# Experimental Study on the Near-Red Flow Characteristics of Alluvial Channel with Seepage
## 2 Experimental Setup and Methodology
In the field, the effect of sediment size, \(d_{50}\), was not considered to be important to the turbulent flow characteristics because of the very large values of the ratio \(b/d_{50}\) (where \(b\) is the channel width) [26]. Figure 1: Schematic diagram of the tilting channel used in this study (not to scale). A four-beam down-looking Acoustic Doppler Velocimeter (ADV) probe, named Nortek(r)Vectrino, was used to measure the instantaneous water velocities at the centerline of the test section. Data were collected with a sampling frequency of 200 Hz. The ADV worked with an acoustic frequency of 10 MHz, having an adjustable cylindrical sampling volume of 6 mm diameter and 1-4 mm high. The sampling volume was 5 cm below the central transmitter; therefore, it was impossible to measure the velocity data at a distance less than 5 cm from the water surface. A time duration of 180 sec was set to collect the instantaneous velocity data. The uncertainty of the ADV data was calculated by collecting 12 samples of instantaneous velocities in the vicinity of the bed region. The uncertainty of the ADV data is given in Figure 2, where, \(u\), \(v\), and \(w\) are the mean velocities in the longitudinal, widthwise, and vertical directions, respectively; \(u^{\prime}\), \(v^{\prime}\), and \(w^{\prime}\) are the corresponding components of velocity fluctuations, and \(\left(\overline{u^{\prime}u^{\prime}}\right)^{0.5}\), \(\left(\overline{v^{\prime}v^{\prime}}\right)^{0.5}\), and \(\left(\overline{w^{\prime}w^{\prime}}\right)^{0.5}\) are the turbulence intensities. The time-averaged (mean) streamwise velocity (\(u\)) and vertical velocity (\(w\)) were computed as  
\[\left. \begin{array}{l}u=\frac{1}{n}\sum\limits_{i=1}^{n}U_{i}\\ w=\frac{1}{n}\sum\limits_{i=1}^{n}W_{i}\end{array} \right\} \tag{1}\]  
where \(U_{i}\), \(W_{i}\), and \(n\) represent the streamwise instantaneous velocity, the vertical instantaneous velocity, and the sample number, respectively. In particular, the Reynolds shear stress (RSS) specifies the exchange of mass and momentum from the flow to the sediment bed. The RSS is measured as  
\[\overline{u^{\prime}w^{\prime}}=\frac{1}{n}\sum\limits_{i=1}^{n}(U_{i}-u)(W_{ i}-w) \tag{2}\]  
One can also write  
\[\tau_{uw}=-\rho\overline{u^{\prime}w^{\prime}} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sand \(d_{50}\)** & **Sand \(\sigma_{g}\)** & **Flow Depth \(h\)** & **Discharge \(Q\)** & **Reynolds Number** & **Seepage Discharge** \\ \hline
[mm] & [-] & [m] & [m\({}^{3}\)/s] & [-] & [\%Q] \\  & & & & 0 \\
0.50 & 1.65 & 0.11 & 0.0242 & 19,836 & 10\% \\  & & & & 15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the sediment mixture and flow conditions for this study. All experiments were characterized by clear-water conditions. Figure 2: Grain distribution curve for the sand used in this study. where \(\tau_{uu}\) is the Reynolds shear stress. Moreover, the turbulence intensity is the root mean square (rms) of the velocity fluctuation. Therefore, the turbulence intensities in the streamwise direction, \(\sigma(u)\), and vertical direction, \(\sigma(w)\), are expressed by  
\[\sigma(u)=\sqrt{\frac{\sum\limits_{i=1}^{n}(U_{i}-u)}{n}} \tag{4}\]  
\[\sigma(w)=\sqrt{\frac{\sum\limits_{i=1}^{n}(W_{i}-w)}{n}} \tag{5}\]  
Finally, the third-order correlation (skewness) is directly linked to the flow coherent features, which would provide essential evidence on the time-based appearances of the velocity fluctuations. The skewness of velocity fluctuations is given by Equation (6):  
\[S(u,w)=\overline{\widehat{u}^{j-k}}\text{ where }j+k=3\] \[\left.\widehat{u}=\frac{u^{\prime}}{(u^{\prime}u^{\prime})^{0.5}},\widehat{w}=\frac{u^{\prime}}{(\widehat{w}^{\prime}u^{\prime})^{0.5}}\right\} \tag{6}\]  
where \(S(u)=\overline{\widehat{u}^{3}}\) signifies the flux of \(u^{\prime}u^{\prime}\) in the flow direction and \(S(w)=\overline{\widehat{w}^{3}}\) characterizes the flux of \(w^{\prime}w^{\prime}\) in the vertical direction.