*3.3. Joint Probability Distribution of Re<sup>τ</sup> and Reb*

Reynolds numbers such as *Re<sup>τ</sup>* and *Reb* are traditionally seen as global parameters characterizing the flow. They are defined based on velocity scales obtained from space-time average. It is straightforward to extend these definitions to the local fields *Reb*p*x*, *z*, *t*q " *ub*p*x*, *z*, *t*q*h*{*ν* and *Re<sup>τ</sup>* " *uτ*p*x*, *z*, *t*q*h*{*ν*, with *<sup>u</sup>τ*p*x*, *<sup>z</sup>*, *<sup>t</sup>*q"p*τ*p*x*, *<sup>z</sup>*, *<sup>t</sup>*q{*ρ*q1{2. Please note that with this definition, Ğ*Reτ* is not strictly equal to the imposed *ReG <sup>τ</sup>*, because of the nonlinear relation between *Re<sup>τ</sup>* and *τ*.

Investigation of the entire transitional regime is provided through a two-dimensional state portrait (*Reb* ´ *Reτ*) constructed from this local definition of the Reynolds number. The joint probability density distribution is constructed in this state space with the space-time data for different *ReG <sup>τ</sup>*. The state space for *ReG <sup>τ</sup>* " 100, 80, 60, 40 is shown in Figure 9. The continuous blue and red lines again correspond to the scalings known analytically for the laminar flow, and empirically for featureless turbulent flows for high enough Reynolds numbers. As expected the most probable values of *Reb* and *Reτ*, follow the same trend as their global counterpart: they match the continuous curve in the featureless turbulent regime, and progressively depart from it to move towards the laminar branch at the lowest *ReG <sup>τ</sup>* explored here. More interesting are the distributions. First, we observe that the relative fluctuations are significantly larger for *Re<sup>τ</sup>* than for *Reb*, the difference being larger for the larger *ReG <sup>τ</sup>*. Secondly the distributions are not simple Gaussians. Even in the featureless turbulent regime, the marginal distribution of *Re<sup>τ</sup>* is already relatively skewed (Figure 9(a3)).

**Figure 9.** p**a1**q p**b1**q p**c1**q p**d1**q Joint probability distribution of the quantities *Reb* and *Re<sup>τ</sup>* for *ReG <sup>τ</sup>* " 100, 80, 60, 40 together with their marginal distribution shown in lin-log scale for *Reb* in p**a2**q p**b2**q p**c2**q p**d2**q and for *Re<sup>τ</sup>* in p**a3**q p**b3**q p**c3**q p**d3**q with the mean value indicated by a vertical/horizontal black line.

As *ReG <sup>τ</sup>* is reduced, the overall width of the distribution decreases, but the shape of the marginal distributions of *Reτ* differs more and more from a Gaussian. More specifically, although the distribution remains unimodal, we note that the marginal distribution of *Reτ* is more and more skewed. We also note that the right wing of the distribution is not convex anymore. To further quantify these observations, a systematic analysis of the moments of this distribution is conducted in the next section.
