Simulation and Analysis of Real-Time Coupling of Free-Surface Flow and Pressure Flow in Hydropower Station Tailrace Tunnels Based on the Finite Volume Method

Abstract

Accurate water hammer calculations are crucial for hydraulic safety and unit stability in hydropower systems with free-surface tailrace tunnels. However, existing models often neglect hydraulic variations in free-surface sections, while the commonly used method of characteristics tends to cause numerical instability and dissipation due to interpolation or wave speed adjustments, leading to significant computational errors. Aiming at the transient process of hydropower stations with free-surface tailrace tunnels and fully considering the influence between pressure and free-surface conditions, this study employs the second-order Godunov scheme to solve the governing flow equations for pressurized and free-surface flows. A generalized boundary of the regulating pool and a variable time step calculation method were proposed to solve the problem of real-time coupling calculation in the pressure–free-surface transition area. The results show that during the large fluctuation transient process, the hydraulic characteristics of the free-surface flow have little impact on the inlet pressure of the unit’s volute and the unit’s rotational speed but have a significant impact on the fluctuation period and extreme value of the inlet pressure of the draft tube. During the small fluctuation transient process, the hydraulic characteristics of open channel flow are beneficial for improving the unit’s regulation quality. This indicates that considering the hydraulic characteristics of free-surface flow is of great significance for realizing an accurate simulation of the transient process of hydropower stations.

1. Introduction

In the generator sets and their flow-passing systems of hydropower stations, the flow changes due to factors like working condition conversion and unit load rejection, resulting in the water hammer phenomenon. The substantial pressure fluctuations generated by this phenomenon can easily affect the power output of the units and may even lead to structural damage to the water conveyance system [1,2]. Accurately simulating these pressure fluctuations is crucial for ensuring the safe and stable operation of the plant’s water conveyance system.

In diversion-type hydropower stations, owing to restrictions imposed by topographic and geological conditions, most hydropower stations are not directly connected to the downstream river channel in the design of their pressure tailrace; instead, they are connected to free-surface tunnels. For the actual complex water conveyance systems of hydropower stations with free-surface water conveyance tunnels, existing calculation models or commercial water hammer calculation software often make the following assumptions or simplifications: (1) In the study of hydraulic transient processes, it is usually assumed that the pressure tailrace section is directly connected to the steady tailwater level or the downstream river channel, without considering the impact of possible water level fluctuations in the free-surface tunnel. This assumption fails to accurately account for water level fluctuations in the tailrace tunnel and the backflow of water in the free-surface tunnel under conditions of changes in the station load or downstream tailwater level, which may lead to large calculation errors. Using such assumptions for water hammer simulations is bound to affect the safe operation of the unit and flow-passing system. (2) In actual hydropower station projects, the method of characteristics (MOC) is usually adopted for modeling and simulation to calculate the hydraulic transients of pressure water-conveyance systems [1]. However, previous research has demonstrated that MOC often requires interpolation calculations or wave speed adjustment in complex pipeline network systems, which leads to problems of numerical instability and reduced accuracy [1]. In addition, the implicit difference method is commonly used in free-surface transient flows; however, it requires solving large-scale simultaneous equations at each time step, resulting in low calculation efficiency [3,4]. Therefore, it is necessary to establish a hydraulic transient process analysis model that fully considers free-surface flow and to conduct simulations and analyses of the transient process characteristics of hydropower station systems with free-surface tailrace tunnels to ensure the safe and reliable operation of the power station.

Many scholars have developed models for pressurized, free-surface, and coupled pressurized–free-surface systems. In the early 1960s, Wylie and Streeter [1] applied the Method of Characteristics to pressurized water hammer calculations. Subsequently, scholars such as Professor J. A. Fox from the United Kingdom, Professor Tokuzo Akimoto from Japan, Soviet scholar G.I. Krivchenko, Swiss researcher C. Jaeger, and Canadian expert M. H. Chaudhry successively published seminal works in the field of water hammer [5,6,7,8,9]. These contributions significantly enriched water hammer theory. Yang Lingxia et al. [10] derived a new water hammer equation through theoretical analysis, which resolved the issue where traditional equations failed to maintain validity under steady-flow conditions. Wan Wuyi et al. [11] developed a transient flow model accounting for laminar-turbulent transitions by comprehensively considering laminar friction effects. Compared with conventional transient flow models, this approach accelerates the attenuation rate of large fluctuations in practical engineering applications. Li Hui et al. [12] used the Preissmann slot method and the characteristic implicit scheme to uniformly solve the free-surface flow and pressure flow. Based on the method of characteristics, Zhang [13] constructed a transient process analysis model for diversion-type hydropower stations, which includes non-automatic regulating diversion free-surfaces, pressure forebays, tail water pools, and tail water free-surfaces. Cui Weijie et al. [14] used the method of characteristics to calculate the transient process of pressure pipelines and the Preissmann implicit difference method to calculate the transient process of free-surfaces.

Most of the above studies are based on the method of characteristics and the Preissmann implicit difference method for solving and calculations. However, existing studies [15,16] have proven that the finite volume method has more advantages than the method of characteristics and the Preissmann four-point implicit difference method in solving pressure flow and free-surface flow. In recent years, the finite volume method has been widely used in computational fluid dynamics [17,18,19]. Guinot [20] first applied the Godunov scheme to calculate transient flow in pressure pipelines and obtained a first-order scheme, which is similar to the interpolated method of characteristics. Zhao [3] applied the finite volume method to water hammer calculations and found that for a given accuracy level, using the same computer, the second-order Godunov required much less calculation time than the first-order Godunov and the MOC with spatial linear interpolation. Hwang and Chung [21] applied the finite volume method to the study of water hammer problems, without neglecting the convective terms but rather adopting the conservative form of the compressible flow equations for their research. Leon et al. [22] implemented the finite volume method in water hammer studies, where they developed and evaluated a second-order Godunov-type scheme. This method employs a ghost-cell technique for boundary treatment, achieving high numerical stability while preserving the conservative properties of the algorithm. The study demonstrates that this scheme effectively avoids non-physical oscillations and exhibits excellent convergence performance in practical computations. Furthermore, when simulating both smooth flows and sharp transients, its computational efficiency significantly surpasses both the MOC method proposed by Zhao and Ghidaoui [3] and traditional second-order finite volume methods. This research provides robust technical support for real-time control of water hammer phenomena in large-scale pipe networks. Yazdi et al. [23] applied an explicit second-order Godunov scheme to water hammer problems. Through experimental data analysis, they found that under conditions with a Courant number less than 1, the simulation results of the traditional Method of Characteristics (MOC) were less stable than those of the second-order Godunov formulation in certain extreme cases. The study demonstrates that this higher-order algorithm can better capture the complexity of the flow, thereby providing a solution with greater practical value compared to conventional methods. Sanchez-Barra et al. [24] applied the aforementioned method to practical hydrodynamic issues in oil wells, validating the effectiveness of the first-order Godunov scheme when dealing with pressure wave propagation in transient flows. Although this scheme is less accurate than its second-order counterpart, its computational efficiency and stability make it an ideal choice for practical engineering applications. Scholars such as Alcrudo et al. [25], Glaister [26], and Toro [27] introduced the classical Godunov computational framework into the study of free-surface pipe hydraulics. Alcrudo et al. [25] innovatively incorporated second-order modification techniques in solving open-channel hydraulic equations, effectively balancing computational accuracy and numerical stability. Fujihara and Borthwick [28] extended this methodological framework to the field of two-dimensional shallow water flow, establishing a high-accuracy numerical model for hyperbolic equations. Leon [29] applied the finite volume method to solve the free-surface flow equations. Zhou et al. [30] added a corresponding number of virtual cells at the boundary according to the requirements of the order in the Godunov scheme, thereby realizing the unified calculation of all cells. This method avoids the need to separately process the boundary values before the calculation and provides a more convenient solution for transient flow calculation. In addition, Zhou et al. [31] conducted simulations and analyses of a pumping station system with free-surface water conveyance and found that the hydrodynamic force of the free-surface water conveyance part affected the pressure part. They suggested that the influence of free-surface flow should be fully considered in hydraulic calculations and analyses.

In summary, accurate water hammer calculation is crucial for ensuring hydraulic safety and unit stability in hydropower stations with free-surface tailrace tunnels. However, existing models frequently neglect hydraulic condition variations in the free-surface sections. Furthermore, the commonly used method of characteristics can introduce numerical instability and dissipation through interpolation or wave-speed adjustments. These limitations may lead to significant computational errors. Therefore, to achieve accurate simulation of water hammer in these complex conveyance systems, numerical models must account for the correlations between different system components and their flow characteristics.

Therefore, this study fully considers the impact of hydraulic transients in the free-surface section on water hammer calculations in the pressurized section and establishes a combined pressure–free-surface hydraulic calculation model. Meanwhile, considering the advantages of the accuracy and stability of the second-order finite volume method, the second-order Godunov scheme is used to solve the flow control equations for the pressurized and free-surface sections. A generalized boundary of regulating pool and a variable time step calculation method are proposed to solve the problem of real-time coupling calculation in the pressure–free-surface transition area. The accuracy of the model was verified using existing experimental data, and the model and its solving method were applied to the complex water conveyance system of an actual hydropower station with a free-surface tailrace tunnel. The hydraulic transient process was calculated and analyzed, and the influence of the transient characteristics of free-surface flow on the hydraulic transient process of the hydropower station was studied to provide a new idea for accurate water hammer simulation of complex water conveyance systems.

2. Mathematical Model and Its Solution

2.1. Governing Equations for Pressure Flow

The basic governing equations for water hammer in pressurized pipelines can be derived in the following matrix form [1,3]:

$${\displaystyle \frac{\partial U}{\partial t}}+A{\displaystyle \frac{\partial U}{\partial x}}=S$$

(1)

In the formula: $U=\left(\begin{array}{c}H\\ V\end{array}\right)$; $A=\left(\begin{array}{cc}V& a^2/g\\ g& V\end{array}\right)$; $S=\left(\begin{array}{c}0\\ -{\displaystyle \frac{fV\left|V\right|}{2D}}\end{array}\right)$, where H is the piezometric head of the water body, m, V is the flow velocity of the water body, m/s, t is time, s, a is the pressure wave velocity in the pipeline, m/s, g is the gravitational acceleration, m/s², x is the distance along the pipe axis, m, D is the pipe diameter; and f is the Darcy–Weisbach friction factor. If the influence of dynamic friction is considered, a dynamic friction model can be added here [4].

The finite volume method performs numerical solutions by dividing the solution domain into N equidistant discrete cells (Δx = L/N) [4]. To handle the boundary conditions, additional auxiliary virtual cells were set at the upstream and downstream boundaries of the computational domain, and the parameter calculation method for these cells is elaborated in the boundary conditions section. The spatial discretization scheme of this model is shown in Figure 1, where the left and right interfaces of each cell i are marked as i − 1/2 and i + 1/2. Equation (1) is integrated along the x-direction within the cell control volume, and the cell-averaged variables $U_i={\displaystyle \frac{1}{\Delta x}}{\displaystyle {\int }_{i-1/2}^{i+1/2}udx}$ are defined (with the integration interval from i − 1/2 to i + 1/2), ultimately deriving the following integral expression:

$$U_i^{n+1}=U_i^n+{\displaystyle \frac{\Delta t}{\Delta x}}(F_{i-1/2}-F_{i+1/2})+{\displaystyle \frac{\Delta t}{\Delta x}}{\displaystyle {\int }_{i-1/2}^{i+1/2}Sdx}$$

(2)

where F_i−1/2 and F_i+1/2 represent the fluxes at the interfaces i − 1/2 and i + 1/2, respectively, n denotes the current time of calculation, and n + 1 denotes the next time of calculation.

In the Godunov scheme, the numerical flux at the boundary of the computational cell is obtained by calculating the exact solution of the local Riemann problem.

$$u^n(x)=\left\{\begin{array}{cc}{U}_L^n,& x<x_{i+1/2}\\ U_R^n,& x>x_{i+1/2}\end{array}\right\}$$

(3)

where $U_L^n$ is the average value of variable U at the left side of the i + 1/2 interface at time n, and $U_R^n$ is the average value of variable U at the right side of the i + 1/2 interface at time n.

Therefore, the flux value at the boundary i + 1/2 can be derived as

$$F_{i+1/2}=\overline{A_{i+1/2}}{U}_{i+1/2}(n)={\displaystyle \frac{1}{2}}\overline{A_{i+1/2}}\left\{\left(\begin{array}{cc}1& a/g\\ g/a& 1\end{array}\right)U_L^n+\left(\begin{array}{cc}1& -a/g\\ -g/a& 1\end{array}\right)U_R^n\right\}$$

(4)

In this paper, the MUSCL-Hancock method [4], which has second-order accuracy in both time and space, is used for numerical reconstruction and time-step advancement. Meanwhile, the slope limiting function MINMOD [4] is selected to improve the calculation accuracy of the Godunov scheme and avoid spurious oscillations in the calculation results.

To ensure that the numerical simulation maintains second-order accuracy at all times, the Runge–Kutta method [4] with second-order accuracy is selected for time integration calculations:

$${\overline{U}}_i^{n+1}=U_i^n-{\displaystyle \frac{\Delta t}{\Delta x}}(F_{i+1/2}^n-F_{i-1/2}^n)$$

(5)

$${\overline{\overline{U}}}_i^{n+1}={\overline{U}}_i^n+{\displaystyle \frac{\Delta t}{2}}S(U_i^{n+1})$$

(6)

$$U_i^{n+1}={\overline{U}}_i^{n+1}+\Delta tS({\overline{\overline{U}}}_i^{n+1})$$

(7)

Equations (5)–(7) represent the steps of a second-order accurate Runge–Kutta time integration method. Equation (5) first considers only the convective transport process, updating the conserved variables by a half-time step through flux differences. Equation (6) introduces source terms (such as friction, gravity, etc.) and semi-implicitly corrects the intermediate results using conserved variables at future time instants, with coefficients ensuring second-order accuracy. Equation (7) then performs an explicit update over the entire time step based on the intermediate results from the previous two steps, combined with the source terms, ultimately yielding the conserved variables at the next time instant. Through these multi-step intermediate corrections, the local truncation error of the time integration is controlled to second order, ensuring the second-order accuracy of the numerical simulation.

The finite volume method numerical simulation model with the second-order accurate Godunov scheme adds two virtual cells, namely −1, 0, N + 1, and N + 2, at both the upstream and downstream boundaries, where N represents the number of cells into which the water body is divided.

2.2. Governing Equations for Free-Surface Flow

Similarly to the treatment of pressure flow in pipelines, the governing equations for unsteady free-surface flow in pipelines can also be written in matrix form [29]:

$${\displaystyle \frac{\partial U}{\partial t}}+{\displaystyle \frac{\partial F}{\partial x}}=S$$

(8)

where $U=\left(\begin{array}{l}A\\ Q\end{array}\right)$, $F=\overline{B}U$, $\overline{B}=\left(\begin{array}{c}Q\\ {\displaystyle \frac{Q^2}{A}}+{\displaystyle \frac{A\mathrm{p}}{\rho }}\end{array}\right)$, $S=\left(\begin{array}{c}0\\ F_w+(S_0-S_f)gA\end{array}\right)$.

Where A is the cross-sectional area of the free-surface section through which water flows, m², Q is the flow rate in the free-surface section, m³/s, p is the hydrostatic pressure at the mid-depth position of the free-surface’s cross-section, in Pascals (Pa), ρ is the density of the water body, kg/m³, g is the gravitational acceleration, m/s², S₀ is the bottom slope of the free-surface; S_f is the frictional resistance of the free-surface, and if the influence of dynamic friction is considered, a dynamic friction model can be added here [4], F_w is the momentum term generated by the longitudinal variation in the free-surface width. The complete explanations of terms are provided in Appendix A.

The finite volume method, same as that used for the pressure flow part, is adopted to process Equation (8), resulting in an expression with the same form as Equation (2). The specific calculation process is consistent with that of the pressure flow part, and the HLL Riemann solver [29] is used for the flux calculation.

2.3. Combined Calculation Strategy for Pressure and Free-Surface Flows

The governing equations for the pressure flow segment are not the same as those for the free-surface flow segment. Therefore, for the combined calculation of the pressure and free-surface flow segments, the key lies in properly handling the junction between the two segments.

At the junction of pressure flow and free-surface flow, this transition zone can be simplified as a regulating pool, as shown in Figure 2. Given that the time step Δt used in the calculation of pressure flow is significantly smaller than the calculation time step ΔT of free-surface flow, the following processing can be performed within each calculation cycle [t₀ + nkΔt, t₀ + (n + 1)kΔt] of pressure flow (where n and k are integers, n = 0, 1, 2…, and k ≤ ΔT/Δt ≤ k + 1):

When the pool serves as the terminal boundary of the pressure flow, it can be assumed that the water level remains stable at the reference value H₀ during this period. When the pool acts as the initial boundary of free-surface flow, it is necessary to consider the cumulative water volume flowing into the pool from the pressure pipeline within the kΔt period. At this time, the instantaneous water level H₁₀ of the pool can be calculated using the water balance equation: H₁₀ = H₀ + kΔt × ∑Q/A_s.

H₁₀ is taken as the initial boundary condition for the calculation of the free-surface flow segment, and H₁ is iteratively solved for the water level H₁ at the next moment, and then the obtained H₁ is used inversely as the boundary input for the next calculation time step of the pressure pipeline. The above steps were repeated to realize the combined calculation of the pressure flow and free-surface flow segments.

3. Model Validation

3.1. Validation of the Pressure Flow Model

The refined experimental platform for long-distance hydraulic transients at Hohai University was used to verify the accuracy of the water hammer model established in this study, as shown in Figure 3. This experimental system mainly consists of three modules: the first is a pressure-stabilized water supply device, which uses a pressure-stabilizing tank to ensure the stability of the water supply; the second is a steel pipe system made of stainless steel; the third is an integrated system for pressure and flow measurement, as well as data acquisition and processing. During each experiment, precise regulation was performed to ensure that the pressure in the upstream pressure-stabilizing tank remained at the set constant value.

In this experiment, the water hammer phenomenon was induced by adjusting the downstream valve position. A pressure sensor installed at the valve was used to monitor the dynamic changes in the water hammer pressure in real time, and the data were transmitted to the LabVIEW data acquisition system for recording and analysis. The parameters of the experimental pipeline were as follows: the total length of the pipeline was 241.52 m, the inner diameter was 5 cm, and the wall thickness of the steel pipe was 3.5 mm. Based on the pressure and flow data measured by the sensors, the average hydraulic loss coefficient of the experimental system was calculated to be 0.0192. Through measurements under multiple working conditions, the average water hammer wave velocity in the steel pipe of this experimental system was calculated as 1305 m/s using the relationship between the time difference between adjacent wave crests and the pipeline length.

In this experimental system, the water hammer phenomenon was generated by quickly closing the valve. The water hammer experimental schemes were selected as shown in

Table 1. Water hammer experiment scheme.

Case	Upstream Reservoir Water Level Hu (m)	Water Flow Velocity V0 (m/s)
1	30.36	0.18
2	24.18
3	27.21	0.22
4	24.17

, and each working condition was repeated at least three times to ensure the consistency of the experimental results.

The finite volume method with a second-order Godunov scheme was used to solve the water hammer problem of the aforementioned water conveyance system, and the results were compared with the experimental values. The specific calculation results are shown in Figure 4.

As shown in Figure 4, the calculation results based on the finite volume method with the second-order Godunov scheme are in good agreement with the experimental measurement data, which verifies the accuracy of the pressure flow model with the second-order Godunov scheme proposed in this study.

3.2. Validation of the Free-Surface Flow Model

To verify the accuracy of the free-surface flow model proposed in this study, a set of free-surface flow experimental devices was designed and built, which was mainly used to study the dynamic characteristics of the water head at various points in the free-surface pipeline over time. The experimental device from upstream to downstream included a reservoir, pipeline, pressure sensors, a manual valve, and a water storage tank. The total length of the experimental pipeline was 20 m, made of organic glass with an inner diameter of 0.15 m, and the pipeline slope was 0.01. Figure 5 shows a schematic of the experimental system and actual photos of some experimental devices.

The initial condition was a steady flow state, where the water depth upstream of the gate was set to 0.254 m, and the water depth downstream was 0.25 m. The upper boundary was a constant inflow rate, and the lower boundary was a constant outflow rate, both at 0.487 m³/s. The dynamic change process of the water depth in the pipeline after the gate was instantly closed was simulated. The specific calculation results are presented in Figure 6.

It can be seen from Figure 6 that, regarding the calculated curve of water depth variation with time at point P3, the calculated values of the model established in this study are in good agreement with the experimental values, which indicates the accuracy of the free-surface flow model established in this chapter.

3.3. Validation with Actual Engineering

In this section, the measured data of a hydropower station project with a free-surface tailrace tunnel under unit load rejection conditions are used to verify the model. A layout diagram of the hydropower station project is shown in Figure 7. In this system, the rated speed of the turbine is 500 rpm, the rated head is 362 m, the rated flow is 18.3 m³/s, the rated output is 61.5 MW, the unit installation elevation is 1367.6 m, and the moment of inertia is 500 tm².

The finite volume method model with the second-order Godunov scheme established in this study was used to calculate the hydropower station system, and the calculation results were compared with the on-site measured results. The calculation results are presented in

Table 2. Comparison between numerical simulation values and on-site test values.

Parameter	Measured Data	Calculated Data (Joint)
Unit status	1#, 2#, 3# unit load rejection
Upper reservoir water level (m)	1761.62
Downstream tailwater level (m)	1376.75
Output before unit load rejection (MW)	61.1/61/61	60.62/61.53/59.4
Opening degree before unit load rejection (%)	76.5/75.8/73.5	76.5/75.8/73.5
Initial pressure at the inlet of the volute (m)	363.12/368.22/365.16	364.41/368.88/367.10
Maximum pressure at the inlet of the volute (m)	517.14/522.24/518.16	518.14/520.35/518.71
Maximum speed rise rate of the unit (%)	35.4/35.2/35.6	36.4/35.2/34.66
Initial water level of the upstream surge chamber (m)	1752.3	1752.3
Maximum water level in the upstream surge chamber during the load rejection process (m)	1765	1765

.

As shown in Table 2 that under the three-unit load rejection condition, with the same measured water level and initial opening data, the finally calculated steady-state output of the unit and the steady-state maximum pressure at the end of the unit’s volute are in good agreement. Additionally, the calculated maximum pressure at the end of the unit’s volute and the maximum speed rise rate of the unit after load rejection were relatively close to the measured values, indicating that the model established in this study is reliable.

4. Analysis of the Impact of the Free-Surface Section of Tailwater on Hydraulic Calculation—A Case Study of an Actual Power Station

The hydropower project is a cascade hydropower station among the four hydropower cascades in the lower reaches of the Jinsha River [32]. The power station adopts the underground head development mode, with eight sets of 1000 MW Francis turbine-generator units arranged on both the left and right banks, with a total installed capacity of 16,000 MW. The rated flow of a single unit is 547.8 m³/s, and the rated head is 202 m. The water diversion tunnels adopt a layout of one tunnel for one unit, and the tailrace tunnels adopt a layout of one tunnel for two units. The last sections of tailrace tunnels No. 2–6 were reconstructed from diversion tunnels, and their vertical layout consisted of a gentle slope section followed by a steep slope section (transition section) and then a flat slope section, as shown in Figure 8. When the downstream tailwater level is low, the tunnel is in a free-surface flow regime. Therefore, to consider the impact of the transient characteristics of free-surface flow on the large fluctuation transition process of the hydropower station, this section considers the No. 6 hydraulic unit on the right bank as the research object, uses the above-established finite volume method numerical model based on the second-order Godunov scheme, selects some typical working conditions involving free-surface flow in the tailrace tunnel, and conducts calculation and analysis on load rejection conditions and small fluctuation conditions.

4.1. Calculation and Analysis of Load Rejection Conditions

Typical working conditions involving free-surface flow in the tailrace tunnel were selected to conduct load rejection condition calculations to analyze the impact of the hydraulic characteristics of the free-surface flow in the tailrace on the large fluctuation transition process. The specific working conditions are listed in

Table 3. Calculation Table of Load Rejection Conditions.

Case	Upper Reservoir Water Level	Downstream Tailwater Level	Load Change
D1	825	582.14	One unit full load rejection
D2	825	581.5	One unit full load rejection
D3	825	579.44	One unit full load rejection
D4	784.14	582.14	One unit full load rejection

.

Considering the influence of free-surface flow in actual situations, a combined pressurized–free-surface flow calculation model based on the second-order Godunov scheme was established, and a comparative analysis was conducted with the fully pressurized flow calculation model based on the second-order Godunov scheme (which equates the water surface of the free-surface section to the downstream reservoir water level). The specific calculation results are shown in Figure 9, Figure 10, Figure 11 and Figure 12.

As shown in Figure 9, Figure 10, Figure 11 and Figure 12, when considering the influence of free-surface flow in the tailrace, the maximum internal water pressure at the volute inlet of the unit and the unit speed are mainly affected by the closing law of the guide vanes of the unit. The hydraulic characteristics of the free-surface flow in the tailrace tunnel had little impact on the results, and the calculation results were not significantly different from those without considering the influence of free-surface flow. Taking Case D1 as an example, the peak internal water pressure at the volute inlet of the unit is calculated as 277.7 m by the pure pressurized model, versus 277.8 m by the coupled pressurized–free-surface model. When the pressure wave propagates to the free-surface section, the propagation speed of the gravity wave on the free water surface (approximately 5–10 m/s) is much lower than the water hammer wave speed in the pressurized pipeline (1000–1200 m/s), which causes part of the reflection energy of the pressure wave in the combined model to be dissipated, thereby inhibiting the growth of extreme pressure values. Taking Case D1 as an example, the peak pressure at the draft tube inlet calculated by the pure pressurized model reaches 18.8 m, whereas the result obtained from the coupled pressurized–free-surface model is 17.0 m, representing a relative reduction of 5.85%. In addition, in the subsequent fluctuation cycles, the fully pressurized model has a longer pressure fluctuation period owing to the characteristic of repeated superposition of pressure waves in the closed system, which fully reflects the importance of considering the influence of the free-surface section for accurately simulating the hydraulic transient process of hydropower stations with free-surface sections.

Therefore, by accurately simulating the pressurized–free-surface coupling effects, it can provide a precise hydraulic transient simulation tool for hydropower stations with complex tailwater systems, establish a theoretical basis for the structural design of transition sections in the water conveyance system, and reduce project costs while ensuring safety.

4.2. Calculation and Analysis of Small Fluctuation Conditions

Some typical working conditions involving free-surface flow in the tailrace tunnel were selected to conduct calculations and analyze small fluctuation conditions to evaluate the impact of free-surface flow in the tailrace on the stability and regulation quality of operating units under small fluctuation conditions. The regulating parameters of the speed governor were set as follows: permanent speed droop coefficient b_p = 0.04, corresponding proportional constant K_B = 2.5, integral constant K_I = 0.3851/s, and differential constant K_D = 2.5 s. The specific working conditions are listed in

Table 4. Calculation Table of Small Fluctuation Conditions.

Small Fluctuation Calculation Cases		Upstream Water Level	Downstream Tailwater Level
X1	One unit is operating normally and shedding 5% of its load.	825	581.5
X2	One unit is operating normally and shedding 10% of its load.	825	581.5
X3	One unit is operating normally and shedding 5% of its load.	784.14	582.14
X4	One unit is operating normally and shedding 10% of its load.	784.14	582.14

.

Considering the influence of free-surface flow in practical situations, a combined pressure–free-surface calculation model based on the second-order Godunov scheme was established, and a comparative analysis was conducted with the fully pressurized calculation model based on the second-order Godunov scheme. The specific calculation results are shown in Figure 13 and

Table 5. Analysis Table of Regulation Quality in Small Fluctuation Transient Process (with Δ = 0.2%).

Case	Maximum Deviation xmax/%	Overshoot σ/%	Adjustment Time Tp/s	Number of Oscillations X
X1 (Excluding free-surface flow)	0.8	11.8	16.8	0.5
X1 (Considering free-surface flow)	0.8	11.2	16.4	0.5
X2 (Excluding free-surface flow)	1.6	10.8	29.9	0.5
X2 (Considering free-surface flow)	1.5	9.6	28.5	0.5
X3 (Excluding free-surface flow)	1.3	11.9	24.1	0.5
X3 (Considering free-surface flow)	1.3	12.3	23.1	0.5
X4 (Excluding free-surface flow)	2.5	11.3	103.4	1.0
X4 (Considering free-surface flow)	2.4	8.0	62.4	0.5

.

As indicated in Figure 13 and Table 4, under small fluctuation conditions, accounting for the transient characteristics of free-surface flow in the tailrace tunnel results in a reduction in the unit’s maximum rotational speed, a decrease in overshoot, and a shortening of adjustment time. This demonstrates that free-surface flow in the tailrace tunnel exerts a certain improvement effect on the regulation quality of operating units and contributes to the stable operation of the units. Therefore, considering the free-surface flow in the tailrace tunnel can provide valuable references for establishing more accurate mathematical models of hydroelectric unit regulation systems.

5. Conclusions

This paper focuses on the water conveyance systems of hydropower stations with free-surface tailrace tunnels. By fully accounting for the impact of hydraulic transients in the free-surface section on water hammer calculations in the pressurized section, a coupled pressure–free-surface hydraulic calculation model is established. Additionally, leveraging the accuracy and stability of the second-order finite volume method, the second-order Godunov scheme is employed to solve the flow control equations for both the pressurized and free-surface sections, respectively. The model’s accuracy is validated using existing experimental data. Furthermore, this model and its solution method are applied to the complex water conveyance system of an actual hydropower station with free-surface tailrace tunnels to conduct calculations and an analysis of hydraulic transient processes, with the aim of investigating the influence of the transient characteristics of free-surface flow on the hydraulic transient processes of hydropower stations. The main conclusions are as follows:

(1)

A coupled calculation model for pressure and free-surface flows, based on the second-order Godunov scheme of the finite volume method, was established. A generalized boundary condition for regulating pools and a variable time-step calculation method were proposed, which addressed the issue of real-time coupled calculation in the pressurized–free-surface transition zone.

(2)

For hydropower stations with free-surface tailrace tunnels, hydraulic fluctuations in free-surface flow exert a certain impact on the hydraulic characteristics of units and other flow-passing systems. During large-fluctuation hydraulic transient processes, when accounting for the influence of transient free-surface flow in the tailrace, the hydraulic characteristics of free-surface flow have minimal effect on the volute inlet pressure and unit speed, yet significantly affect the fluctuation period and extreme values of the draft tube inlet pressure. This is because the fully pressurized numerical solution method, relying on the simplified assumption of a closed system (which ignores the influence of the free water surface in the free-surface section), restricts pressure wave propagation within the pressurized pipeline, thereby preventing pressure fluctuation energy from being released through the free-surface section. Consequently, the calculation results are often more conservative. In contrast, the results obtained via the coupled pressure–free-surface solution method are more consistent with actual conditions and can reduce the construction volume of surge tanks and other surge-regulating structures to a certain extent, yielding better economic efficiency.

(3)

During small-fluctuation hydraulic transient processes, when the influence of the transient characteristics of free-surface flow in the tailrace tunnel is taken into account, the unit exhibits better regulation quality. This suggests that the free-surface flow in the tailrace tunnel is conducive to the stable operation of the unit, thereby providing new perspectives for operational optimization.