## 2 Governing Equations
### Theoretical Equations
#### 2.2.3. Equations for the Gradually Varied Section  
The airflow analysis of the gradually varied section is shown in Figure 3.  
Figure 2: Airflow analysis of three specific nodes: (**a**) node I, (**b**) node II, and (**c**) node III.  
Considering the air entrapment on the free surface of water flow and the air escaping from the surface, the mass conservation equation in this gradually varied section can be rewritten as  
\[v_{t1}A_{a1}-q_{aw}+q_{wu}=v_{t2}A_{a2}, \tag{11}\]  
where \(q_{aw}\) is the air quantity entrapped in the water flow and \(q_{wu}\) is the air quantity escaping from the water flow.  
Taking into account the drag force imposed by the water flow, the friction resistance near the wall and the pressure between the inlet and outlet of the air intake well, the aerodynamic equilibrium equation can be expressed as  
\[p_{t1}+\frac{1}{2}\rho_{a}{v_{t1}}^{2}+\tau_{w}\frac{\overline{b}}{\overline{A} _{a}}\Delta L-\tau_{a}\frac{\overline{\chi}_{a}}{\overline{A}_{a}}\Delta L-( \zeta_{z}+\zeta_{k})\frac{1}{2}\rho_{a}{v_{t1}}^{2}=p_{t2}+\frac{1}{2}\rho_{a}{v _{t2}}^{2}, \tag{12}\]  
\[\tau_{w} =f_{w}\cdot\frac{1}{2}\rho_{a}\Big{|}\overline{v}_{w}-\overline{v}_{t} \Big{|}(\overline{v}_{w}-\overline{v}_{t}) \tag{13}\] \[=f_{w}\cdot\frac{1}{2}\rho_{a}\Big{|}\frac{v_{t1}+v_{w2}}{2}- \frac{v_{t1}+v_{t2}}{2}\Big{|}\Big{(}\frac{v_{t1}+v_{t2}}{2}-\frac{v_{t1}+v_{ t2}}{2}\Big{)}\,\ \text{and}\] \[\tau_{a}=f_{a}\cdot\frac{1}{2}\rho_{a}\overline{v}_{t}^{2}=f_{a} \cdot\frac{1}{2}\rho_{a}\Big{|}\frac{v_{t1}+v_{t2}}{2}\Big{|}\Big{(}\frac{v_{ t1}+v_{t2}}{2}\Big{)}, \tag{14}\]  
where \(\overline{b}\) and \(\overline{A}_{a}\) are the average width and the average area of the cross-sections upstream and downstream of the gradually varied section, respectively; \(\overline{\chi}_{a}\) is the average length of the boundary between the airflow and the wall of the cross-sections upstream and downstream of the gradually varied section, expressed as \(\overline{\chi}_{a}=h_{a1}+h_{a2}+(b_{1}+b_{2})/2\) (in which \(h_{a1}\) and \(h_{a2}\) are the cross-sectional heights of the water-free space upstream and downstream, respectively, and \(b_{1}\) and \(b_{2}\) are the cross-sectional widths of the water-free space upstream and downstream, respectively); \(\Delta L\) is the length of the gradually varied section along the flow direction; \(\tau_{a}\) and \(\tau_{w}\) are the friction resistance and the drag force, respectively; \(f_{a}\) is the wall friction coefficient, which can be expressed as the friction resistance coefficient, \(f_{a}=\lambda/4\); \(f_{w}\) is the drag coefficient of the water flow to the airflow; \(v_{w1}\) and \(v_{w2}\) are the water velocities in the gradually varied section upstream and downstream, respectively; \(\overline{v}_{w}\) is the average water velocity in the gradually varied section; and \(\overline{v}_{t}\) is the average air velocity in the gradually varied section.  
On the whole, the above three parts of the airflow equations constitute the governing equations describing the air motion inside the spillway tunnel with the multi-intake-well air supply system. The water depth and the water velocity can be obtained by theoretical calculations or physical model  
Figure 3: Airflow analysis of the gradually varied section.  
tests. It should be noted that parameters such as the local resistance coefficient and the drag coefficient have large influences on the theoretical analysis results.