*3.4. Higher-Order Statistics*

The higher-order statistics of *Reτ*, *Reb* and *Ec f* are presented in this section. For any field *A* " *A*p*x*, *z*, *t*q, we compute the spatio-temporal average *m* " <sup>Ě</sup>*A*, the variance *<sup>σ</sup>*<sup>2</sup> " p*<sup>A</sup>* Ğ´ *m*q2 and the *kth* standardized higher-order moment - p*A* Ğ´ *m*q*<sup>k</sup>* {*σ<sup>k</sup>* (for *k* ě 3).

Their mean values of *Reb* and *Re<sup>τ</sup>* (Figure 10a) simply follow the trends described above for the most probable value of the distribution, connecting the turbulent and the laminar branch, when *ReG τ* decreases. Away from the turbulent and laminar branches *Re<sup>τ</sup>* is linearly related to *Reb*, in agreement with the observation of a constant *Cf* . The standard deviation *σ* (Figure 10b) for *Re<sup>τ</sup>* and *Reb* decrease together with *ReG <sup>τ</sup>*. This decreasing trend agrees well with the experimental wall shear stress data reported in Reference [29]. The standard deviation for *Ec f* is found to increase with decreasing *ReG τ*, matching the trend reported in Reference [34].

The variation of the 3rd and 4th moments *m*<sup>3</sup> and *m*4, i.e., the Skewness (*S*) and Kurtosis (*K*), versus *ReG <sup>τ</sup>* for the observable *Re<sup>τ</sup>* and *Ec f* is shown in Figure 10c. These moments exhibit a strongly increasing trend with reducing *ReG <sup>τ</sup>* for both quantities. This similarity in behavior leads to *K*9*S*<sup>2</sup> as shown in Figure 10e. This correlation between the third and fourth statistical moments was first noted in Reference [48] for the fluctuating velocity in turbulent boundary layers at high Reynolds number. In the transitional regime, the same relationship has been found to hold in the experiments of Agrawal et al. [29] from wall shear stress data. We therefore confirm this yet-to-be-understood extension of a high Reynolds number scaling down to the spatio-temporal intermittent regime. Furthermore, we observe that the same scaling also holds for the turbulent kinetic energy *Ec f* (Figure 10e). In contrast it does not apply to *Reb* (inset of Figure 10e). The reason is that while the Kurtosis follows the same trend as for the two other observables, (Figure 10d), the skewness shows a markedly different behavior: it is non-monotonous, changes sign twice and exhibit a maximum in the core of the spatio-temporal intermittent regime.

**Figure 10.** *Cont*.

(**e**)

**Figure 10.** (**a**) Mean values (*xm*) of *Reb* and *Reτ*. (**b**) Variation of the Standard deviation (*σ*) of *Re<sup>τ</sup>* (red), *Reb* (green), *Ec f* (blue) (indicated in the legend) vs. *ReG <sup>τ</sup>* . The *σ*p*Reb*q and *σ*p*Reτ*q are scaled as indicated in the legend in order make them comparable. (**c**) Variation of Skewness (*y*-axis on left, filled symbols) and kurtosis (right *y*-axis, open symbols) vs. *ReG <sup>τ</sup>* for the observables *Re<sup>τ</sup>* (red) and *Ec f* (blue) (**d**) Variation of Skewness (left *y*-axis on the left, filled symbol) and kurtosis (*y*-axis on right, open symbols) vs. *ReG <sup>τ</sup>* for the observable *Reb* (green). (**e**) Kurtosis vs. squared skewness for *Re<sup>τ</sup>* (red), *Reb* (green, inset), *Ec f* (blue).
