**4. Identifying Low- and High-Drag Events**

Figure 2a shows the PDF (probability density function) of wall shear stress fluctuations (*τ w*) obtained at *Re<sup>τ</sup>* = 180 using experiments. The PDF of wall shear stress has a longer positive tail which means that the PDF is positively skewed. This shows that some of the positive fluctuations have much larger magnitude than the negative fluctuations. In the present study, the wall shear stress is representative of the skin-friction drag. Previously, Gomit et al. [41] used the PDF of wall shear stress to divide low- and high-wall shear stress events in a turbulent boundary layer. They divided the PDF into four quartiles, where each quartile contains one-fourth of the realisations. In this study, to define the low- and high-drag "events", two significant parameters are considered: the magnitude of the wall shear stress fluctuations and the duration of time the fluctuations stay below or above the time-averaged value.

**Figure 2.** (**a**) PDF of wall shear stress fluctuations for *Re<sup>τ</sup>* = 180 obtained using experiments. Blue dashed lines represent the threshold criteria for the low- and high-drag events, i.e., *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* > 1.1, respectively. (**b**) Distribution of negative and positive wall shear stress fluctuations per hour for *Re<sup>τ</sup>* = 180 obtained using experiments. The black dotted lines cover the region of low-drag events based on the criteria: *τw*/*τ<sup>w</sup>* < 0.9 and Δ*t* + *cr* = 200 and the black dashed lines cover the region of high-drag events based on the criteria: *τw*/*τ<sup>w</sup>* > 1.1 and Δ*t* + *cr* = 200.

The PDF of wall shear stress fluctuations, as shown in Figure 2a, provides statistical information about the magnitude of the fluctuations but information regarding the time-duration of the fluctuations cannot be inferred. Therefore, it is necessary to find a way to visualise all the positive and negative fluctuations as a function of the magnitude and time-duration. This is carried out by calculating the distribution of all the fluctuations (*τ <sup>w</sup>*) about the time-averaged value (*τw*) with their corresponding time durations (Δ*t*). Figure 2b shows this distribution for *Re<sup>τ</sup>* = 180. Here, inner scaling (*u*<sup>2</sup> *<sup>τ</sup>*/*ν*) is used to scale the time-duration of the negative and positive wall shear stress fluctuations. The strength of the wall shear stress fluctuations is given by *τ <sup>w</sup>*/*τw*. The number of these fluctuations is higher for the lower strengths and lower time-durations.

In this study, to detect a low-drag or a high-drag event, a magnitude threshold criterion and a time duration criterion are employed on the wall shear stress signals. For the threshold criteria, values less than 0.9*τ<sup>w</sup>* for the low-drag events and greater than 1.1*τ<sup>w</sup>* for the high-drag events have been typically employed previously by Kushwaha et al. [22]. Whalley et al. [24] used the same threshold criteria for the low-drag events, but for the high-drag events they employed a less stringent criteria of greater than 1.05*τw*, in order to obtain more data points to carry out the statistical analysis. In the present study, the same values for the threshold criteria as used by Kushwaha et al. [22] are employed to detect the conditional events; however, the effect of varying the threshold criteria will also be discussed. For the time-duration criteria, Kushwaha et al. [22] and Whalley et al. [23,24] employed a mixed scaling (Δ*t* <sup>∗</sup> = Δ*tuτ*/*h*) to detect conditional events in channel flows. They typically used Δ*t* ∗ = 3 as the time-duration criterion while discussing the sensitivity of the value of the time-duration criterion on the conditional quantities. Unlike these previous studies, in the present investigation, an inner scaling is used for the time-duration criterion for the conditional events: Δ*t* <sup>+</sup> = 200 is used as the minimum time-duration to detect conditional events. The reasons for, and implications of, choosing this scaling will be discussed in detail in the next section. The effect of varying the length of the time-duration criterion on the conditional quantities will be discussed in Section 6. To further understand the definition of these conditional events, examples of instantaneous wall shear stress signals meeting the above-mentioned criteria for the low-drag and the high-drag events are shown in Figure 3. This figure shows the instantaneous normalised wall shear stress during the low-drag (Figure 3a) and the high-drag (Figure 3b) events. In Figure 3, the acquisition time of the wall shear stress is shifted such that *t* <sup>+</sup> = 0 indicates the beginning of a low- or a high-drag event. Each event is shown to act longer than the minimum time duration (for "low-drag" ~230 units and for "high-drag" ~320 units).

**Figure 3.** Time history of normalised wall shear stress at *Re<sup>τ</sup>* = 180 during (**a**) a low-drag and (**b**) a high-drag event obtained using experiments. Blue solid lines highlight the low-drag and the high-drag events in panels (**a**,**b**), respectively. Black dotted lines show mean value of normalised wall shear stress *τw*/*τ<sup>w</sup>* = 1. Black dashed lines show *τw*/*τ<sup>w</sup>* = 0.9 and *τw*/*τ<sup>w</sup>* = 1.1 in panels (**a**,**b**), respectively. Red dashed line indicates the time-duration criteria of Δ*t* + *cr* = 200. In panels (**a**,**b**), *t* <sup>+</sup> is shifted such that *t* <sup>+</sup> = 0 indicates the beginning of a conditional event.
