**Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Determination of the Distance from the Wall Measured in Wall Units in the Diffuser Flow for Smagorinsky Model**

Using a linear interpolation, let us assume the friction velocity *<sup>u</sup>τ*(*x*) in the diffuser is as follows:

$$u\_{\tau}(\mathbf{x}) = \begin{cases} u\_{\tau}0, & \mathbf{x} \le \mathbf{0} \\ u\_{\tau}0 + \mathbf{x}(u\_{\tau}\mathbf{e} - u\_{\tau}\mathbf{0})/H, & \mathbf{0} < \mathbf{x} \le 21H \\ u\_{\tau}e, & \mathbf{x} > 21H, \end{cases} \tag{A1}$$

where *<sup>u</sup>τ*0 denotes the inlet friction velocity, 407; and *uτe* denotes the friction velocity in down stream extension.

Reynolds number *Rem* based on the bulk velocity *Um* is constant in the diffuser because of the continuity:

$$\frac{\imath \jmath \jmath\_{m} H}{\nu} = \frac{\frac{\imath \jmath}{4} \not\jmath \bullet \jmath\_{m}}{\nu} . \tag{A2}$$

Thus, *Reτ* is also constant due to the Dean's suggested correction [35] *Reτ* = 0.191*Re*0.875 *m* , which was validated by Kim et al. [4] in fully developed channel flow. Then, *Reτ* in diffuser can be approximately expressed as:

$$\text{Re}\_{\text{\textquotedblleft}} 0 = \frac{\mu\_{\text{\textquotedblleft}} 0 \cdot H}{\nu} = \frac{\mu\_{\text{\textquotedblleft}} \cdot 4.7H}{\nu} = 407,\tag{A3}$$

thus we ge<sup>t</sup> *uτe* = *<sup>u</sup>τ*0/4.7. Using Equation (A1), *y*<sup>+</sup> can be approximately defined as:

$$y^{+}(x,y) = \begin{cases} \text{Re}\_{\tau} 0 \cdot \min(y, 1-y) / H, & x \le 0 \\\text{Re}\_{\tau} 0 \cdot (1 - 3.7 / 4.7 \cdot x / H) \cdot \min[y, \Upsilon(x) - y] / H, & 0 < x \le 21H \\\text{Re}\_{\tau} 0 / 4.7 \cdot \min(y, 4.7H - y) / H, & x > 21H. \end{cases} \tag{A4}$$
