Therefore, more research studies have been devoted to the profound analysis and exact modeling of the free-surface-pressurized flow based on experimental research and numerical simulation, leading to many achievements. Free-surface-pressurized flow is free surface flow in which the conduit is pressurized during the transient state, and often occurs in sewers and in the conduits of hydroelectric power plants or pumped storage projects [

2]. Based on an overall review and analysis, it was pointed out that the free-surface-pressurized flow in storm water systems is difficult to capture in modeling, and rapid pipe filling or emptying of water mains and sewer systems is accompanied by transitions between free surface and pressurized flow regimes and subatmospheric unsteady flow [

3,

4]. Focusing on the free-surface-pressurized flow, based on different experimental setups and further data analysis, the obtained experimental results were not only used for verification of numerical models but also revealed some obvious phenomena involved in air–water interactions; for example, the air near the pipe crown may pressurize and lead to consequent intense pressure oscillations [

5,

6,

7,

8]. Furthermore, considering the obvious air–water interaction in the free-surface-pressurized flow, the effect of air cavity intrusion into horizontal and inclined pipes on flow behavior was also experimentally investigated, enabling different degrees of ventilation by various orifices [

9,

10,

11]. Considering the typical flat ceiling tail system in

Figure 1 in particular, for the possible free-surface-pressurized flow, the types of air–water interactions observed in the combined diversion tunnel are divided into single air pocket motion, multiple air pocket motion, interfacial instability, and negligible interactions [

12].

With a clear understanding of the free-surface-pressurized flow provided by experimental research, recent research studies have emphasized that it is more important to perform the exact numerical simulation for the free-surface-pressurized flow in different water systems, and in most cases the given prototype systems are too difficult or uneconomical to be modeled in the lab. For numerical simulation of the free surface flow, a family of well-balanced, semi-implicit numerical schemes was proposed and further proved to be reliable for solving engineering problems [

13]. In simple hydraulic systems containing a channel, considering the partial free surface and partial pressurized flow, fundamental numerical simulation models were derived and presented by means of the control volume method or other methods [

14,

15]. More practically, based on the traditional Preissmann model [

2], some numerical simulation models were constructed for drainage networks, including a model based on an introduced virtual slot on the crown of the pipe to treat a separated gas–liquid flow [

16,

17]; a model capable of simulating transient flows in closed conduits, ranging from free surface flows to mixed flows and fully pressurized flows [

18]; the storm water management model (SWMM), which has wide practical applications [

19]; a model based on the first-order Roe’s scheme within the framework of finite volume methods [

20]; and a discrete model with a four-point linear implicit format [

21]. From the viewpoint of engineering applications, considering the possible transient mixed free-surface-pressurized flow in tailrace tunnels of large hydropower stations, a new method was proposed—namely, the implicit method of characteristics based on an implicit finite difference scheme [

22]—and was used for detailed characteristic analysis of the free-surface-pressurized flow for all kinds of tunnel network topologies, including large fluctuation computation, hydraulic disturbance analysis, and small disturbance analysis, mostly for changing top-altitude tail tunnels [

23,

24,

25]. Particularly, based on the assumption of a rigid incompressible water column and a compressible air bubble, a numerical model was derived to simulate pressure fluctuation, void fraction, air–water flow rate, and water velocity in a closed conduit [

26]. Focusing on the special free-surface-pressurized flow with a clear and regularly moving interface region in the changing top-altitude tail tunnel, a three-dimensional computational fluid dynamics research code with the volume of fluid (VOF) model was applied [

27]. In addition, for some other specified water systems with possible free-surface-pressurized flow, the numerical methods mainly include the computational fluid dynamics (CFD) models [

28], the high-precision discontinuous Galerkin finite element method [

29], a model used to simulate vertical water level fluctuations with coupled liquid and gas phases [

30], and a mathematical model based on dam break theory and bore flow theory [

31]. For the irrigation distribution system, a fully implicit time scheme for free-surface-pressurized water flow was developed and validated using a typical test system and prototype experiment [

32]. Most recently, an explicit smoothed particle hydrodynamics model for incompressible fluid was presented to simulate flow in conduits during transitions between free surface and pressurized flow [

33]. A semi-implicit numerical model with a linear solver was proposed for mixed free surface and pressurized flow in hydraulic systems [

34]. A novel 1D–2D coupled model was presented for accurate simulation of transient flow hydrodynamics in urban drainage systems, which was further confirmed by some test cases using comparative analysis with other existing models [

35]. Using comparative analysis of all the aforementioned numerical models, most of them are presented for different draining networks and specified waterworks, while for modeling of the free-surface-pressurized flow in the hydropower system in

Figure 1, the characteristic implicit method [

22,

23,

24,

25] can be appropriately used for the hydraulic transient analysis, with further mathematical modeling of necessary algorithm solutions and boundary conditions.

This paper aims to numerically model the free-surface-pressurized flow in a hydropower system with a flat ceiling tail tunnel, along with further hydraulic characteristics analysis. Considering that other state-of-the-art models are basically used for typical water systems and considering the difficulty in introducing these into the numerical simulation for an entire hydropower system with various and complex boundaries, the characteristic implicit format, which is an improved slot model, is preferred. Therefore, based on the Newton–Raphson linearization of the basic equations for the characteristic implicit method, the corresponding mathematical models for necessary boundary conditions are built according to the characteristic implicit format, and then the mathematical model for the hydraulic characteristics in the connecting tunnel and flat ceiling tail tunnel, which is presented by linear algebraic equations with a band coefficient matrix, is constructed with appropriate algorithm methods. Next, combined with the method of characteristics for pressurized pipelines, the mathematical model of a downstream surge tank and surge unit’s motion equation, and their detailed hydraulic characteristics, a unified mathematical model is established for hydraulic transient analysis of the given hydropower systems. For the tail tunnel system with possible free-surface-pressurized flow, experimental research is preferred to investigate its complex hydraulic characteristics. Based on the experimental setup in the lab and further data analysis, particularly on the wave speed for the free-surface-pressurized flow, the corresponding wave speed for the computation of Preissmann slot in the unified model is corrected together with sensitivity analysis, and then the detailed hydraulic characteristics of the free-surface-pressurized flow in the flat ceiling tail tunnel are further revealed, accompanied by comparative analysis with experimental data.