## 1 Introduction  
High-head spillway tunnels are important structures to regulate the water surface of a reservoir in hydraulic engineering. A spillway tunnel is usually divided by a shuice gate. Downstream of the gate, the free surface flow with typical flow velocities of 30-60 m s\({}^{-1}\) drags away large amounts of air. If ventilation structures such as air vents are designed too small and cannot meet the air demand, the vacuum negative pressure cannot be reduced to desired level, which can result in secondary disaster, including gate vibration [1], cavitation damage, and unbearable noise [2, 3]. Therefore, the simulation of the air-water two phase flow and design of air ventilation structures is a critical issue for a spillway tunnel. In the past decades, engineers have designed air vent size by air demand prediction mainly on the basis of the empirical formulas based on prototype observation or model test [4, 5, 6, 7, 8, 9]. Most formulas have similar forms, such as \(\beta=\alpha_{1}(Fr_{c}-1)^{\alpha_{2}}\), where \(\beta\) is the ratio of air and water flow rate, \(Fr_{c}\) is the Froude number at the contracted flow section downstream of the gate, and \(\alpha_{1}\) and \(\alpha_{2}\) are the constant coefficients. However, these formulas usually underestimate air demand and result in unexpected high air velocity in the actual project [10, 11]. Although new empirical formulas [12, 13] and one-dimensional theoretical model [3] have been successfully proposed recently to improve the accuracy of air demand prediction, the traditional methods can only provide limited information about the ventilation characteristics of a spillway tunnel. Numerical simulation can obtain more detailed information of the air-water two-phase flow field in spillway tunnel. [PERSON] et al. [14] used the volume of fluid (VOF) method to simulate the ventilation quantity of a free flow pipe and studied the velocity distribution of airflow in the pipe, on the basis of which they proposed a calculation method for the design of air duct. [PERSON] and [PERSON] [15] conducted a numerical analysis of a transient process of air entrainment into bottom spillway flows when a spillway gate is open. [PERSON] and [PERSON] [16] investigated the effect of air duct area on the air demand of a spillway tunnel by VOF method. To simulate the water-air interaction, [PERSON] et al. [17] tested several different operating situations of Susqueda dam's bottom outlet in Spain with the method of particle finite element method (PFEM), then obtained various flow regimes observed in the experiments of [PERSON] [6]. [PERSON] et al. [18] analyzed the air supply amounts of a spillway with fully opened radial gates and at the full pool water level. [PERSON] et al. [13] analyzed the effects of approach water velocity and area residual of tunnel cross-section on air demand by using the VOF method. Nevertheless, the spray flow or air entrainment were not simulated in previous results. In theory, the VOF method can be used for direct simulation of dispersed bubble or droplet only if the mesh is 10 times smaller than the smallest bubble. However, smaller timesteps (order of \(1\times 10^{-7}\) s), which is not feasible for realistic turbulent multiphase flows, is required for stable solution according to the limitation of the Courant number [19]. For better prediction of air demand including air entrainment, the Euler-Euler two-fluid model, which allows fluid mixing and solves equations for each phase with couple phases through interaction forces, is more competitive [20; 21; 22]. This study is organized as follows. In Section 2, the prototype observation of Jinping-I spillway tunnel is introduced briefly. The basic equations of two-fluid model and the CFD model setup are briefly reviewed. In Section 3, the simulation results are verified including grid convergence, time independence, air demand, and velocity profile. Furthermore, a hybrid drag model is proposed to improve the air demand prediction.