*3.2. Turbulence Intensity and Pressure Turbulence Intensity*

The time series of the streamwise velocity, *U*, obtained at the midplane by HWA are just straight lines superimposed by background noise at low Reynolds numbers, e.g., *Re* = 652 in Figure 4a. When a turbulent band or spot passes through the measuring point, the time series show a velocity defect, i.e., the midplane streamwise velocity decreases first along with the time, then oscillates strongly with high frequencies before increasing abruptly to recover its laminar level. The velocity fields of the spots and turbulent bands are measured by PIV, and their consistencies with the direct numerical simulations are confirmed and shown in [39]. The present study mainly focuses on the statistical kinematic and dynamic properties of the transitional flow. It is shown in Figure 4d that the widths and amplitudes of the velocity defects are comparable for different entrance disturbances and different Reynolds numbers, indicating that the statistical properties of localized structures are weak functions of *Re* and external disturbances during the transition. Such a streamwise velocity defect appears more and more frequently with the increase of *Re*, as shown in Figure 4.

**Figure 4.** *Cont*.

**Figure 4.** The time series of velocity, *U*, measured at (*x*, *z*) = (780, 0) for (**a**) Case\_1, (**b**) Case\_2, and (**c**) Case\_3. Typical signals of localized turbulent structures for different cases at different *Re* and turbulence fraction, *FT*, are shown in (**d**).

The development of turbulence may be described by the turbulence intensity of streamwise velocity *Iu* = *u*2 1/2 <sup>=</sup> (*U* − *Uc*) 2 1/2 at the midplane (*<sup>y</sup>* <sup>=</sup> 0) and the pressure turbulence intensity *IP* = *Prms*/(*dP*/*dx*) − [*Prms*/(*dP*/*dx*)]*r*, where means the time averaged quantity, and the subscripts *r* and *rms* represent a reference value and the root mean square. In this paper, [*Prms*/(*dP*/*dx*)]*<sup>r</sup>* is the value at *Re* = 600, corresponding to a laminar flow with background noise. When *Re* is smaller than 850, *IP* remains a small value and is almost independent of the entrance disturbances, the downstream position, and the Reynolds number as shown in Figure 5a. When *Re* is larger than 850, *IP* of Case\_1 increases obviously and reaches a peak at about *Re* = 950 before decreasing. The corresponding *Re* of *IP* peaks for Case\_2 and Case\_3 is around 980 and 1020, respectively. In the right column of Figure 5, it is shown that the turbulence intensity, *Iu*, has peak values at the same *Re* as *IP* for all three cases. The existence of these peaks is explained in Section 3.5, with an intermittent structure model.

**Figure 5.** Pressure turbulence intensity, *IP* (left column), and turbulence intensity, *Iu* (right column) measured at different locations. (**a**,**b**), (**c**,**d**), and (**e**,**f**) are for Case\_1, Case\_2, and Case\_3, respectively.
