## 3 Results
### Verification
#### 3.1.4 Improvement of the Drag Model  
As can be seen in Figure 6, the gate panel and its bottom boundary were able to strongly disturb the water surface. As the gate opening percentage became larger, this effect  
Figure 5: Water velocity and air volume fraction on the center line of typical sections (50% gate opening ratio) (**a**) downstream the gate, section marked 0 + 773.1 m; (**b**) downstream the aerator no. 2, section marked 1 + 141.90 m; (**c**) downstream the aerator no. 3, section marked 1 + 221.87 m; (**d**) downstream the aerator no. 4, section marked 1 + 360.52 m.  
waned. Obviously, when the gate opening is small, the drag effect of the free surface will be reduced because it is strongly disturbed and broken. In this case, the following features of airflow were poor, and thus the result of the symmetric model was more reasonable. On the contrary, when the gate was totally open, the drag effect of water on the air was stronger because the surface of the water was not disturbed and maintained better integrity. In this case, the following features of airflow were good, and therefore the result of the Schiller and Naumann model was more reasonable. Furthermore, although the disturbance effect of the gate may vanish far downstream from the gate, the air entrainment by forced-aeration or self-aeration in the tunnel will also destroy the continuity of water bodies, especially for the condition where gate opening percentage or water flow rate is small. Here, a hybrid drag model was proposed on the basis of a parameter related to the gate opening to integrate the Schiller and Naumann model and symmetric model by a blending function:  
\[K_{D,hyb}=\left(1-\varepsilon^{2}\right)K_{D,sym}+\varepsilon^{2}K_{D,s-n}, \tag{9}\]  
where \(K_{D,hyb}\) are the interphase exchange coefficients of hybrid drag model, \(K_{D,sym}\) are the coefficients calculated by the symmetric model, \(K_{D,s-n}\) are the coefficients calculated by the Schiller and Naumann model, and \(\varepsilon\) is the percentage of gate opening. In this formula, when the gate is totally open, the hybrid model degenerates into the Schiller and Naumann model, while the hybrid model is closer to the symmetric model for small gate opening conditions. In the commercial solver Fluent 15.0, the interphase exchange coefficients \(K_{D,hyb}\) in every cell can be calculated and applied to Equation (2) through the macro named DEFINE_EXCHANGE_PROPERTY. The air demands of different gate-opening ratios were calculated on the basis of the hybrid drag model. As can be seen in Figure 7, the relative error of air demand prediction from the hybrid drag model was less than 20% for all conditions. Similarly, several typical cross-sections were selected. The water velocity, air velocity, and air volume fraction \(\alpha_{a}\) on the center line of the sections were plotted in Figure 8. It is consistent with the previous analysis that the distribution of air velocity in the water body was eventually consistent with the profile of water velocity even for the hybrid drag model. Nevertheless, the air velocity profiles above the water of the hybrid model were between that of the Schiller and Naumann model and the symmetric model. It is suggested that the new model is more applicable for the prediction of air demand and air velocity profile under the complicated flow pattern in a high-head spillway tunnel. Therefore, the numerical simulation results based on the hybrid drag model are discussed further. Figure 6: The disturbance effect of gate. Figure 8: Water velocity and air volume fraction on the center line of typical sections (50% gate opening ratio) (**a**) downstream from the gate, section marked 0 + 773.1 m; (**b**) downstream from the aerator no. 2, section marked 1 + 141.90 m; (**c**) downstream from the aerator no. 3, section marked 1 + 221.87 m; (**d**) downstream from the aerator no.