*3.2. Global Variables: Moody Diagram*

The mean velocity profile - Ę*u*` *x* is defined as the average of *ux* over *x*,*z* and *t*, expressed in units of *uτ*. It is shown in Figure 7 as a function of *y*` " *yuτ*{*ν* and compared with the classical DNS data by Kim et al. [45] obtained at higher *Re<sup>G</sup> <sup>τ</sup>* " 180. The whole figure is similar to figures 3 and 10 in Reference [17,46], respectively. As expected for the present low values of *Re<sup>G</sup> <sup>τ</sup>* , the velocity field matches the linearized profile *u*` *<sup>x</sup>* " *y*` next to the wall but does not develop a logarithmic dependence with respect to *y*`.

**Figure 7.** Mean flow profile *u*` *<sup>x</sup>* p*y*`q for *Re<sup>G</sup> <sup>τ</sup>* from 100 down to 39. Blue: law of the wall *u*` *<sup>x</sup>* " *y*`, red: logarithmic law of the wall *u*` *<sup>x</sup>* " 2.5 logp*y*`q ` 5.5, black: DNS by Kim, Moin and Moser from Reference [45].

At a global level of description, the laminar and turbulent flow are traditionally represented in the classical Moody diagram in which the Fanning friction factor *Cf* defined as the ratio between the pressure drop along the channel length and the kinetic energy per unit volume based on the mean bulk velocity *Ub* " Ě*ub*,

$$C\_f = \frac{|\Delta p|}{1/2 \,\rho \, \text{l} \text{l}\_b^2} \frac{h}{\text{L}\_\text{x}} = \frac{\overline{\langle \pi \rangle}}{1/2 \,\rho \, \text{l} \text{l}\_b^2} = \frac{2 \, Re\_\text{r}^{c2}}{Re\_b^2},\tag{8}$$

is traditionally plotted versus *Reb* as shown with plain symbols in Figure 8. Another way to express *Cf* is to use inner units, in which case *Cf* " 2{p*u*` *<sup>b</sup>* <sup>q</sup>2, with *<sup>u</sup>*` *<sup>b</sup>* " *ub*{*uτ*. *Cf* is then linked only to the integral of the mean profile displayed in Figure 7.

For the laminar flow, the dependence of *Cf* vs. *Reb* is analytically given by *C flam* " 6{*Reb* (blue continuous line). In the featureless turbulent regime, it is known empirically as the Blasius' friction law scaling *Re* Ě*<sup>b</sup>* ´1{<sup>4</sup> (red continuous line). For intermediate values of *Reb*, *Cf* clearly deviates from the turbulent branch, and remains far from the laminar value [47]. Here we notice, in agreement with [30] and [34] that *Cf* « 0.01 remains essentially constant in this transitional regime. What is remarkable is that this regime of constant *Cf* coincides with the patterning regime observed for 50 ď *ReG <sup>τ</sup>* ď 90, corresponding to 690 ď *Reb* ď 1225, *as if the respective amount of turbulent and laminar domains was precisely ensuring Cf* " *cst*. As the pattern fractures, *Cf* increases and approaches the laminar curve. We note that the observation of this property requires large computational domains to be observed, which explains why it had not been noticed until recently, even in experiments.

**Figure 8.** Friction coefficient *Cf* vs. *Reb*, with horizontal and vertical error bars indicating the fluctuations these quantities would inherit from that of the field *ub* (see text for details).

Given the complex spatio-temporal dynamics in the transitional regime, the bulk velocity *ub* is expected to strongly fluctuate both in space and time. We also report in Figure 8, how these fluctuations would translate on *Reb* and *Cf* , if the latter were computed using the locally fluctuating field *ub* instead of its mean value *Ub*. These fluctuations are significant (up to 10–15%) and suggest to further explore them, which is the topic of the next section and the main focus of the present work.
