Numerical investigation on the transient gas–liquid flow in the rapid switching process of pump turbine

The frequent switching of pump mode and turbine mode of the pump turbine leads to frequent transient phenomena. To ensure the safe and stable operation of the unit, a detailed study on the exhaust and pressurization process when the unit switches from turbine mode to the pump mode has been carried out. Based on the Shear Stress Transfer model (SST) k–ω turbulent model, the numerical simulations are processed both in a steady and unsteady state. The visualization results of gas–liquid two phases distribution in the dynamic process of exhaust and pressurization are given, and the characteristic references of each stage are also carried out. The transient characteristics of the torque, axial, and radial force of the runner and guide vane are analyzed by combining the short‐time Fourier transform. The results show that the main frequencies in this transient process are the blade passing frequency and its harmonic frequency. This process also presents a high amplitude band at all frequency values, which may be caused by the entrainment and centrifugal action of the runner on the free liquid surface.


| INTRODUCTION
Energy is crucial to global economic development. 1,2 With the steady development of economic globalization, the goal of environmental sustainability has also become a concern of many countries in the world. The choice of energy exploitation focuses more on clean energy, such as wind energy, solar energy and hydro energy. 3,4 The Sustainable Development Goals Report 2022 issued by the United Nations also points out that, "When we take action to invest in clean energy, we have dealt with the fundamental problems of increasingly serious environmental degradation and climate change." 5 In recent years, as an efficient and clean energy conversion method, pump storage has been widely used in the field of hydropower exploitation, it has also caused a series of challenges to researchers. 6,7 Pump storage is a way of mechanical energy storage. 8,9 By the end of 2020, the cumulative installed capacity of pump storage units has reached 172.5 GW in the world. 10 Pumped storage has also become the most mature, cost-effective, green, low-carbon, clean and highly flexible power system with the most large-scale development conditions in the current technology. 11 Therefore, as a highly advantageous energy with great development potential in the future, the research and promotion of pumped storage related issues plays a vital role in achieving global sustainable development. 12,13 As a key component of the pumped storage system, the working principle of pump turbine units is contained in two working modes. 14 Under the pump mode, the fluid is pumped from the downstream reservoir to the upstream reservoir by using the electric energy at the low power load, while under the turbine mode, the fluid is discharged from upstream reservoir to downstream reservoir for power generation to relieve peak pressure of power load. 15 For the long-term development of pump turbine units, experts and scholars have focused on the following three aspects: (1) Hydraulic performance improvement of pump turbine units. Dai et al. 16 studied the influence of the runner blades geometry on the energy performance of pump turbines, such as head and efficiency. Zhao et al. 17 showed the influence of guide vane opening on the hydraulic performance of the axial pump operating in the pump-as-turbine mode, and analyzed the energy loss to get relevant verification.
(2) Operation stability improvement of pump turbine units. Yin et al. 18 improved the "S" curve in the operation curve of the pump turbine by optimizing the distribution of blade load, and the operation stability of the pump turbine is improved as well. Li et al. 19 summarized the research on positive slopes in the performance characteristics of pump turbine in the past two decades and proposed a method to improve the positive slopes in the performance characteristics. (3) Rapid switching process of pump turbine units. Zeng et al. 20 studied the rapid switching process of pump turbines through model experiments and improved the "S" characteristics by setting different guide vane opening schemes. Guo et al. 15 analyzed the effects of hysteresis on the dynamic characteristics of the pump turbine in detail by using numerical simulation. Due to the complexity of transient flow and the coexistence of liquid and gas in the rapid switching process for pump turbine units, there are few existing research results on this process.
For pump turbine units, the process of rapid mode switching is very important. 21,22 In this process, the degree of confusion between liquid and gas phases and the component change of signal frequency are of great significance to judge whether the mode switching is stable and whether it can work normally. 23 Corresponding to the two modes of pump turbine units, its mode switching process can also be divided into two types: turbine mode to pump mode (generation mode to storage mode), pump mode to turbine mode (storage mode to generation mode). Figure 1 shows the process of mode switching from turbine mode to pump mode. First, the motor is disconnected from the power grid, and the turbine is unloaded. At this time, the rotation speed of runner (n) starts to drop, and the ball valve and guide vane are closed at the same time. After the guide vane is fully closed, the n value is still quite high. At this time, the impact of the water flow in the runner chamber makes the runner decelerate. Next, the electric braking is applied when the n value drops to 30% of rated rotation speed (n d ). When the n value drops to 15% of n d value, pressurized air is injected into the runner chamber until the unit totally stops, and the liquid surface is pressed out of the runner chamber, while the liquid level in the draft tube is at the same level with the outlet tank. At this time, there is no liquid in the runner chamber, the guide vane is in a fully closed state, and the unit reaches zero rotation speed state. Then, the unit is grid-connected and started at full voltage to make the rotation speed of runner reach the rated rotation speed in pump mode. Conduct exhaust and pressurization for the unit, that is, open the exhaust port to balance the pressure, and the liquid surface begins to rise until the gas inside the unit is completely exhausted. Finally, open the guide vane and ball valve to pump water at full load, the process of switching from turbine mode to pump mode is completed. 24 For the phenomenon of the dynamic process of exhaust and pressurization for pump turbine units, the time frequency signal analysis of the transient process is the focus of the research to reveal the operating mechanism. 25 Time frequency analysis is to combine the time domain and frequency domain to analyze the F I G U R E 1 Sketch of storage mode switch to pump mode for pump turbine, where n is rotation speed of runner in turbine or pump mode, nd is rated rotation speed of runner in turbine or pump mode. n/nd is a dimensionless parameter. α0 is the opening degree of guide vane and ball valve, the value 1 represents which are totally open, while the value 0 represents which are totally close. signal. 26 For the time frequency analysis of linear transient signals, short-time Fourier transform (STFT) is widely used to extract and analyze the signal characteristics in pump turbine unit. 27 Su et al. 28 used STFT to analyze the pressure pulsation signals in different components of the pump turbine. Han et al. 29 used STFT to analyze the pressure pulsation signal in the process of guide vane movement of the pump turbine in pump mode. It was found that the main pressure pulsation frequency came from the blade passing frequency and its harmonic frequency. The research shows that STFT has unique advantages in analyzing transient signals by analyzing the local sine frequency and amplitude strength of dynamic signals when the signal changes with time.
The exhaust and pressurization process of the pump turbine unit is of great significance for its pump mode startup. During the whole process, the runner rotates at high speed, and the guide vane is always closed. During this process, the liquid level in the draft tube is sucked and thrown onto the guide vane wall, which leads to very complex force conditions of the guide vane. However, there is little research on this process in existing studies, and there is almost no research to optimize this process. Nevertheless, if gas is not completely discharged or the guide vanes are fatigued due to excessive force or torque during the exhaust process, it can lead to safety issues in the operation of the unit.
This study mainly aims at the process of exhaust and pressurization during the mode switching of pump turbine. Based on the unsteady-state numerical simulation and STFT method, the torque, axial, and radial force of runner and guide vane are monitored and analyzed to reveal the connotation mechanism of exhaust and pressurization process. Contours of water volume fraction are carried out to visualize the transient process. In general, the phenomenon of the exhaust and pressurization process is proposed, and the transient characteristics which include transient flow and frequency characteristics are analyzed. The complex change process of gas-liquid two phases in the exhaust and pressurization process is revealed, it lays a certain foundation for the study of the mode switching process for pump turbine unit. It also ensures the operational safety and stability of pumped storage power stations.

| Objective model
In view of the rapid mode switching process of pump turbine unit, this study research the process of exhaust and pressurization when the unit operation mode is converted from turbine mode to pump mode. It is defined as: the guide vane and ball valve are fully closed, when the unit is connected to the grid and the runner rotation speed reaches the rated rotation speed of pump mode, open the exhaust port to balance the pressure, the liquid level in the unit will rise and the transient flow will occur in runner chamber. In this study, the flow and force characteristics in the runner and guide vane during this process will be analyzed in detail.
The fluid domain of the objective unit is shown in Figure 2A. The fluid domain includes a draft tube, runner, guide vane, stay vane, and spiral casing. Figure Table 1. The rated rotation speed n d in pump mode is 1200 r/min.

| Simplified model
For the process of exhaust and pressurization studied here, the guide vane is completely closed. At this time, the outer boundary of the objective fluid domain is equivalent to the working surfaces of the guide vane blades. To speed up the numerical calculation, this study simplifies all components that cannot pass the water, including half of the guide vane, stay vane, and spiral casing. The simplified model is shown in Figure 3. Based on this simplified model, the gas-liquid two phases simulation of the exhaust and pressurization process will be carried out in detail to prepare for model experiments in the future.

| Turbulence model and numerical setup
CFD based on fluid mechanics and numerical calculation has become an important simulation method for flow field and performance analysis of various fluid machinery. Based on the commercial software ANSYS CFX turbulence solver and computational fluid dynamics theory, numerical simulation of a pump turbine unit is carried out, and Navier-Stokes equation is used to close the turbulence model. By comparing k-ε, Renormalization Group (RNG) k-ε, k-ω, and Shear stress transport (SST) k-ω turbulence model, where SST k-ω turbulence model has the characteristics of good convergence and high stability. 30 SST k-ω turbulence model not only enhances the wall function, but also can avoid too dense grid near the fixed wall, for too dense grid will make the solution result too sensitive to the near wall grid. 31 Therefore, SST k-ω turbulence model is selected as the turbulence prediction model in this study. The turbulent kinetic energy equation and dissipation rate transport equation of the SST k-ω turbulence model is shown in below equations where P k is the production term of turbulent kinetic energy, S is the invariant measure of the strain rate. Parameters β, β*, σ k , and σ ω are constants of the turbulence model, μ is the dynamic viscosity, ω is angular velocity and F 1 is the blending function, μ t is turbulent eddy viscosity.
In the whole fluid domain, it is a two-phase flow medium of water and air at 25°C, and the reference pressure in the fluid domain is 1 atm. After setting the position of the free liquid surface, the step function is used to define air above the free liquid surface and water below the free liquid surface. By analyzing the volume fraction of the water phase, the two-phase state is analyzed. In addition, the effect of gravity is considered in the simulation process of exhaust and pressurization process. The gravity direction is along the Z-direction, and the gravity acceleration is g = 9.8 m/s 2 . Also, all walls in the fluid domain are set as nonslip walls. The inlet of the draft tube is set as the inlet of the whole fluid domain, and its boundary condition is the free boundary type of open pressure and flow direction. The outlet of the exhaust clearance is set as the outlet of the fluid domain, and its boundary condition is the flow rate type.
According to the Mei's publication, the exhaust rate in outlet is 0.06 kg/s. 32 Except that the runner is a rotating component, the rest are all fixed. Set the dynamic-static interface type at the interface between the runner and other fixed parts, and set the static-static interface type at every two fixed interfaces. To ensure the accuracy and reliability of data transmission at the interface, the boundary grid connection mode is set as general grid interface (GGI) which permit nonmatching of node location, element type, surface extent, surface shape and even nonmatching of the flow physics across the connection.
In the whole simulation process, the convergence criteria of momentum equation and continuity equation is 10 −5 . In the steady simulation, the maximum number of iteration steps is set as 1000. In the unsteady simulation, every calculated step spend 0.005 s, and the total simulation time is 26 s. The maximum number of iteration steps is set as 5-20 for each time step of unsteady simulation. In this study, based on the above numerical simulation settings, the torque and force in the runner and guide vane of the exhaust and pressurization process are monitored, the visual research of the exhaust and pressurization process is also carried out.

| Mesh scheme and independence check
In the process of numerical simulation, as the smallest computing element, the choice of grid scheme has an important impact on the simulation results. In this study, the commercial software ICEM is used to mesh the fluid domain of the simplified model. In consideration of the calculation time and fitness between gird and geometry, tetrahedral unstructured mesh is adopted for half guide vane, and hexahedral structured mesh is adopted for draft tube, runner and exhaust clearance. To use the wall function, the boundary layer mesh of each mesh scheme is densified, and the y + value on the wall boundary is controlled between 30 and 100, which effectively ensured the convergence of the wall function solution.
Before determining the final calculation mesh scheme, it is necessary to check the independence of the mesh schemes to ensure the accuracy of the numerical simulation results. The grid convergence method (GCI) based on Richardson's extrapolation method, the independence of three mesh schemes with different grid numbers is analyzed here. 33 Among them, the total number of grids in the three mesh schemes (N1: Coarse, N2: Medium, and N3: Fine) is N1 = 1,014,954, N2 = 2,384,831, and N3 = 5,361,851, respectively. The refinement factor of N1 and N2 is expressed by r 21 , and its value is 1.329, similarly, the refinement factor of N2 and N3 is expressed by r 32 , and its value is 1.310. The runner torque τ run is taken as the evaluation index, and the details are shown in Figure 4. The GCI value of coarse grid is 0.36%, and that of fine grid is 0066%, according to the mesh independence check requirements, the GCI values is less than 5% to meet the convergence requirements, the mesh independence is verified.
In comprehensive consideration of computing resources and mesh convergence indicators, the second set of mesh scheme N2 is finally selected to calculate. The final mesh scheme is shown in Figure 5.

| CFD simulation scheme
For the exhaust and pressurization process studied, due to its gas-liquid two phases and unstable simulation characteristics, it brings certain difficulties to the setting of the CFX numerical simulation. To completely simulate the whole exhaust and pressurization process, combined with the simulation setup experience and contours of the liquid phase volume fraction of the simulation results, according to the complexity degree gas-liquid two phases in different components and the stability of numerical simulation, the whole simulation process is divided into five stages: (a) stable development stage in draft tube, (b) gas-liquid two phases confusion stage in draft tube, (c) liquid phase entry stage in runner chamber, (d) gasliquid two phases confusion stage in runner chamber, (e) stable development stage in runner chamber.
On the basis of the exhaust rate, the entire exhaust time is 26 s. According to the confusion degree of gas-liquid two phases in each stage, the timestep distribution and simulation settings in each stage are shown in Table 2. The simulation process of in this study adopts the method of combining serial computing and parallel computing. The dynamic maximum iterative step setting is used in the calculation process to ensure that the calculation can converge in every two phases confusion stage.

| PREPARATION OF FLOW ANALYSIS
During the process of exhaust and pressurization, the liquid level in the draft tube rises gradually from the liquid level in the inlet tank. To further save the calculation resources, the steady simulation is used to determine the initial liquid level height for unsteady simulation at which the rotation of the runner does not produce entrainment on the free liquid level. As shown in Figure 6 Figure 6A), and the contour of liquid phase volume fraction f v inside the model are carried out. It can be seen that when the liquid level is not less than −0.2 m, the runner chamber is completely in the gas phase, that is, the runner has no entrainment effect on the free liquid level in the draft tube. Therefore, it is reasonable to select the liquid level height of −0.2 m as the initial liquid level height for unsteady simulation.

| Visualization
The initial liquid level height selected by the steady simulation is taken as the initial condition for the unsteady simulation. Based on the numerical simulation settings in Section 3.1, the contours of the liquid phase volume fraction for the whole exhaust and pressurization process are shown in Figure 7. All the contours are the cross section of fluid domain where the liquid and gas phases are located, taking the Y-Z plane at X = 0 as the cross section. Contours show the transient results for every 2 s. The curve of the liquid phase volume fraction over time is shown in Figure 8.
During the whole process of free liquid level rising, the liquid phase in the guide vane gradually increases, but before the liquid surface enters the runner chamber, the liquid phase in the guide vane is mostly accumulated water by the rotation and centrifugation of the runner. When t = 6 s, the free liquid level begins to enter the runner chamber, and the confusion of the flow pattern in the runner and guide vane is further enhanced. Since t = 14 s, the free liquid level has reached a certain height in the runner chamber. At this time, serious entrainment phenomenon will occur in the runner chamber, which will lead to the formation of cavities containing gas in the draft tube. The cavities will rupture with the increase of F I G U R E 5 Mesh scheme for objective simplified model. pressure and further exhaust gas. When t = 16 s, most of the gas in the fluid domain has been discharged, and the process of exhaust and pressurization has been basically completed. From the curve, it can also be seen that, consistent with the contour conclusion observed, the stable liquid phase volume fraction fluctuates around 0.95. When t = 18 s, with the further progress of the exhaust process, the cavities containing gas in the draft tube will break one after another, release gas, and further exhaust. This exhaust process will make the gas in the runner chamber and draft tube mainly concentrated at the runner outlet, and this part of gas can be discharged with the opening of the guide vane. After t = 22 s, the gas phase content in the cavity of the draft tube is relatively low. At this time, the guide vane and ball valve can be opened to operate the pump turbine in pump mode, and the gas collected at the runner outlet can be discharged to achieve pressure balance. In addition, with the increase of the liquid phase in the guide vane, a cavity containing gas will be gradually generated in the guide vane. For this part of the cavity is relatively close to the inlet of the guide vane, it will be washed out of the guide vane instantaneously with the opening of the guide vane, which will hardly increase the gas content in the unit. The contour results are similar to the experiment results monitored by Elena. 34 There are indeed quite complex changes in the composition of the liquid and gas phases, as well as the force and torque on the runner and guide vane during the exhaust and pressurization actually. The complex changes in this process can cause an increase in unit power and abnormal vibration or failure of the unit.

| Torque and force analysis
To obtain the transient torque and force characteristics in the process of exhaust and pressurization, in the unsteady simulation process, the torque of runner τ run , the axial force of runner F z-run , the radial force of runner F r-run , the torque of guide vane τ gv , the axial force of guide vane F z-gv , the radial force of guide vane F r-gv , the torque of single guide vane blade τ gvb , the axial force of single guide vane blade F z-gvb , the radial force of single guide vane blade F r-gvb are monitored. The STFT is used to analyze different monitoring signals, and the results are shown in Figures 9-17. STFT is a Fourier correlation transform. 35 Its principle is to select a time frequency localized window function to analyze the local sine frequency and amplitude intensity of the dynamic signal when the signal changes with time. Through STFT analysis, the energy density and frequency of the signal can be obtained, and the resolution of the signal depends on the selection of the window function. The nine physical signals monitored in this research are analyzed by STFT and the results are shown in Figures 9-17. The horizontal coordinates of these result graphs represent time, and vertical coordinates of these result graphs represent frequency (f), f b is blade passing frequency and its value is 180 Hz, f/f b is a dimensionless quantity characterizing frequency characteristics. Color represents amplitude information, and the amplitude increases gradually from violet to red and black. The Hanning window has a smaller side lobe peak value and larger side lobe spectrum peak attenuation speed, which    can make the side lobes cancel each other, eliminate high-frequency interference and energy leakage, greatly reduce spectrum leakage, and has better adjustment performance. Therefore, in this study, Hanning window is selected as the window function. In this study, the width of the Hanning window is 256. Based on Heisenberg's uncertainty principle: the time domain and frequency domain characteristics of signals cannot be accurately obtained at the same time. 36 The narrower the window function is, the higher the time domain resolution is. On the contrary, the higher the frequency domain resolution is. To achieve the balance between time resolution and frequency resolution, the frequency resolution is 720 Hz, and the time resolution is 0.001389 s.
As shown in Figure 9, the STFT transformation result of runner torque signal is shown. Before t = 3 s, the amplitude of different frequencies appeared as several high values. At t = 3.3-6.6 s, the main frequencies are 1.2f b and 1.6f b , and the amplitudes corresponding to these two main frequencies gradually decrease. During the whole process of exhaust and pressurization of pump turbine unit, the variation of torque signal is very complex, but its main frequency is mainly related to the blade passing frequency. Figure 10 shows the STFT transformation result of the runner axial force signal. At t = 3 s, there is a strong high amplitude band at each frequency. At t = 3.3-6.6 s, the main frequency is 0.4f b , at which the corresponding amplitude intensity is the largest, that is to say, the axial force signal of runner reaches its maximum at the main frequency related to the blade passing frequency. Figure 11 shows the STFT transformation results of the radial force signal of runner, high amplitude value occurred many times from t = 1.1 to 3.3 s. It can be seen that the radial force signal of runner has no characteristic frequency, which means that the radial force of the whole runner offsets each other among the blades and gets a relative balance.
It can be seen from the STFT transformation results of the three monitoring signals of the runner that before the free liquid surface completely enters into the runner chamber, the torque and force of the runner are complicated due to the entrainment effect, but all signal results indicate that the variation of torque and axial force are related to the blade passing frequency. Figure 12 shows the STFT transformation results of the guide vane torque signal. When t = 1.4 s, the amplitude of different frequencies appears high values. At t = 3.3-6.6 s, the main frequencies are 1.0f b and 1.6f b , and the amplitudes corresponding to these two main frequencies gradually decrease. As shown in Figure 13, the STFT transformation result of the guide vane axial force signal is shown. At t = 2.3 s, the amplitude of different frequencies has a super-high value. At t = 3.3-6.6 s, the main frequencies are 0.4f b , 1.0f b , and 1.6f b , and the amplitude corresponding to the main frequency decreases gradually. Figure 14 shows the STFT transformation result of the guide vane radial force signal, and the high amplitude value appears many times from t = 1.1 to 3.3 s. From this result, the radial force signal of the guide vane has almost no characteristic frequency, which means that the radial force of the entire guide vane offsets each other among the blades and gets a relative balance.
It can be seen from the STFT transformation results of the three monitoring signals of the guide vane that the STFT transformation results are similar to those of the runner: Before the free liquid surface completely enters into the runner chamber, the entrainment and centrifugal action of the runner causes the liquid phase in the guide vane to increase gradually. Its complex action process causes the torque and force of the guide vane to be F I G U R E 17 Short-time Fourier transform result of F r-gvb .
complex, but all signal results are related to the blade passing frequency.
As shown in Figure 15, the STFT transformation result of torque signal of a single guide vane blade is shown. When t = 2.7 s, the amplitude of different frequencies appears high value. At t = 3.3-6.6 s, the main frequencies are 0.4f b , 0.8f b , 1.2f b , and 1.6f b , and the amplitudes corresponding to these main frequencies gradually decrease. Among them, when f = 0.4f b , there is obvious high amplitude characteristic in the whole window. Figure 16 shows the STFT transformation results of the axial force signal of a single guide vane blade. At t = 2.7 s, the amplitude of different frequencies has a super-high value. At t = 3.3-6.6 s, the main frequencies are 0.4f b , 0.6f b , and 1.0f b , and the amplitude corresponding to the main frequency decreases gradually. Figure 17 shows the STFT transformation result of the radial force signal of a single guide vane blade. When t = 2.7 and 3.1 s, the amplitude of different frequencies appears higher, but the amplitude value is not high. At t = 3.3-6.6 s, the main frequencies are 0.4f b , 0.6f b , 0.8f b , 1.2f b , and 1.6f b , and the amplitude corresponding to the main frequency decreases gradually. This indicates that the result of radial force on a single guide vane blade is related to the blade passing frequency.
From the STFT transformation results of three monitoring measurements of a single guide vane blade, it shows that the torque and force of a single guide vane are complex with the gradual increase of the liquid phase in the guide vane caused by the entrainment and centrifugal action of the runner, but all the signal processing results are related to the passing frequency.
In general, in the process of exhaust and pressurization, the torque and force of the runner and guide vane for pump turbine become complicated due to the irregular and turbulent flow pattern. Transient frequency characteristics are obtained based on STFT transform. The results show that the main characteristic frequencies of the torque and stress of the runner and guide vane are the blade passing frequency and its harmonic frequency. It showed that the guide vanes are likely to experience fatigue damage during this process, which also indicates that during the process, abnormal vibration and guide vane accidents can be reduced by optimizing the rate of exhaust and the suction effect of runner. Due to the entrainment and centrifugal effect of the runner before the free liquid surface completely enters into the runner, a high amplitude band at full frequency occurs. Combined with the contours of liquid phase volume fraction in unsteady simulation and the torque and force of the runner and guide vane, the high frequency of which may come from the complex vortex motion in the flow channel caused by the entrainment and centrifugal action of the runner.

| CONCLUSIONS
In this research, a detailed unsteady numerical simulation of the exhaust and pressurization process of pump turbine is carried out to analyze the complex transient characteristics of the unit in this process. The achievements are summarized as follows: (1) Based on the analysis of the mechanism of the exhaust and pressurization process of pump turbine unit, the objective model is simplified, and the initial free liquid surface of the unsteady simulation is determined based on the steady state numerical simulation, which provides references and methods for improving the efficiency of the unsteady numerical simulation. (2) Based on the analysis of the contours of transient liquid phase volume fraction during the exhaust and pressurization process, the development process of the internal flow pattern is carried out, and the process is divided into five stages. It provides a reference for the total time and numerical simulation settings of the exhaust and pressurization process. (3) It shows the dynamic phenomenon of exhaust and pressurization process of pump turbines. By monitoring the torque, axial and radial force signals of the runner, guide vane and single guide vane blade based on STFT, it is revealed that the torque and force signals in the runner and guide vane are related to the blade passing frequency and harmonic frequency.
In this study, the dynamic phenomenon of exhaust and pressurization process is simulated in detail, and the transient characteristics, including transient flow and frequency characteristics, are analyzed, providing a reference for the simulation and analysis methods of similar process. However, for practical applications, the dynamic process is not always the same as this process. Therefore, the different change rules of exhaust rate still need further research, and more details of simulation should be added to better approach the actual situation.

NOMENCLATURE B
guide vane height CFD computational fluid dynamics D 1 inlet diameter of runner D 2 outlet diameter of runner f frequency F 1 blending function f b blade passing frequency F r-gv radial force of guide vane F r-gvb radial force of single guide vane blade F r-run radial force of runner f v liquid phase volume fraction F z-gv axial force of guide vane F z-gvb axial force of single guide vane blade F z-run axial force of runner g gravity acceleration GCI grid Convergence Method GGI general Grid Interface H r rated hydraulic head l liquid level height n rotation speed of runner n d rated rotation speed of runner P k production term of turbulent kinetic energy r 21 refinement factor r 32 refinement factor RNG renormalization group S invariant measure of the strain rate SST shear stress transport STFT short-time Fourier transform t time Z G number of guide vane blades Z r number of runner blades ω angular velocity μ dynamic viscosity β constant of the turbulence model α studied guide vane opening α 0 opening degree of guide vane and ball valve τ gv torque of guide vane τ gvb torque of single guide vane blade σ k constant of the turbulence model τ run torque of runner μ t turbulent eddy viscosity σ ω constant of the turbulence model β * constant of the turbulence model