# Experimental Study of the Air Demand of a Spillway Tunnel with Multiple Air Vents
## 1 Introduction
\begin{table}
\begin{tabular}{l l l} \hline
**Reference** & **Equation** & **Flow Pattern** \\ \[PERSON] [8] & \(Q_{a}=0.0066(\text{Fr}_{o}-1)^{1.4}Q_{w}\) & Hydraulic jump \\ \[PERSON] [9] & \(Q_{a}=0.04(\text{Fr}_{c}-1)^{0.85}Q_{w}\) & Free-surface flow \\ \[PERSON] [10] & \(Q_{a}=0.040(\text{Fr}_{c}-1)^{0.85}Q_{w}\) & Hydraulic jump \\  & \(Q_{a}=0.014(\text{Fr}_{c}-1)^{1.4}Q_{w}\) & Hydraulic jump \\  & \(Q_{a}=0.024(\text{Fr}_{c}-1)^{1.4}Q_{w}\) & Free-surface flow \\ \[PERSON] [11] & \(Q_{a}=0.0066(\text{Fr}_{c}-1)^{0.85}Q_{w}\) & Free-surface flow \\  & \(Q_{a}=0.0066(\text{Fr}_{c}-1)^{0.85}Q_{w}\) & Hydraulic jump \\ \[PERSON] [2] & \(Q_{a}=0.022\frac{H_{\text{F}}}{g\hbar^{2}}\left(\frac{h}{h_{c}}\right)^{0.167} \left(\frac{h_{o}}{h_{c}}\right)^{0.5}(gB^{3})^{0.5}\) & Free-surface flow \\ \[PERSON] [12] & \(Q_{a}=0.037\text{Fr}_{c}^{1.3}\frac{a}{(A+(B+1)^{0.8})}^{0.8}(\frac{L}{h})^{0. 25}(1+S)^{-1.5}Q_{w}\) & Free-surface flow \\ \hline [PERSON] [13] & \(Q_{a}=\frac{l_{a}A_{a}}{1+|A_{a}/(l/lB)|^{23}A_{a}/a}\) with \(f=\frac{2}{(5.836\text{Er}_{o}^{1.3}+1)-1.76)^{2}}\) & Free-surface flow \\ \hline [PERSON] [14] & \(Q_{a}=\frac{l_{a}A_{a}}{1+|A_{a}/(l/lB)|^{23}A_{a}/a}\) with \(f=\frac{l_{a}^{2}}{450\text{Er}_{o}^{2}A_{a}}\) & Free-surface flow \\ \hline [PERSON] [15] & \(Q_{a}=\frac{l_{a}A_{a}}{1+|A_{a}/(l/lB)|^{23}A_{a}/a}\) with \(f=\frac{\omega l_{a}^{2}}{(l-h_{c})^{2}}\) & Free-surface flow \\ \hline \end{tabular}
\begin{tabular}{l l} \hline \(\text{Fr}_{o}=U_{w}/\sqrt{gh_{c}}\); where \(U_{w}=\) average water velocity, \(g=\) gravitational acceleration, \(h_{o}=\) gate opening, \(\text{Fr}_{c}=U_{w}/\sqrt{gh_{c}}\); where \(h_{c}=\) water depth at the vena contracta section. \(Q_{w}=\) average water discharge. \(H_{E}=\) total energy head at the gate. \(\delta\) = variable loss coefficient of the air vent. \(a=\) cross-sectional area of the air vent. \(L=\) tunnel length. \(h=\) tunnel clearance height. \(h_{max}=\) maximum allowable gate opening. \(A=\) cross-sectional area of the tunnel. \(S=\) tunnel bed slope. \(A_{a}=\) cross-sectional area of airspace above the free surface. \(f=\) air-water interfacial drag coefficient. \(h_{h}=\) airspace height above the free surface. \(h_{w}=\) flow depth in the water section. \(\omega=\) correction factor for the interfacial drag coefficient \(f\).  
\end{table} 
Table 1: Overview of existing air demand (\(Q_{a}\)) equations for spillway tunnel. CFD simulations have been widely adopted by many researchers to investigate the air characteristics of spillway tunnels [7; 15]. [PERSON] and [PERSON] employed the VOF model on both circular and rectangular tunnels. Using the results of the numerical model, they calculated the air demand and validated it with available experimental data [17]. However, such models grossly overestimate the turbulent viscosity of the free surface and mispredict the velocity close to the air-water interface [18; 19; 20]. Therefore, a proper two-phase model is necessary to accurately determine the drag forces to study the air demand of a spillway tunnel. [PERSON] employed the Euler-Euler approach to simulate air discharge through tunnels. The results revealed that the Euler-Euler approach is more accurate than the VOF model in determining drag forces and calculating air demand, which was validated by available experimental data [21]. [PERSON] used a particle finite element method to model the air-water interaction at a bottom outlet at Susqueda Dam, which was validated by the information gathered on site during gate operation tests [22]. While CFD methods require high-fidelity modeling of drag-induced air entrainment and substantial computational effort--particularly for free-surface flows with multiple ventilation ports--the experimental methodology proposed herein enables rapid, precise air demand estimation, offering practical advantages for real-time engineering assessments.