3.1.2. Skewness

Despite many studies on the hydrodynamics of the flows around bridge piers and downward seepage, there is still a lack of knowledge about higher order moments of velocity fluctuations, which contain information related to the flux of the Reynolds normal stress, retaining the sign details [32].

The third order correlation, *Mlm*, is expressed as [33] follows:

$$M\_{lm} = \overline{\hat{a}^l \hat{w}^m}$$

where, *u*ˆ = *u u*<sup>2</sup>0.5 , *w*ˆ = *w w*<sup>2</sup>0.5 and (*l* + *m*) = 3.

In particular, *M*30 (*u*ˆ3)and *M*03 (*w*<sup>ˆ</sup> 3) express the longitudinal and vertical flux of streamwise RNS (*uu*) and vertical RNS (*ww*), respectively, and *M*12 and *M*21 indicate the advection of (*ww*) and (*uu*), in *x*-direction and *z*-direction, respectively. They are shown in Figure 3, together with the vertical profiles of the skewness factors in the streamwise (*Su* = *<sup>u</sup>*<sup>3</sup>/*<sup>u</sup>*∗3) and vertical (*Sw* = *<sup>w</sup>*<sup>3</sup>/*<sup>u</sup>*∗3) direction, respectively.

Upstream of the pier (U), near the bed, positive *M30* and *M12* and negative *M03* and *M21* show the flux of streamwise RNS and the diffusion of vertical RNS in streamwise direction, and the flux of vertical RNS and the diffusion of streamwise RNS in a downward direction, respectively. By increasing *h*+, *M30* and *M12* become negative, and *M03* and *M21* become positive. The positivity of *M30* and *M12* decreases in the seepage runs, showing that the flux of the streamwise RNS and the di ffusion of the vertical RNS results in a lower mobility of the bed material.

Near the bed, the positive and negative values of the streamwise and vertical skewness factors confirm that the transport of the turbulent kinetic energy are in streamwise and downward directions, respectively.

In section D, near the bed, *M30* is negative, while *M03* is positive, suggesting that the streamwise and vertical RNS fluxes are in an opposite direction to the flow, and in an upward direction, respectively, showing ejections. By increasing *h*+, *M30* and *M03* attain an opposite sign, becoming positive and negative, respectively. Near the bed, *M12*, is positive, while *M21*, is negative, showing that the di ffusion of vertical and streamwise RNS is in the flow and in downward directions, respectively. By increasing *h*+, *M12* becomes negative, while *M21* becomes positive, showing the di ffusion of vertical and streamwise RNS in the opposite direction to the flow, and in the upward directions, respectively. In section D, near the bed, positive *Sw* and negative *Su* show that the transport of the turbulent kinetic energy is in the opposite direction to the flow, and in an upward direction, respectively, increasing the seepage percentage. The results confirm the dominance of the secondary currents downstream from the piers, because of the formation of wake vortices; however, downward seepage limits the strength of the wake vortices.

In the Section S1, near the bed, *M30* and *M12* increase with the seepage percentage, because of the increased particle mobility in the seepage runs. Similarly, the increased *M03* and *M21* in the seepage runs, correspond to the increasing vertical flux of RNS, and the advection of streamwise RNS in a downward direction, respectively. In the section of S1, in the seepage runs, the distributions of the skewness that factor their increase with *h*+, clearly show the increased movement of bed particles. These results show the prevalence of a high turbulent activity, resulting in enhanced sediment transport, laterally to the pier, in the case of seepage runs.

**Figure 3.** *Cont.*

92

**Figure 3.** Non-dimensional distribution of third order moment and skewness factors in streamwise and vertical direction for without seepage (NS), 10% seepage (S), and 15% S runs in the following sections: (**a**) upstream the pier (U); (**b**) downward the pier (D); (**c**) laterally to the pier (S1).

#### 3.1.3. Turbulent Kinetic Energy Flux (TKE-Flux)

TKE-flux significantly describes the bed material movement in a sandy channel. The streamwise and vertical TKE-flux are calculated as follows [34]:

$$\begin{cases} f\_{ku} = 0.75(\overline{\overline{u\nu\mu\nu}\nu} + \overline{\overline{u\nu\tau}\nu\overline{\tau}})\\ f\_{kw} = 0.75(\overline{\overline{u\nu\mu\tau}\nu} + \overline{\overline{u\tau}\nu\tau}\overline{\overline{\nu\tau}}) \end{cases} \tag{1}$$

and are scaled with the shear velocity *u*\*, thus obtaining (*Fku* = *fku*/*u*\* 3) and (*Fkw* = *fkw*/*u*\* 3), respectively. The vertical profiles of the TKE-flux around the pier, for all of the experimental conditions, are shown in Figure 4. In particular, upstream of the pier (U), near the bed (i.e., *h*+ < 0.2), *Fku* is positive and *Fkw* is negative, in agreemen<sup>t</sup> with the bed erosion as a result of the flow separation upstream of the pier. However, in the case of seepage runs, the absolute values of *Fku* and *Fkw* decrease, because of the lower erosive capacity of the reversal flow. Downward the pier (D), near the bed, *FKu* and *FKw* show slightly negative and positive values, respectively, with absolute values decreasing with the seepage percentage. This trend shows the hindered flow and the effects of low-speed incoming fluid particles. Moving towards the free surface, approximately at the edge of the scour hole (*h*+ ≈ 1), *FKu* and *FKw* become positive and negative, respectively.

**Figure 4.** Non-dimensional distributions of turbulent kinetic energy flux (TKE-flux) in the following sections: (**a**) U; (**b**) D; (**c**) S1.

Laterally to the pier (S1)1, near the bed, the increased positive and negative values of *Fku* and *Fkw* with a downward seepage show the enhanced erosion of the bed material in respect to the other sections.

## 3.1.4. Turbulence Production, Dissipation, and Diffusion

To deeply understand the trend of the velocity fluctuations, it is essential to consider the production, dissipation, and diffusion of the turbulent kinetic energy, expressed as follows:

$$\begin{array}{ll}\text{Turbulent production} & t\_p = -\overline{\mu\nu\nu}\frac{\partial f}{\partial z} \\ \text{Turbulent dissipation} & \varepsilon = \frac{158}{\overline{\mathcal{U}}} \left(\frac{\partial \nu}{\partial t}\right) \\ \text{Turbulent diffusion} & t\_D = \frac{\partial f\_{\text{fix}}}{\partial z} \end{array} \tag{2}$$

where *tp*, ε, and *tD* were scaled with *h*/*u\**2, and are denoted by *TP*, *ED*, and *TD*, respectively. In Figure 5, the vertical profiles of the production (*TP*), diffusion (*ED*), and dissipation (*TD*) of the turbulent kinetic energy around the pier, for all of the experimental conditions, are shown.

**Figure 5.** Non-dimensional distributions of turbulent production (*TP*), turbulent kinetic energy dissipation (*ED*), and diffusion (*TD*) for NS, 10% S, and 15% S in the following sections: (**a**) U; (**b**) S1; (**c**) D.

The turbulence production (TP) comes from the exchange of energy between the mean flow velocity and the velocity fluctuations. Therefore, the positive TP expresses the energy flowing from the mean flow velocity to the velocity fluctuations, while, on the contrary, the negative TP shows the

energy flowing from the velocity fluctuations to the mean flow velocity. In Figure 5, it is possible to observe that in the section of U, near the bed, in the seepage runs, the decreased TP, with respect to sections D and S1, is evidence of lower turbulence intensities and moment transfer, as the reversal flow is hindered by the seepage flow. On the contrary, in sections D and S1, when the downward seepage is applied, a higher TP shows the enhanced turbulence level in the sections, because of more energy being converted into the turbulent fluctuations. Consequently, in section U, near the bed, in the seepage runs, the ED is higher than in sections D and S1. Moreover, in sections D and S1, near the bed, in the seepage runs, lower TD values correspond to a gain in TP, and a consequent reduction in ED.
