*3.3. Skewness and Kurtosis*

Though *IP* and *Iu* reflect the mean levels of fluctuation amplitudes or strengths, they cannot describe the intermittency and asymmetry of the signals. In this subsection, the skewness *S*(*u*) = *u*3 / *u*2 3/2 is calculated based on the streamwise fluctuation velocity, *u*, measured at the midplane, representing the asymmetric distribution of the velocity. The kurtosis or flatness *F*(*u*) = *u*4 / *u*2 2 is computed as well, reflecting the intermittency and the deviation from the random distribution. At low Reynolds numbers, the laminar velocity signal mixed with the background white noise conforms to the normal distribution, and hence *S*(*u*) = 0 and *F*(*u*) = 3. When the localized turbulent spots or bands emerge intermittently in the flow, the velocity defects appear, leading to a negative skewness and a positive flatness, e.g., *Re* < 700 for Case\_1 shown in Figure 6, while the corresponding turbulence intensity (Figure 5) and the friction coefficient (Figure 3) remain nearly unchanged. Specially, it is shown in

Figure 6 that the skewness and the kurtosis reach a minimum and a maximum during the transition, respectively, and the corresponding underlying mechanisms are discussed in Section 3.5.

**Figure 6.** (**a**) Skewness and (**b**) kurtosis of the streamwise velocity measured at (*x*, *y*, *z*) = (780, 0, 0) for different disturbance cases and Reynolds numbers.

The transition process is triggered by the entrance disturbances, the abundant vortex structures shed from the beads placed at the inlet. It has been shown that, at *ReD* = 3700 (based on the free-stream velocity and the sphere diameter *D*), the turbulence intensity, *Iu*, along the wake centerline of a sphere quickly reduces to 0.05 at *x*/*D* = 12 [40]. Based on the centerline velocities measured for *Re* = 600~1200, the corresponding *ReD* for the present inlet beads can be estimated to be 720~1920. Considering that the working section is 500*D*~666*D* long, the strong turbulence intensity, *Iu*, around 0.1, as shown in Figure 5, should be caused by the localized turbulent patches triggered by the remnants of the bead wakes rather than the remnants themselves. According to Figure 6, the Reynolds number intervals where the skewness and the kurtosis deviate from the normal distribution are [660,960], [780,1000], and [910,1060] for Case\_1, Case\_2, and Case\_3, respectively. It is interesting to note that the upper limits of these *Re* intervals are close to the corresponding peak *Res* for *IP* and *Iu* shown in Figure 5. The lower limits indicate the onset of turbulence, and the minimum lower limit of tested cases is about 660, which is consistent with the threshold determined numerically for the oblique turbulent bands [24,25] and the value obtained by flow visualization [27]. In numerical simulations, the computation may last long enough, e.g., ~10<sup>4</sup> time units, to observe the transient growth and eventual decay of the patterns near the critical state, while, in experiments, the channel length is limited and the traveling turbulent patches may grow transiently but have no time to experience the final decay. This factor may cause a mild underestimate of the threshold value in experiments. It is shown in the insets of Figure 6 that, when *Re* > 1100 and *FT* is close to 1, the skewness and the kurtosis of streamwise velocity continue to evolve, deviating from 0 and 3 (the values for white Gaussian noise) and remain at about −0.5 and 3.5 after *Re* > 1750, respectively, the values for fully developed turbulence [41]. Consequently, the threshold for fully developed turbulence may be defined as *Re* ≈ 1750.
