## 1 Introduction  
Cavitation damage is a common problem of high-velocity water in hydraulic engineering [1; 2; 3]. An economical and effective solution to reducing the possibility of cavitation is to aerate the water flow [4; 5; 6; 7; 8]. Aeration devices are usually installed along a spillway to make use of the air above the water surface. However, in a spillway tunnel that is a half-closed system and not connected directly to the atmosphere, the air can be dragged away by high-velocity water and needs supplementation. Therefore, air intake wells are usually installed along the tunnel. If these wells are not designed sufficiently large and cannot meet the air demands, extreme negative air pressures will occur. Negative pressure can increase the possibility of cavitation and cause many secondary disasters, such as structure vibration [9; 10] and low frequency noise [11; 12]. It is clear that the prediction of the air demand of a spillway tunnel is fundamental work for the reliability of project design. Research on the prediction of air demand can be classified into three categories, namely, theoretical analysis, empirical formulas derived from fitting experimental data and numerical simulation. The original theoretical model for calculating the air demand of spillway tunnels was proposed by [PERSON] [13]. The dynamic balance of the whole airflow above the water surface was considered in the model. In addition, the empirical equation was subsequently improved by [PERSON] [14], which took the drag force into consideration based on their experiments and prototype data. [PERSON] [15] also proposed a theoretical method to calculate the air demand, but the theoretical results had significant errors compared to the data from prototype observations. [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) proposed a mathematical model based on the simplified Navier-Stokes equation set, which was only applicable to uniform flow or gradually varying flow. [PERSON] and [PERSON] [PERSON] and [PERSON] (1977) were the pioneers who proposed the empirical formula that correlated the air demand to the Froude number based on their experiments in the laboratory. Then, a similar empirical formula based on prototype data was proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (1988). Nevertheless, the air demand is much more complicated and should be a function of multiple parameters, not only the Froude number [PERSON] and [PERSON] (1988). It was suggested that the flow regime would influence the applicability of formulas and should be initially judged, which means that an empirical formula should be selected for a specific flow regime. Several empirical formulas have been proposed for different flow regimes by [PERSON] (2000) based on their experiments. In addition, [PERSON] and [PERSON] [PERSON] and [PERSON] (1988) studied the air demand in a rectangular tunnel. Their experimental results indicated that the air demand is related not only to the Froude number of water flow but also to the structures of the spillway tunnel and the gate [PERSON] et al. (2000); [PERSON] et al. (2001). [PERSON] et al. [PERSON] et al. (2001) proposed a different empirical formula based on a large amount of prototype data, which considered more parameters, including types of sluice gates, lengths of tunnels and ranges of Froude numbers. However, all of the previous empirical formulas were based on experimental or prototype data of a specific project, which limited their application to a wider extent. Recently, numerical simulation technology has been greatly developed due to the improvement in computer technology [PERSON] et al. (2000); [PERSON] and [PERSON] (2000); [PERSON] and [PERSON] (2000); [PERSON] and [PERSON] (2000); [PERSON] et al. (2001). The results from numerical simulations can give full details of the flow field, so many researchers have conducted numerical simulations on spillway tunnels for investigation.