*3.1. Friction Coe*ffi*cient*

The friction coefficient *Cf* = 8 *h*<sup>∗</sup> *dP*<sup>∗</sup> *dx*∗ / -9ρ∗ *U*∗<sup>2</sup> *b* is measured at different Reynolds numbers, with different entrance disturbances, where *dP*∗ /*dx*∗ is the mean pressure gradient calculated based on the pressure difference between *x* = 660 and 740, and the bulk velocity, *U*∗ *b* , is obtained from the mean velocity profile. *Cf* is calculated for every 10-s sample, and the averaged *Cf* for 20 samples (totally 104~105 time units at the transition stage) are shown in Figure 3, where the error bars represent the standard deviation. It is shown that when *Re* < 600 or there are no entrance artificial disturbances (Baseline), the present experimental data agree well with the laminar value *Cf* = 4/*Re*. The previous results [5,6,22,24] are shown as well for references. When *Re* is greater than 1750, *Cf* data for different entrance disturbance cases tend to agree with the "optimum log-law" labeled by the dashed line for developed turbulence, where *Re* = 2 *Cf* exp 0.41 <sup>8</sup> <sup>9</sup>*Cf* <sup>−</sup> 2.4 [22,37]. During 950 <sup>&</sup>lt; *Re* <sup>&</sup>lt; 1010, *Cf* in three disturbed cases increases abruptly, reflecting a strong development of turbulence. As shown in the inset of Figure 3b, such an abrupt increase of *Cf* occurs as well in the previous direct numerical simulations, where the turbulent band split occurs, i.e., parallel split to form a new band parallel to the original one and transverse split to sprout new branch (as shown by Figure 6 of Reference [24]). Recent systematical simulations [22] revealed that the transition from "one-sided" (all localized turbulent bands point to the same direction) to "two-sided" (the bands may grow in different directions) propagations takes place at *Re* ≈ 924. By simulations in tilted slender domains, a critical Reynolds number is defined as 950, where the statistically estimated mean lifetimes for band decay and splitting

coincide with each other [38]. All of these numerical results explain, to some degree, why *Cf* increases abruptly as *Re* > 950.

**Figure 3.** (**a**) The friction coefficient, *Cf*, as a function of *Re*. The previous experimental and numerical data are illustrated in (**b**) for references.
