## 4 Hybrid Modeling
### Physical Model Tests  
The fixed-bed model is based upon the Froude law of gravity similitude. The reservoir bathymetry is acquired via echo-sounding from a remotely controlled vehicle. The reservoir length is 820 m. The shuice outflow being affected by the tailwater, a 380 m river reach downstream is also included. The model scale is \(\lambda=L_{\text{pt}}/L_{\text{m}}=40\), where \(L=\) length and subscripts pt and m refer to prototype and model. Figure 7 shows a photograph over the model. Measured along the river centerline, the model is \(\sim\)30 m long and \(\sim\)4 m wide in cross-section. At the upstream inlet, a honeycomb is installed to calm the inflow. At the outlet, a flip gate regulates tailwater levels in the light of the discharge-water stage relationship at the location. The shuice structure is manufactured with accuracy. Each gate opening is 0.2 m wide. Including all the piers, the shuice structure is 1.5 m long in the \(Y\) direction. Figure 6: Schematic layout of guide vanes, placed in an array downstream of the bend. (a) Plan view; (b) cross-sectional view, looking downstream. From prototype to model, the conversion scales for [PERSON]'s roughness coefficient (\(n\)), flow pressure (\(p\)), time (\(t\)), flow velocity (\(u\)), and flow rate (\(Q\)) are \(\lambda_{\text{n}}=\lambda^{1/6}\), \(\lambda_{\text{p}}=\lambda\), \(\lambda_{\text{t}}=\lambda^{0.5}\), \(\lambda_{\text{u}}=\lambda^{0.5}\), and \(\lambda_{\text{Q}}=\lambda^{2.5}\), respectively. The errors in terrain production are, on average, below \(\pm 4\) mm vertically and \(\pm 7\) mm horizontally. Efforts are made to achieve corresponding surface roughness so that [PERSON]'s criterion is approximately followed. Figure 8 shows the experimental setup. The water is supplied from a circulation system. A calibrated 90\({}^{\circ}\) sharp-crested weir measures flow rates, with a relative error below \(\pm\)(2-3)%. From up- to downstream, six cross-sections, denoted as CS1-CS6, are defined to facilitate the documentation. Measured along the river centerline, Table 2 lists their distance (denoted as \(L_{0}\)) to the \(Y\) axis. All except CS1 are parallel to \(Y\). Near the left bank at each cross-section, a point gauge monitors the water levels (denoted as \(Z\)). The measurement inaccuracy is in the order of \(\pm 0.2\) mm. The water levels at CS4 and CS5 correspond to \(Z_{\text{u}}\) and \(Z_{\text{d}}\). Under the roof of the laboratory building, a particle image velocimetry (PIV) system, comprising nine industrial digital cameras at \(\sim\)10 m height above the floor, is set up to capture surface flow velocities. The measurement error is 0.10 cm/s. The tracers are customized plastic particles 2.5 cm in diameter. Instantaneous flow images are captured at 10 Hz frequency, and the duration for data collection is 10 s. The time-averaged results are used when the flow fields are compared. Figure 7: Laboratory model of the shuice with six bulkhead gates constructed in scale \(\lambda=\) 1:40, with a 30 m length measured along the river centerline and a 4 m width in cross-section. (**a**) View from the right bank; (**b**) shuice structure, looking downstream. Scale effects of viscous and surface-tension forces are always an issue of discussion in physical modeling of free-surface flows [35; 36], defined by Reynolds number \(\text{R}=(UH)/\
u\) and Weber number \(\text{W}=U/(\sigma/\rho H)^{0.5}\), where \(U\) (m/s) = depth-averaged velocity, \(H\) (m) = flow depth, \(\rho\) (kg/m\({}^{3}\)) = water density, \(\sigma\) (N/m) = surface-tension coefficient, and \(v\) (m\({}^{2}\)/s) = water kinematic viscosity. Along the main stream in the model, \(\text{R}=(0.5\)-\(2.0)\times 10^{4}\) and \(\text{W}=3.5\)-\(5.0\) in case 1, and \(\text{R}=(2.0\)-\(9.0)\times 10^{4}\) and \(\text{W}=4.0\)-\(5.5\) in case 2. In the flow circulation zones, they are lower. As indicated by [PERSON] [35; 37; 38] and [PERSON] and [PERSON] [35; 37; 38], if \(\text{R}>(0.4\)-\(4.0)\times 10^{4}\) and \(\text{W}>3.3\)-\(10\), their effects become insignificant.