*3.4. Turbulence Fraction*

An important parameter to describe the pattern evolution and intermittency during the subcritical transition is the turbulence fraction, *FT*, whose determination relies on the identification of the boundaries between the laminar and the turbulent regions. Different from the previous experiments, where *FT* was mostly calculated based on flow visualization images, in this paper, the time series of velocity are used to define *FT* as *FT* = *tT*/*tTotal*, where *tT* and *tTotal* are the turbulent period and the total sampling time, respectively. As shown in Figure 7a, the time series of the midplane streamwise velocity includes many velocity defects, which correspond to the traveling localized turbulent patches

and include high-frequency components, as illustrated by the wavelet power spectrum shown in Figure 7b. Consequently, high-pass filtering is used to extract these components, as shown in Figure 7c, whose time intervals are defined as the turbulent period, *tT*. Different cutoff frequencies, *fc*, are tested, and the corresponding *FT* values vary in the same trend, as shown in Figure 8a, though a higher *fc* leads to a lower *FT*. By comparing Figure 7a,c, the cutoff frequency of 45 Hz is found to capture the turbulent periods reasonably well, and hence is used in the following analyses.

**Figure 7.** (**a**) The time series of streamwise velocity *U* measured at (*Re*, *x*, *y*, *z*) = (935, 780, 0, 0) for Case\_1 and (**b**) its wavelet power spectrum. (**c**) The high-frequency component, *u* , after high-pass filtering of the signal shown in (**a**). Localized turbulent patches are marked with shadowed areas in (**a**,**c**).

**Figure 8.** (**a**) *FT* calculated with different cutoff frequencies, *f*<sup>∗</sup> *<sup>c</sup>* , for Case\_1, and (**b**) data calculated with *f*∗ *<sup>c</sup>* = 45 Hz for different entrance disturbances. Inset of (**b**): the growth steepness σ versus *Re*.

*FT* shown in Figure 8 is computed from the midplane streamwise velocity signals sampled at six locations, i.e., (*x*, *z*) = (700, −40), (700, −20), (700, 0), (780, −40), (780, −20), and (780, 0). Each time series lasts 2000 s (105~106 time units at the transition stage), and the error bar represents the standard deviation. As *Re* < 850, the localized patches are far from each other, as shown in Figure 4, and *FT* increases slowly with *Re* and is less than 0.1 for all three cases. When *Re* is larger than 1050, the localized turbulent structures almost occupy the whole flow field and are arranged nearly side by side, as shown by the case of *Re* = 1155 in Figure 4b, and hence *FT* is close to 1, as shown in Figure 8. The growth steepness σ = *dFT*/*dRe* is calculated and is found to reach its maxima (as shown in the inset of Figure 8b) at *Re* = 950, 975, and 1005 for Case\_1, Case\_2, and Case\_3, respectively, where *FT* is around 0.6. It is interesting to note that the Reynolds numbers of the σ peaks are almost the same as those of the *IP* and *Iu* peaks shown in Figure 5, confirming the intrinsic relation between the turbulence intensity and the growth steepness of the turbulence fraction.

According to Table 1, the beads' diameters are different for Case\_1 and Case\_2, representing different localized disturbance intensities, and the wire diameter of Case\_3 is about one order larger than that of Case 1, denoting different entrance disturbance forms, i.e., the entrance disturbances of Case\_3 are more uniform in the spanwise direction due to the vortex shedding of the thicker wire. As shown in Figure 8b, *FT* data for different entrance disturbances vary in the same manner but do not collapse with each other as 850 < *Re* < 1050, reflecting the sensitivity of transition to the external forcing, and the reason lies in several aspects. Firstly, *FT* data collapse will occur when *FT* is a single valued function of *Re*, e.g., at laminar state or the equilibrium state, which is found to be retrieved only as *Re* > 924 in long-term simulations [22]. In other words, when the upstream or initial disturbances are different, *FT* may be different from case to case as *Re* < 924 even for simulations with the same computational configurations, e.g., domain size and mode numbers. Secondly, in reality, the lengths of experimental channels are finite, and at moderate Reynolds numbers, the turbulent structures may have no enough time to spread completely before leaving the outlet. Consequently, *FT* will depend on the entrance disturbances. Thirdly, the effectiveness to trigger the transition are different for different types of perturbations. The turbulence fractions obtained based on flow visualization by Sano and Tamai [21] are shown in Figure 8b, as well, and are different from the present data: *FT* does not increase with *Re* as *Re* > 1000 but maintain at about 0.7. In Sano and Tamai's experiments, turbulent flow was excited in a buffer box by a grid and injected from the inlet, and hence the entrance perturbations occupied the span of the channel and are different from the localized disturbances used in this paper. In addition, different approaches applied to identify the laminar–turbulent boundaries and different data (e.g., the two-dimensional images of flow visualization and the one-dimensional velocity series measured by HWA) may lead to different *FT* values, as well.
