*3.5. Intermittent Structure Model*

In order to understand the peaks and valleys of turbulence intensity and high-order moments during the transition, an intermittent structure model is constructed as follows. For convenience, the characteristic velocity is chosen as 1.5*U*∗ *<sup>b</sup>* instead of *U*<sup>∗</sup> *<sup>c</sup>* in this subsection. The velocity during the turbulent period is decomposed into two parts: the turbulent mean velocity, *UT*, representing the behavior of low-frequency and large-scale structures, and the turbulent perturbation velocity, *uT* (relative to *UT*), denoting the high-frequency and small-scale components. *U* = *UT* + *uT*, and it is assumed that *uT* satisfies Gaussian distribution, i.e., the time averaged values *uT* <sup>=</sup> 0, *u*3 *T* = 0, and *u*4 *T* = 3 *u*2 *T* 2 , but its temporal and spatial distribution is strongly asymmetric and aperiodic just like the measured velocity (gray curve) shown in Figure 9a. Assuming that *UT* and *u*2 *T* are the same for all localized turbulent patches in a given case and *FT* is known, it can be derived that the mean velocity *Uc* = *U*<sup>0</sup> − *FT*(*U*<sup>0</sup> − *UT*) and the fluctuation velocity relative to *Uc* is as follows:

$$
\mu = \mathcal{U} - \mathcal{U}\_c = \begin{cases}
\, \, \, \_T (\mathcal{U}\_0 - \mathcal{U}\_T)\_\prime & \text{laminar periods}, \\
\, \, (\mathcal{U}\_0 - \mathcal{U}\_T) (\mathcal{F}\_T - 1) + \mu\_T & \text{turbulent periods}.
\end{cases} \tag{1}
$$

**Figure 9.** (**a**) The simplified velocity signal (thick solid line) of the intermittent structure model at midplane, and the time averaged (**b**) *U*<sup>0</sup> − *UT* and (**c**) *u*2 *T* sampled at the midplane during the turbulent periods. A measured midplane velocity signal is shown in (**a**) by the gray curve for a reference.

Consequently, the turbulence intensity and the high-order moments can be derived as follows:

$$\begin{cases} I\_{\rm u} = \frac{\sqrt{\{\mu^2\}}}{\mathcal{U}\_{\rm c}} = \frac{\sqrt{F\_T(1-F\_T)(\mathcal{U}\_0 - \mathcal{U}\_T)^2 + \{\mu\_T^2\}F\_T}}{\mathcal{U}\_0 - F\_T(\mathcal{U}\_0 - \mathcal{U}\_T)}\\ \{\mu^3\} = 3F\_T(\mathcal{U}\_0 - \mathcal{U}\_T) \{\mu\_T^2\}(F\_T - 1) - F\_T(\mathcal{U}\_0 - \mathcal{U}\_T)^3 \{2F\_T^2 - 3F\_T + 1\} \\ \{\mu^4\} = F\_T(1-F\_T) \left(1 - 3F\_T + 3F\_T^2\right) \left(\mathcal{U}\_0 - \mathcal{U}\_T\right)^4 + 3F\_T \{\mu\_T^2\}^2 + 6F\_T \{\mu\_T^2\}(F\_T - 1)^2 \left(\mathcal{U}\_0 - \mathcal{U}\_T\right)^2 \end{cases} \tag{2}$$

*UT* is estimated by the mean value of low-pass filtered midplane velocity during the turbulent periods at each *Re*, and the cutoff frequency, *fc*, used for the filtering is the same as those used for calculating *FT*. It is shown in Figure <sup>9</sup> that *<sup>U</sup>*<sup>0</sup> <sup>−</sup> *UT* increases with *FT*, while the variance *u*2 *T* increases first then decreases with the growth of *FT*, reflecting the fact that the localized turbulent structures are influenced to some degree by the entrance disturbances, *FT*, and then *Re*. *<sup>U</sup>*<sup>0</sup> <sup>−</sup> *UT* and *u*2 *T* may be fitted as follows:

$$
\Delta l\_0 - \mathcal{U}\_T = 0.06 \left( 1 + F\_T^4 \right)\_\prime \quad \left\langle u\_T^2 \right\rangle = 0.0026 + 0.01 \left( F\_T - 0.64 F\_T^7 \right)\_\prime \tag{3}
$$

which are shown in Figure 9b,c as solid curves.

According to the previous studies [42], the characteristics of localized turbulent bands, e.g., the band's tilt angle, width, and convection velocity, do not change much during the transition. Similar properties are shown in Figure 4d, as well: The midplane velocity defects of localized turbulent structures are similar and not very sensitive to the Reynolds number, the entrance disturbances, and the turbulence fractions. Therefore, these localized turbulent structures may be simplified to a unified structure, whose statistical dimensionless properties are independent of time, *FT*, and the initial or upstream disturbances. This unified structure is referred as turbulence unit hereafter. Consequently, *<sup>U</sup>*<sup>0</sup> <sup>−</sup> *UT* and *<sup>u</sup>*<sup>2</sup> *<sup>T</sup>* are chosen for mature structures and are set as the values when *FT* reaches 1, and then Equation (3) is simplified as follows:

$$\begin{aligned} lL0 - lL\tau &= 0.12, \quad \left\langle u\_T^2 \right\rangle = 0.006. \end{aligned} \tag{4}$$

For all three test cases, it is shown in Figure 10a–i by the solid lines that the main features of the second-, third-, and forth-order moments predicted by the model are consistent acceptably with the experimental results when the relations between *FT* and *Re* shown in Figure 8b are applied. The variance of the midplane streamwise velocity *u*2 is *FT*(1 − *FT*)(*U*<sup>0</sup> − *UT*) <sup>2</sup> + *uT* 2 *FT*, where the contribution of fluctuations (the second term) increases with *FT*, while the first term increases first and then decreases with *FT* due to the fact that the mean velocity, *Uc*, leaves *U*<sup>0</sup> for *UT*, leading to a peak value of *u*2 . Consequently, there exist peak values of *Iu* and *u*4 during the transition. Furthermore, when *FT* is close to 1 and the flow field is nearly fully occupied by the localized turbulent structures, *Uc* is almost as low as *UT*, and *u*2 and *u*3 are close to *uT* 2 and *uT* 3 , respectively. Therefore, at the

late transition stage, *u*3 should be close to zero again, and then there must exist a minimum *u*3 during the transition. Similarly, the asymptotic values for *Iu* and *u*4 should be finite ( *uT*<sup>2</sup> /*UT* and 3 *uT* 2 2 in the model), just as shown by the experimental data in Figure 10. The consistencies of the model curves with the experimental data indicate that, not only the turbulence fraction, but also the characteristics of localized structures is required in order to describe properly the statistical properties of transitional flows.

**Figure 10.** Turbulence intensity (**a**–**c**), the third (**d**–**f**) and the fourth (**g**–**i**) order moments of the midplane velocity, and the friction coefficient (**j**–**l**) for different disturbance cases. The symbols of different cases shown in (**a**–**i**) are experimental data measured at (*x*, *y*, *z*) = (780, 0, 0), and *Cf* symbols shown in (**j**–**l**) are the same as those shown in Figure 3a. The solid curves are the results of the intermittent structure model.

Recently, it is found that, for a channel flow with constant pressure gradient, the kurtosis of the bulk velocity, which fluctuates during the transition and is represented by *Re*<sup>b</sup> in the simulations [34], increases abruptly as the Reynolds number decreases to the threshold value. However, the kurtosis obtained in experiments is close to zero near the onset of turbulence, as shown in

Figure 6. This discrepancy may be explained to some degree with the present model. Considering that, in simulations, the velocities in the laminar periods are as clean as the present model and have no background random noise, an inevitable factor in experiments, then when *FT* is close to 0, *u*4 ∼ *FT* while *u*2 2 ∼ *FT* <sup>2</sup> according to Equation (2), and hence the kurtosis will increase sharply.

Next, we use this model to study the dynamic property. Considering a turbulence unit with volume, *V*, mean velocity, *UT*(*y*), and mean pressure, *PT*, the perturbation velocities are *uT*, *vT*, and *wT*, and then the volume averaged friction coefficient is obtained from the mean x-momentum equation:

$$\mathcal{L}\_{fT} = -\frac{2}{V} \int \frac{\partial \mathcal{P}\_T}{\partial \mathbf{x}} \mathrm{d}\mathbf{V} = -\frac{2}{\mathrm{Re}V} \int \frac{d^2 \mathcal{U}\_T}{dy^2} \mathrm{d}\mathbf{V} + \frac{2}{V} \int \left[ \frac{\partial \left< u\_T^2 \right>}{\partial \mathbf{x}} + \frac{\partial \left< \mu\_T \mathbf{u} \mathbf{r} \right>}{\partial \mathbf{z}} \right] \mathrm{d}\mathbf{V}.\tag{5}$$

Note that <sup>1</sup> −1 ∂*uTvT* <sup>∂</sup>*<sup>y</sup> dy* = 0. Since the velocity fluctuations are strongly asymmetric and there is nearly a velocity discontinuity at the later edge of time series (upstream edge) of the structure and the present model (Figure 9a), the Reynolds stresses, e.g., *u*2 *T* , are different at the upstream and the downstream edges of the turbulence unit. In fact, the Reynolds stresses of a localized turbulent band are aperiodic in both the streamwise and the spanwise directions, as shown by the disturbance velocity structures in Figure 2b of Reference [23], due to its oblique manner. Since the transition occurs at relatively high Reynolds numbers and the properties of turbulence unit are assumed to be weak functions of *Re*, <sup>−</sup> <sup>2</sup> *ReV <sup>d</sup>*<sup>2</sup>*UT dy*<sup>2</sup> dV may be expanded with 1/*Re* as <sup>4</sup> *Re* <sup>−</sup> <sup>2</sup> *Re*- *A*<sup>0</sup> + *A*<sup>1</sup> <sup>1</sup> *Re* <sup>+</sup> *<sup>A</sup>*<sup>2</sup> <sup>1</sup> *Re*<sup>2</sup> <sup>+</sup> ... , where <sup>4</sup> *Re* corresponds to the laminar state, and the constants *Ai* represent the contribution of mean flow modification. Similarly, the Reynolds stress term (the second term on the right hand side of Equation (5)) is expanded as *B*<sup>0</sup> + *B*<sup>1</sup> <sup>1</sup> *Re* <sup>+</sup> *<sup>B</sup>*<sup>2</sup> <sup>1</sup> *Re*<sup>2</sup> + ..., where the constants *Bi* reflect the aperiodicity of the Reynolds stress. Consequently, Equation (5) can be expressed as follows:

$$C\_{fT} = B\_0 + \frac{1}{Re}(4 - 2A\_0 + B\_1) + \frac{1}{Re^2}(B\_2 - 2A\_1) + \dots = B + \frac{A}{Re} + O\left(\frac{1}{Re^2}\right) \tag{6}$$

where *A* and *B* are constants for the turbulence unit. For a transitional flow with a turbulence fraction, *FT*, the total friction coefficient can be obtained as follows, after ignoring the higher orders terms in Equation (6):

$$C\_f = (1 - F\_T)\frac{4}{Re} + C\_{fT}F\_T = \left(1 - F\_T + \frac{A}{4}F\_T\right)\frac{4}{Re} + F\_TB. \tag{7}$$

It is shown in Figure 10j–l and that Equation (7) describes well the variations of *Cf* data for different entrance disturbance cases when the measured relation between *FT* and *Re* are applied. *A* and *B* are determined by fitting the data between *Re* = 1300 and 2000 as 0.78 and 0.00426, respectively.

At the initial and middle stages of transition, *Cf* may have different variation scenarios. If the external disturbances are not effective to trigger the turbulent patches and the transition starts at high Reynolds numbers, - <sup>1</sup> <sup>−</sup> *FT* + *<sup>A</sup>* <sup>4</sup> *FT* 4 *Re* may become smaller than *FTB* after a short *Re* range, and then there will be a stage where *Cf* increases with *FT* and *Re*, as shown in Figure 10. Note that *A* < 4 and - <sup>1</sup> <sup>−</sup> *FT* + *<sup>A</sup>* <sup>4</sup> *FT* 4 *Re* decreases with the increase of *FT* and *Re*. Consequently, there will be a maximum of *Cf* during the transition as illustrated by the present data shown in Figure 10l and the data of Patel and Head [6] shown in Figure 10k. If the transition begins at low Reynolds numbers, the variation of - <sup>1</sup> <sup>−</sup> *FT* + *<sup>A</sup>* <sup>4</sup> *FT* 4 *Re* may be comparable with that of *FTB*. Depending on the variation feature of *FT*, the stage of *Cf* growth may be short or even disappear, and a *Cf* plateau may appear, where *Cf* remains nearly constant in a finite range of *Re*. The *Cf* plateaus were observed in the previous numerical simulations [22,24,34] and are shown in Figure 10k for references. According to Equation (7), provided that the decrease of - <sup>1</sup> <sup>−</sup> *FT* + *<sup>A</sup>* <sup>4</sup> *FT* 4 *Re* is balanced by the rise of *FTB*, *Cf* will keep constant, though this constant value may be different for different entrance or initial disturbances, domain sizes, and computational periods. At the late stage of transition, *FT* tends to 1, and *Cf* is close to *A*/*Re* + *B* according to Equation (7) and then decreases with *Re*. The dashed lines in Figure 10j–l, *Re* = 2 *Cf* exp 0.41 <sup>8</sup> <sup>9</sup>*Cf* <sup>−</sup> 2.4, represent the fully developed turbulence [22,37], where the Reynolds stresses are assumed to be uniform in the streamwise direction. According to the experiments, *FT* is close to 1 as *Re* > 1100, but *Cf* still deviates from the dashed line as *Re* < 1750, indicating a moderately developed turbulent state. By extrapolating *A*/*Re* + *B* to the laminar value 4/*Re*, as shown by the dot-dash line in Figure 10l, we get *Re* = 756, corresponding to an asymptotic threshold for the moderately developed turbulence.
