# Experimental Study of the Air Demand of a Spillway Tunnel with Multiple Air Vents
## 1 Introduction  
Spillway tunnels are placed inside dam bodies to regulate the level of the reservoir water and flush the sedimentation of the reservoir [1]. Spillway tunnels are normally divided into a pressurized portion controlled by a shuice gate and an outlet tunnel that discharges the flow into the atmosphere [2]. [PERSON] and [PERSON] defined six flow patterns for an outlet tunnel: (i) subcritical free-surface flow with a small Froude number; (ii) supercritical free-surface flow with a large Froude number; (iii) hydraulic jump with a free-surface flow; (iv) hydraulic jump with a pressurized flow, leading to choking; (v) pressurized flow in the full tunnel, occurring for a hydraulic jump close to the shuice gate; and (vi) pressurized flow in the front part and free-surface flow in the rear part, leading to unstable flow on steep slopes [3, 4]. High-velocity free-surface flows under turbulent conditions induce substantial air entrainment, generating pronounced negative pressure gradients in confined tunnel environments. Insufficient ventilation to offset this aerodynamic demand leads to critically low-pressure zones, intensifying operational risks such as structural gate oscillations, cavitation erosion, and transient water hammer effects. These issues introduce greater uncertainties and complexities in dam engineering [5, 6, 7]. In practical hydraulic engineering applications, spillway tunnels with multiple vents are systematically optimized through aerodynamic design modifications to maintain stable free-surface flow. Therefore, it is essential to study the factors influencing the air demand and air pressure fields under freesurface flow and optimize the design of air vents to avoid these problems and excessive project costs. [PERSON] and [PERSON] conducted the first systematic study on the air demand (\(Q_{a}\)) in circular conduits. Their work examined the influence of the pipe slope on air entrainment and proposed a semi-empirical relationship linking air discharge to the Froude number at the gate (Fr\({}_{o}\)) [8]. Over the subsequent three decades, researchers including [PERSON], [PERSON], and [PERSON] systematically developed empirical air demand equations for distinct flow regimes, utilizing combined datasets from scaled model experiments and field prototype measurements. These formulations explicitly incorporated the Froude number at the vena contracta (Fr\({}_{c}\)) as a governing hydraulic parameter [9; 10; 11]. We estimate an error of \(\pm 15\) to 30% by applying the listed formulas, which still leaves a lot of variability unexplained, and the remaining differences in air demand are attributed primarily to the effects of the characteristics of the air vent and the geometry of the tunnel. [PERSON] and [PERSON] established a comprehensive empirical formula to predict the air demand of a single air vent, summarizing the other governing parameters as the variable loss coefficient (\(\delta\)) and area (\(a\)) of the air vent, tunnel length (\(L\)), and slope (\(5\)) [2; 12]. The empirical models mentioned above cannot accurately clarify the effects of hydraulic and structural factors on air demand, so few researchers have established analytical models through the conservation of mass, momentum, and energy. [PERSON] revised the expression for the interfacial drag coefficient (\(f\)) on the basis of prototype observations from \(11\) projects, and introduced an equation for calculating air demand on the basis of the momentum equation, including the influence of hydraulic and structural factors [13; 14]. [PERSON] et al. proposed an air-water stratified flow model to calculate the air demand of air vents, and the model was verified with prototype data from the spillway tunnel of the Jinping-I hydraulic project and numerical CFD results [15; 16]. The relationships between air discharge and the effects of hydraulic and structural factors can be quantified on the basis of the empirical and analysis equations listed in Table 1.