*2.2. Experimental Methods and Validations*

Eighteen static pressure holes with 0.5 mm diameter are drilled on the lower wall along the line *z* = 0 with an interval of *l* = 200 mm, and the first hole is located at 300 mm from the entrance of the working section. Consequently, the pressure gradient along the streamwise direction can be monitored by using micro differential pressure transducers (Alpha M168, range: 0~25 Pa, accuracy: ±0.25% FS). A low-noise hot-wire anemometer (HWA, Dantec StreamLine Pro.) with 3 channels is used to measure the velocity with a relative error less than 1.5%. The stainless-steel probe stem is mounted on a two-dimensional traversing mechanism with a positioning resolution of 5 μm. In order to minimize the interference, the probes are inserted through the outlet of the working section.

We checked that, except the region very close to the entrance, the streamwise pressure gradients remained constant at low Reynolds numbers and agreed with the theoretical values for laminar PPF as reflected by the friction coefficients, which are discussed in Section 3.1. As shown in Figure 2a, the uniform distribution of, *U*∗ *<sup>c</sup>*, in the spanwise direction indicates that the velocity field in the central part of the cross-section is hardly affected by the sidewalls. When the flow is laminar at *Re* = 1096, it is shown in Figure 2b that the velocity profiles at five different spanwise positions agree well with the theoretical parabolic distribution. When *Re* is increased to 7543, the time averaged velocity profiles are all close to the 1/8 power law curve, confirming that the sidewall effect is still negligible in the central region. Without the entrance artificial disturbances, it is checked that the flow can remain laminar for *Re* up to 3500, and hence the present setup is appropriate to study the subcritical transition of PPF.

**Figure 2.** Streamwise velocities measured at *x* = 780. (**a**) Spanwise distributions of the time averaged velocity in the midplane *U*∗ *<sup>c</sup>*, and (**b**) the time averaged velocity profiles at different spanwise positions. The measurements of Reference [33] are added in (**b**) as references.

Nine plastic beads evenly spaced with an interval of 100 mm along a thin iron wire are placed at the centerline of channel inlet to introduce entrance disturbances. Different bead diameters, *D\**, and wire diameters, *d\**, are used in four cases and are listed in Table 1.


**Table 1.** Dimensions of the entrance disturbances.

#### **3. Results and Discussions**
