for the inner scaling, *tu*<sup>2</sup> *<sup>τ</sup>*/*ν* = 100, 200 and 300 are used. For the low-drag events the threshold criterion is kept constant as *τw*/*τ<sup>w</sup>* < 0.9 and for the high-drag events the threshold criterion is kept constant as *τw*/*τ<sup>w</sup>* > 1.1.

Figure 4 shows the fraction of time spent in low- and high-drag for different Reynolds numbers and the time-duration criteria. Results are shown for both the experimental as well as DNS data. It can be observed that the fraction of time spent in low-drag or high-drag decreases with increasing Reynolds numbers when mixed or outer scaling is used for the time duration criteria. This is similar to the result obtained using the mixed scaling for the time-duration criteria by Whalley et al. [24]. However, the fraction of time spent in the conditional events remains almost independent of the Reynolds number for 70 ≤ *Re<sup>τ</sup>* ≤ 250 for the experimental data, when the time-duration criteria is kept constant in inner units. DNS data shows a qualitatively consistent behaviour (i.e., show a similar trend for all three scalings) in the fraction of the conditional events compared to the experimental data although for a smaller range of Reynolds numbers. One possibility for the differences observed between DNS and experiments here is that these very rare low- or high-drag events involve flow structures that are much longer in the streamwise direction than usual, and that a domain size that is adequate for the vast majority of the turbulent dynamics might not be long enough to quantitatively capture the frequency of these rare events. Alternatively, subtle differences caused by the finite aspect ratio of the experimental set-up in comparison to the periodic boundary conditions used in the simulations, or the inherent uncertainties associated with the calibration of the hot-film signals maybe the cause of these differences. Based on this observation, inner scaling is chosen for the time-duration criteria in the

remainder of this paper. Figure 4e,f also shows that increasing the value of the time-duration criteria (100 ≤ <sup>Δ</sup>*tu*<sup>2</sup> *<sup>τ</sup>*/*ν* ≤ 300) decreases the fraction of time spent in these conditional events. The fraction of time spent in the intervals of low-drag is found to be greater than the intervals of high-drag for the same values of the time-duration criteria for 100 ≤ <sup>Δ</sup>*tu*<sup>2</sup> *<sup>τ</sup>*/*ν* ≤ 300, and where the threshold criteria is kept the same in terms of the magnitude (*τw*/*τ<sup>w</sup>* < 0.9 for the low-drag events and *τw*/*τ<sup>w</sup>* > 1.1 for the high-drag events).

**Figure 4.** Reynolds number variation of fraction of time spent in low-drag events with using (**a**) mixed scaling, (**c**) outer scaling and (**e**) inner scaling for the time-duration criteria. Reynolds number variation of fraction of time spent in high-drag events with using (**b**) mixed scaling, (**d**) outer scaling and (**f**) inner scaling for the time-duration criteria. Open symbols represent the experimental data and filled symbols represent the DNS data. The threshold criteria to detect a low- and high-drag event are *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* > 1.1, respectively. Note that the *y*-axis is not the same between low- and high-drag data. Error bars obtained by dividing the sample size into two halves and calculating the respective fraction are found to be within the size of the symbols and are therefore removed to avoid cluttering of data. Dotted lines in panels (**e**,**f**) highlight the average value of fraction (%) for 70 ≤ *Re<sup>τ</sup>* ≤ 250 at different values of Δ*tu*<sup>2</sup> *<sup>τ</sup>*/*ν* obtained using experiments.

A similar observation was also made previously by Whalley et al. [24] while using mixed scaling for the time-duration criterion.

Figure 4 shows that the fraction of time spent in the conditional events decreases with increasing the value of the time-duration criterion. A further investigation of this phenomenon is made by studying the dependence of the occurrence of conditional events as a function of their durations. Figure 5 shows the distribution of the occurrence of low- and high-drag events as a function of Δ*t* + for *Re<sup>τ</sup>* = 180. The threshold criteria to detect a low- and high-drag events are *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* > 1.1, respectively. The probability of occurrence of both low- and high-drag events decreases almost exponentially (as the *y*-axis is in log scale) with increasing Δ*t* <sup>+</sup>. For Δ*t* <sup>+</sup> - 400, *P*(Δ*t* +) does not seem to be well resolved because of the lower occurrence of low- and high-drag events for higher Δ*t* <sup>+</sup>, thus leading to lower number of events to carry out the statistical analysis. The distribution of high-drag events is observed to be different to the distribution of low-drag events. There is a higher probability of occurrence of high-drag events for lower Δ*t* <sup>+</sup> as compared to the low-drag events and vice versa. The crossover Δ*t* <sup>+</sup>, where the behaviour of the low- and high-drag events becomes opposite, is about 60. The decay of the probability of the low- and high-drag events is then fitted with an exponential relationship for 100 ≤ Δ*t* <sup>+</sup> <sup>≤</sup> 300, given by *<sup>P</sup>*(Δ*<sup>t</sup>* +) = *Ae*−*λ*Δ*<sup>t</sup>* + . Here, *λ* indicates the rate of decay. The decay rate is calculated for all the Reynolds numbers. Exponential distributions like this arise in so-called Poisson processes, also called memoryless processes. The exponential decay implies that the probability of the interval ending between time Δ*t* <sup>+</sup> and time Δ*t* <sup>+</sup> + *d*(Δ*t* +) is independent of Δ*t* <sup>+</sup>, i.e., the probability of the low- or high-drag intervals ending are independent of how long they have lasted. Avila et al. [42] observed a similar memoryless process with regards to puff splitting during transition in a pipe flow. After an initial formation time, the distribution of puff splitting were exponential and therefore memoryless, thus showing that the probability of a puff splitting does not depend on its age. Table 2 shows the rate of decay obtained for low- and high-drag events at various Reynolds numbers. The rate of decay is found to be almost independent of the Reynolds numbers for both low- and high-drag events, and the *λ* values are lower for the low-drag than the high-drag for the 100 ≤ Δ*t* <sup>+</sup> <sup>≤</sup> 300. A slight discrepancy is observed for *Re<sup>τ</sup>* <sup>=</sup> 70, which can be attributed to the presence of transitional effects at this Reynolds number, as discussed in Agrawal et al. [27]. These results are also consistent with the results shown in Figure 4e,f that the fraction of the conditional events are almost independent of the Reynolds number and the fraction of time spent in low-drag events is higher than for the high-drag events. This is the first evidence that the "low-drag" hibernating turbulent events exist significantly above the Reynolds numbers close to transition [24] and well into the regime where the flow is usually considered to be "fully-turbulent", i.e., *Re<sup>τ</sup>* ≥ 180 [26].

**Figure 5.** PDF of occurrence of low- and high-drag events as a function of Δ*t* <sup>+</sup> for *Re<sup>τ</sup>* = 180 where the threshold criteria for low- and high-drag events are *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* > 1.1, respectively. Here, x-axis (Δ*t* <sup>+</sup>) represents the lifetime or duration of a conditional event.

**Table 2.** Rate of decay (*λ*) of the PDF of occurrence of conditional events for 100 ≤ Δ*t* <sup>+</sup> <sup>≤</sup> 300 at various Reynolds numbers. Numbers in brackets correspond to the *R*<sup>2</sup> value. The threshold criteria for low- and high-drag events are *τ*/*τ<sup>w</sup>* < 0.9 and *τ*/*τ<sup>w</sup>* > 1.1, respectively.

