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Pressure Analysis in the Draft Tube of a Pump-Turbine under Steady and Transient Conditions

: Pumped-storage power stations play a regulatory role in the power grid through frequent transition processes. The pressure pulsation in the draft tube of the pump-turbine under transient processes is important for safe operation, which is more intense than that in the steady-state condition. However, there is no effective method to obtain the exact pressure in the draft tube in the transient ﬂow ﬁeld. In this paper, the pressure in the draft tube of a pump-turbine under steady-state and transient conditions are studied by means of CFD. The reliability of the simulation method is veriﬁed by comparing the real pressure pulsation data with the test results. Due to the distribution of the pressure pulsation in the draft tube being complex and uneven, the location of the pressure monitoring points directly affects the accurate judgement of cavitation. Eight monitoring surfaces were set in the straight cone of the draft tube and nine monitoring points were set on each monitoring surface to analyze the pressure differences on the wall and inside the center of the draft tube. The relationships between the pressure pulsation value inside the center of the draft tube and on the wall are studied. The “critical” wall pressure pulsation value when cavitation occurs is obtained. This study provides references for judging cavitation occurrences by using the wall pressure pulsation value in practical engineering.


Introduction
Different from conventional power and pumping stations, a pumped-storage power station has a long water conveyance system, which means the transition process's effect on the flow in the pump-turbine is more significant. The field test results show that there is a significant amount of pressure pulsations at the draft tube during the transition process [1,2]. Some power stations even suffer from structural damage due to severe unit vibration [3].
Luo X. et al. [4] calculated and analyzed the runaway transient process of a tubular turbine, and found that a strong pressure pulsation was generated in the draft tube, at up to 104% of the test head, which threatened the safe and stable operation of the unit. Liu J. et al. [5] simulated the flow of a Francis turbine under runaway conditions and found that the pressure pulsation in the channel was as high as 16%. Furthermore, the characteristic curve changes significantly in the case of load rejections under different heads based on the characteristic curve of the pump-turbine. Thus, the pulsations in the flow field must be different. All these studies confirm that the pressure pulsation in a transient flow field is significantly stronger than that in a steady flow field, and the study of pressure pulsation in transient processes is an important aspect to ensure the safe and stable operation of the unit. However, it is difficult to measure the pulsation in a field unit operation, and more effective methods are needed to study the pulsation in the extreme conditions of the transition process.
Three dimensional CFD calculation is an effective method to obtain the pressure pulsation in the channel. At present, it is widely used in steady-state calculation and three-dimensional transient flow research of conventional water turbines [6], such as a pump [7], as well as in tubular, axial-flow, and Francis turbines with a short water delivery channel. However, when the water conveyance system is long, considering the three-dimensional calculation of the whole system, especially the cavitation model, the calculation will consume a lot of resources. In order to obtain the transient flow characteristics of the pipeline and the unit at the same time, the one-dimensional and three-dimensional coupling method is more reasonable. In this method, three-dimensional CFD is used for the pump-turbine, and one-dimensional CFD is used for other parts of the pipe system. The interface is set for real-time data exchange to ensure that the boundary conditions of the three-dimensional and one-dimensional models can be updated to advance the calculation.
Ruprecht [8] also realized the influence of draft tube flow on the hydraulic system under part load by using one-dimensional and partial three-dimensional simulation methods. You J. et al. [9] studied the cavitation characteristics of the pump-turbine and the pressure pulsation characteristics of the pump's powered-off runaway transition process by using the three-dimensional CFD method and one-dimensional three-dimensional coupling method, respectively. Li Z. [10] and Zhang X. et al. [11] used this method to simulate the runaway transition process of a pump-turbine, and the results showed that the pressure pulsation obtained by this method is in good agreement with the model test. Cherny S. et al. [12] also studied the transient flow of a hydraulic turbine by the coupling method. In the three-dimensional calculation, the reflection and superposition of the water hammer pressure in the one-dimensional pipeline were considered in the coupled calculation model. Therefore, in the process of guide vane closing, the calculated inlet pressure of the volute will rise higher, and the pressure pulsation in three-dimensional unit flow field is more severe, which can more truly reflect the interaction between the pipeline transient flow and three-dimensional transient flow in the unit.
Because the transition process will pass through a variety of operation areas in a short time, the actual process is often accompanied by the rise and fall of the pressure in the flow channel. The low pressure in the draft tube may cause a short-term cavitation phenomenon, which will significantly change the wave velocity of the local flow field and affect the stability of the whole hydraulic system of the hydropower station. For example, when Yang J. [13] conducted numerical studies on cavitation in a draft tube, it was found that the wave velocity in the cone tube may also be reduced to about 50 m/s when the volume of the vortex band accounts for a large proportion. However, we usually choose a fixed value of 1200 m/s in the calculation of the one-dimensional transition process, which is far from the same. Nicolet [14,15] and Alligné [16] et al. predicted the abnormal pressure vibration and surge phenomenon of power stations considering the influence of draft tube cavitation on the one-dimensional transient process. Meng L. [17] explored the influence of cavitation on the start-up process of a centrifugal pump by coupling the one-dimensional calculation software Flowmaster with the three-dimensional calculation software Fluent. The results show that the transient characteristics of the cavitation are more complicated than the results of an ordinary steady-state cavitation calculation when cavitation occurs in the transient process, because the transient process will pass through many operating regions in a short time.
In addition, the dynamic characteristics and collapse of the cavitation chamber may cause extreme accidents, such as "lift", which has been confirmed by many researchers. For example, Zhang X. [18] et al. have studied the cavitation of a draft tube during the transition process. They think that the larger the volume of the cavitation chamber in the draft tube, the more significant the sudden increase in pressure caused by collapse, and the greater the additional impact on the runner in the axial and radial directions. Pejovic S.  [19] pointed out that in the reverse-S region of the pump-turbine, the phenomenon of liquid column separation and bridging may occur due to cavitation, and there will be a large counter water pressure. The phenomenon of pump-turbine backwater hammer shows that when the suction height of the pump-turbine is large, local cavitation will appear at the bottom of the discharge cone, and the cavitation chamber will expand and collapse, which will form a large backwater hammer pressure.
It can be seen that the cavitation flow field in the draft tube affects the large pulsation results of the flow field of the pump-turbine. The influence of cavitation on stability should also be considered in the analysis of the transition process. At present, few studies focus on the cavitation characteristics of the transient process, with the research on draft tube cavitation mainly aimed at the modeled runner calculation and experimented with in a steady flow field [20].
In engineering, the draft tube pressure pulsation is often monitored by setting a pressure transducer on the wall to judge whether cavitation occurs in a pump-turbine and hydro-turbine. However, due to the existence of a vortex, the pressures on the wall and inside the center of the draft tube are different. Furthermore, according to the results of steady cavitation research, the cavitation vortex rope appears first in the central region of the draft tube. The pressure inside the center of the draft tube is a more accurate reference to judge cavitation, which is impossible to get.
Therefore, in order to accurately reflect the internal flow characteristics of the unit in the transient process, it is necessary to further study the pressure pulsation in the transient process of a pumped-storage power station through one-dimensional and threedimensional coupling methods, and analyze the pressure differences in the central region and wall of the draft tube under the transient process. This also forms the basis for improving the runner optimization design and putting forward reasonable control measures to ensure the safe and stable operation of the power plant.

Research Object
In this paper, a pump-turbine is taken as the research object. The calculation domains, including the spiral casing, guide vane, stay vane, runner, and draft tube were modeled, as shown in Figure 1a. The computing domain of the one-dimensional calculation is shown in Figure 1b, which includes an upstream reservoir, downstream reservoir, gate shaft, diversion tunnel, high-pressure pipeline, pump-turbine, and tailrace tunnel. of liquid column separation and bridging may occur due to cavitation, and there will be a large counter water pressure. The phenomenon of pump-turbine backwater hammer shows that when the suction height of the pump-turbine is large, local cavitation will appear at the bottom of the discharge cone, and the cavitation chamber will expand and collapse, which will form a large backwater hammer pressure. It can be seen that the cavitation flow field in the draft tube affects the large pulsation results of the flow field of the pump-turbine. The influence of cavitation on stability should also be considered in the analysis of the transition process. At present, few studies focus on the cavitation characteristics of the transient process, with the research on draft tube cavitation mainly aimed at the modeled runner calculation and experimented with in a steady flow field [20].
In engineering, the draft tube pressure pulsation is often monitored by setting a pressure transducer on the wall to judge whether cavitation occurs in a pump-turbine and hydro-turbine. However, due to the existence of a vortex, the pressures on the wall and inside the center of the draft tube are different. Furthermore, according to the results of steady cavitation research, the cavitation vortex rope appears first in the central region of the draft tube. The pressure inside the center of the draft tube is a more accurate reference to judge cavitation, which is impossible to get.
Therefore, in order to accurately reflect the internal flow characteristics of the unit in the transient process, it is necessary to further study the pressure pulsation in the transient process of a pumped-storage power station through one-dimensional and three-dimensional coupling methods, and analyze the pressure differences in the central region and wall of the draft tube under the transient process. This also forms the basis for improving the runner optimization design and putting forward reasonable control measures to ensure the safe and stable operation of the power plant.

Research Object
In this paper, a pump-turbine is taken as the research object. The calculation domains, including the spiral casing, guide vane, stay vane, runner, and draft tube were modeled, as shown in Figure 1a. The computing domain of the one-dimensional calculation is shown in Figure 1b, which includes an upstream reservoir, downstream reservoir, gate shaft, diversion tunnel, high-pressure pipeline, pump-turbine, and tailrace tunnel.  The rated speed of the pump-turbine is 300 rpm, the specific speed is 167.3 (specific

Numerical Calculation Method
In the three-dimensional simulation, the quality of the grid has a great influence on the calculation results. Considering the calculation resources, calculation accuracy, and grid quality, after the grid independence test, the number of total cells used was 6,823,783. The mesh statistics are shown in Table 1. In this study, the grid density for the runner and draft tube are set higher to capture a more detailed flow for further analyzing. The corresponding measuring points were arranged in 8 sections of the draft tube, as shown in Figure 2a. A total of 9 measuring points were set at each section, as shown in Figure 2b. The monitoring point marked as P1-1 indicates the number of the monitoring point P1 on Plane 1. The specific parameters of the section position are shown in Table 2, where z is the Z-axis position of the monitoring section, the reference value is the center line of the guide vanes, and r is the radius of the plane. reservoir is 400 m, and the minimum water level is 376.5 m; the normal water level of the lower reservoir is 103.7 m, and the minimum water level is 65 m. The length of the diversion pipeline is about 1078 m, and that of the tailrace pipeline is about 472 m.

Numerical Calculation Method
In the three-dimensional simulation, the quality of the grid has a great influence on the calculation results. Considering the calculation resources, calculation accuracy, and grid quality, after the grid independence test, the number of total cells used was 6,823,783. The mesh statistics are shown in Table 1. In this study, the grid density for the runner and draft tube are set higher to capture a more detailed flow for further analyzing. The corresponding measuring points were arranged in 8 sections of the draft tube, as shown in Figure 2a. A total of 9 measuring points were set at each section, as shown in Figure 2b. The monitoring point marked as P1-1 indicates the number of the monitoring point P1 on Plane 1. The specific parameters of the section position are shown in Table 2, where z is the Z-axis position of the monitoring section, the reference value is the center line of the guide vanes, and r is the radius of the plane.  In this paper, three-dimensional calculation is the main method for the analysis of the steady-state flow. The one-dimensional and three-dimensional coupling method was adopted for the study of the transient condition. Considering the calculation resources  In this paper, three-dimensional calculation is the main method for the analysis of the steady-state flow. The one-dimensional and three-dimensional coupling method was adopted for the study of the transient condition. Considering the calculation resources and calculation time, an indirect coupling method was adopted in this study. A onedimensional numerical method was used to simulate the load rejection transient process, and the results were compared with the test results. According to the one-dimensional calculation results, the pressure values at the spiral casing inlet and draft tube outlet used in the three-dimensional simulations were obtained. The pressure value was calculated according to the water level upstream and downstream of the reservoir and the pipeline loss in the one-dimensional calculation.
Sliding mesh was used at the interface of the guide vane and runner and runner and draft tube. The SST k-ω model was selected as the turbulence model. The time step of the unsteady calculation under the steady-state condition was a 1/200 rotation period, t = 0.001 s, and the maximum number of iterations at each time step was set to 20, and the convergence residual set as 1 × 10 −4 .
For the calculation of cavitation multiphase flow in the hydraulic machinery, the homogeneous model is generally used, which assumes that all of the phases share the turbulent field, using a cavitation model to describe the mass transfer rate between the phases simulated. In this study, the ZGB (Zwart-Gerber-Belamri) cavitation model [21] was selected as the cavitation model, and the saturated vapor pressure was set at 3540 Pa.
The ZGB cavitation model uses the following evaporation and condensation terms: .
The bubble radius R B is replaced by the gas core radius R nuc , which represents the volume fraction of the gas core; F vap and F cond are the evaporation and condensation coefficient, respectively. The default values of each parameter are R nuc = 1 × 10 −6 m, α nuc = 5 × 10 −4 , F vap = 50, and F cond = 0.01.

Analysis of the Results of the Steady-State Conditions
Firstly, the steady-state condition of a 100% load was numerically simulated, and the upstream and downstream reservoir water level was 388.7 and 92.23, respectively. The calculated guide vane opening of 31.63 • was given according to the real machine test data.
In order to verify the pressure pulsation calculation results in this paper, the pressure monitoring points at the real machine were set at the corresponding locations during simulation, which are shown in Figure 3.     Table 3. It can be seen that the mean pressure at the spiral casing inlet is in good agreement with the calculated value, and the mean pressure pulsation at the draft tube inlet has an error of about 8%. The difference is related to the wall roughness of the model and the inaccurate estimation of pipeline loss.

Steady-State Condition without Cavitation
The pressure distribution in the draft tube in a rotation cycle under 100% load condition is shown in Figure 4, in which T is the rotation period of the runner. It can be seen that there is an obvious low-pressure area at the cone of the draft tube, and the pressure difference at the wall and at the center of draft tube is large. We also analyzed the pressure at Plane 4, which is shown in Figure 5. It can be seen that there is a low-pressure area in the center of the straight cone section of draft tube. On the same cross section, the pressure difference at the different measuring points is about 1 × 10 5 Pa. Thus, it is necessary to study the pressure pulsation at different locations.

Steady-State Condition without Cavitation
The pressure distribution in the draft tube in a rotation cycle under 100% load condition is shown in Figure 4, in which T is the rotation period of the runner. It can be seen that there is an obvious low-pressure area at the cone of the draft tube, and the pressure difference at the wall and at the center of draft tube is large. We also analyzed the pressure at Plane 4, which is shown in Figure 5. It can be seen that there is a low-pressure area in the center of the straight cone section of draft tube. On the same cross section, the pressure difference at the different measuring points is about 1 × 10 5 Pa. Thus, it is necessary to study the pressure pulsation at different locations.

Steady-State Condition of Cavitation
In order to ensure the operation safety of the unit, the pumped-storage power station is usually designed with no cavitation operation. Therefore, it is impossible to simulate cavitation under the given upstream and downstream water level according to field test results. In order to study the pressure pulsation in the cavitation flow field, the pressure value at the outlet boundary conditions was reduced according to the Thoma number shown in (3), to simulate the cavitation flow. Then, the pressure pulsation of the measuring points when serious cavitation occurs in draft tube and the pressure values, obtained from the field test, were analyzed.
where Pout is the outlet pressure in draft tube; Pv, the saturated vapor pressure (3540 Pa); hz, the suction height benchmarked against the center of guide vane; and H, the net head.
To compare the pressure near the wall and at the center point in the draft tube for the forecast of cavitation, the pressure pulsation of each monitoring point was processed. Firstly, the mean pressure of each monitoring point P2-P9 on a monitoring surface within the whole period (5 rotation cycles were taken in this paper) was calculated. Then, the mean pressure of P2-P9 in a section was averaged again. Finally, the averaged wall pressure and the center point pressure on each section without cavitation and with cavitation are shown in Figure 6. It can be seen that the central monitoring point obviously is the most accurate point to judge cavitation, no matter the cavitation being present or not. Furthermore, the pressure distribution in the draft tube is quite different with or without cavitation. When severe cavitation occurs in the draft tube, the pressure value of the center point is always the saturated vapor pressure of 3540 Pa. However, the average pressure on the wall is quite different. It is necessary to find a reasonable way to judge the pressure in the draft tube.

Steady-State Condition of Cavitation
In order to ensure the operation safety of the unit, the pumped-storage power station is usually designed with no cavitation operation. Therefore, it is impossible to simulate cavitation under the given upstream and downstream water level according to field test results. In order to study the pressure pulsation in the cavitation flow field, the pressure value at the outlet boundary conditions was reduced according to the Thoma number shown in (3), to simulate the cavitation flow. Then, the pressure pulsation of the measuring points when serious cavitation occurs in draft tube and the pressure values, obtained from the field test, were analyzed.
where P out is the outlet pressure in draft tube; P v , the saturated vapor pressure (3540 Pa); h z , the suction height benchmarked against the center of guide vane; and H, the net head.
To compare the pressure near the wall and at the center point in the draft tube for the forecast of cavitation, the pressure pulsation of each monitoring point was processed. Firstly, the mean pressure of each monitoring point P2-P9 on a monitoring surface within the whole period (5 rotation cycles were taken in this paper) was calculated. Then, the mean pressure of P2-P9 in a section was averaged again. Finally, the averaged wall pressure and the center point pressure on each section without cavitation and with cavitation are shown in Figure 6. It can be seen that the central monitoring point obviously is the most accurate point to judge cavitation, no matter the cavitation being present or not. Furthermore, the pressure distribution in the draft tube is quite different with or without cavitation. When severe cavitation occurs in the draft tube, the pressure value of the center point is always the saturated vapor pressure of 3540 Pa. However, the average pressure on the wall is quite different. It is necessary to find a reasonable way to judge the pressure in the draft tube.

Analysis of Transient Condition Results
During the operation of the unit, when the load does not change, the main parameters of the unit, such as the water head, flow, speed, torque, shaft power, pressure, guide vane opening, etc., do not change, which is a stable working condition. When the hydraulic unit suddenly rejects the load, the flow rate will decrease sharply, the unit speed will rise rapidly from the rated speed, and then decrease after reaching the extreme value. After several pulsations, the unit will be stable in no-load operation, which is a load rejection process. During the load rejection process, the minimum pressure in the draft tube inlet is most likely to occur in the speed maximum moment, in which the cavitation is easy to present. Thus, three-dimensional CFD simulation at this moment was also conducted to analyze the pressure pulse inside the draft tube based on the calculation of the one-dimensional transient process.
In order to verify the accuracy of the calculation method, the 100% load rejection condition, testing the downstream water level, was calculated and analyzed. The working head is 296.47 m, and the guide vane is closed normally.
In the one-dimensional calculation, the influence of the spiral casing and draft tube were considered. The spiral casing and draft tube were treated as equivalent. The equivalent volute diameter is the inlet diameter, which is 2.8 m. The equivalent length is the distance from inlet to tongue, which is 24 m. Draft tube equivalent is divided into three parts: straight cone section, elbow section and diffusion section. The diameter and length of the straight cone part are 2.7 m, as well as the height of the straight cone section. The diameter of the bent shaft section is the average value of the equivalent diameter of the inlet and outlet section, which is 3.116 m, and the length is the length of the elbow section, which is 6.354 m. The diameter of the diffusion section is the average value of the equivalent diameter of inlet and outlet section of 4.645 m, and the length is the length of the diffusion section.
When the unit loses the load, the guide vane starts to close after 10 s delay. The unit speed begins to rise after load rejection. The speed reaches the maximum value in 7.65 s, and the maximum relative value of the speed is 1.407 times that of the rated speed. The speed begins to decline after the guide vane closes, as shown in Figures 7 and 8. In Figure  7, the relative opening degree means the ratio of the actual opening of the guide vane to the maximum opening. The outlet pressure of the ball valve is the first to increase in the

Analysis of Transient Condition Results
During the operation of the unit, when the load does not change, the main parameters of the unit, such as the water head, flow, speed, torque, shaft power, pressure, guide vane opening, etc., do not change, which is a stable working condition. When the hydraulic unit suddenly rejects the load, the flow rate will decrease sharply, the unit speed will rise rapidly from the rated speed, and then decrease after reaching the extreme value. After several pulsations, the unit will be stable in no-load operation, which is a load rejection process. During the load rejection process, the minimum pressure in the draft tube inlet is most likely to occur in the speed maximum moment, in which the cavitation is easy to present. Thus, three-dimensional CFD simulation at this moment was also conducted to analyze the pressure pulse inside the draft tube based on the calculation of the one-dimensional transient process.
In order to verify the accuracy of the calculation method, the 100% load rejection condition, testing the downstream water level, was calculated and analyzed. The working head is 296.47 m, and the guide vane is closed normally.
In the one-dimensional calculation, the influence of the spiral casing and draft tube were considered. The spiral casing and draft tube were treated as equivalent. The equivalent volute diameter is the inlet diameter, which is 2.8 m. The equivalent length is the distance from inlet to tongue, which is 24 m. Draft tube equivalent is divided into three parts: straight cone section, elbow section and diffusion section. The diameter and length of the straight cone part are 2.7 m, as well as the height of the straight cone section. The diameter of the bent shaft section is the average value of the equivalent diameter of the inlet and outlet section, which is 3.116 m, and the length is the length of the elbow section, which is 6.354 m. The diameter of the diffusion section is the average value of the equivalent diameter of inlet and outlet section of 4.645 m, and the length is the length of the diffusion section.
When the unit loses the load, the guide vane starts to close after 10 s delay. The unit speed begins to rise after load rejection. The speed reaches the maximum value in 7.65 s, and the maximum relative value of the speed is 1.407 times that of the rated speed. The speed begins to decline after the guide vane closes, as shown in Figures 7 and 8. In Figure 7, the relative opening degree means the ratio of the actual opening of the guide vane to the maximum opening. The outlet pressure of the ball valve is the first to increase in the whole process and reaches its maximum value of 4.37 × 10 6 Pa in 8.224 s, and then decreases, as shown in Figure 9. The inlet pressure of the draft tube first decreases, and then increases, as shown in Figure 10. From the comparison between the calculation results and the test data, the calculation results are consistent with the test data in the transient process. The calculated error of the maximum speed is 1.4%, and the pressure error at the outlet of the ball valve is 7.17%. On the whole, the one-dimensional calculation results are reliable and effective.
Energies 2021, 14, x FOR PEER REVIEW 9 of 14 whole process and reaches its maximum value of 4.37 × 10 6 Pa in 8.224 s, and then decreases, as shown in Figure 9. The inlet pressure of the draft tube first decreases, and then increases, as shown in Figure 10. From the comparison between the calculation results and the test data, the calculation results are consistent with the test data in the transient process. The calculated error of the maximum speed is 1.4%, and the pressure error at the outlet of the ball valve is 7.17%. On the whole, the one-dimensional calculation results are reliable and effective.   whole process and reaches its maximum value of 4.37 × 10 6 Pa in 8.224 s, and then decreases, as shown in Figure 9. The inlet pressure of the draft tube first decreases, and then increases, as shown in Figure 10. From the comparison between the calculation results and the test data, the calculation results are consistent with the test data in the transient process. The calculated error of the maximum speed is 1.4%, and the pressure error at the outlet of the ball valve is 7.17%. On the whole, the one-dimensional calculation results are reliable and effective.    According to the results of the one-dimensional calculation, in the process of load rejection, the lowest pressure of the draft tube inlet appears at the maximum speed, so the instantaneous value at this moment is taken to analyze the flow in the draft tube; that is, the three-dimensional simulation boundary conditions at the spiral casing inlet and draft tube outlet are set according to the calculated one-dimensional result at 6.9 s in the process of load rejection.  According to the results of the one-dimensional calculation, in the process of load rejection, the lowest pressure of the draft tube inlet appears at the maximum speed, so the instantaneous value at this moment is taken to analyze the flow in the draft tube; that is, the three-dimensional simulation boundary conditions at the spiral casing inlet and draft tube outlet are set according to the calculated one-dimensional result at 6.9 s in the process of load rejection. According to the results of the one-dimensional calculation, in the process of load rejection, the lowest pressure of the draft tube inlet appears at the maximum speed, so the instantaneous value at this moment is taken to analyze the flow in the draft tube; that is, the three-dimensional simulation boundary conditions at the spiral casing inlet and draft tube outlet are set according to the calculated one-dimensional result at 6.9 s in the process of load rejection.
Due to the minimum pressure at the draft tube inlet being lower than the saturation pressure in the one-dimensional calculation (see Figure 10), the cavitation must be present in the draft tube during the load rejection, which was verified by the three-dimensional simulation (see Figure 11). The cavitation volume is represented by the iso-surface of the vapor volume fraction of 0.1, which has been proven to have the most similar shape with the reality vortex rope [22]. A long vortex rope presents in the draft tube cone at time t = 6.9 s; the cavitation is serious in the draft tube at this time, and the cavitation mainly occurs at the center of the draft tube.
Due to the minimum pressure at the draft tube inlet being lower than the saturation pressure in the one-dimensional calculation (see Figure 10), the cavitation must be present in the draft tube during the load rejection, which was verified by the three-dimensional simulation (see Figure 11). The cavitation volume is represented by the iso-surface of the vapor volume fraction of 0.1, which has been proven to have the most similar shape with the reality vortex rope [22]. A long vortex rope presents in the draft tube cone at time t = 6.9 s; the cavitation is serious in the draft tube at this time, and the cavitation mainly occurs at the center of the draft tube. The pressure distributions at different sections in the draft tube at this dangerous moment are also shown in Figure 12. It shows that the pressure at each section center is lower than the saturated vapor pressure of 3540 Pa, which means that cavitation appears. However, the pressure on the side wall is much larger than the saturated vapor pressure.  The pressure distributions at different sections in the draft tube at this dangerous moment are also shown in Figure 12. It shows that the pressure at each section center is lower than the saturated vapor pressure of 3540 Pa, which means that cavitation appears. However, the pressure on the side wall is much larger than the saturated vapor pressure. Due to the minimum pressure at the draft tube inlet being lower than the saturation pressure in the one-dimensional calculation (see Figure 10), the cavitation must be present in the draft tube during the load rejection, which was verified by the three-dimensional simulation (see Figure 11). The cavitation volume is represented by the iso-surface of the vapor volume fraction of 0.1, which has been proven to have the most similar shape with the reality vortex rope [22]. A long vortex rope presents in the draft tube cone at time t = 6.9 s; the cavitation is serious in the draft tube at this time, and the cavitation mainly occurs at the center of the draft tube. The pressure distributions at different sections in the draft tube at this dangerous moment are also shown in Figure 12. It shows that the pressure at each section center is lower than the saturated vapor pressure of 3540 Pa, which means that cavitation appears. However, the pressure on the side wall is much larger than the saturated vapor pressure.   indicating that the vortex rope is located at Planes 1-5. It can be concluded that when there is a vortex rope passing through the section, the average pressure on the draft tube wall is quite different, with an average value of 2.8 × 10 4 Pa to 5 × 10 4 Pa. At the Plane 6 section, the central point pressure is at the critical value of saturated vapor pressure, which is 3540.681 Pa, which means the vortex rope has just disappeared. Thus, the average wall pressure at this section can be seen as the critical value of cavitation. That is to say, the maximum average pressure near the wall of the section to judge cavitation is about 5.1 × 10 4 Pa. Based on the analysis in this paper, it can be seen that the critical average value of the wall monitoring pressure is about 5.25 × 10 4 ± 1000 Pa under this condition. When the actual measured wall pressure is lower than the critical value, the occurrence of cavitation can be preliminarily judged. Further confirmation can be carried out by CFD simulation.  Figure 13 presents the mean pressure of the wall point and the center point at each section at 6.9 s of transient process. The minimum pressure values at the center point of Planes 1-5 are 3540 Pa, while the minimum pressure values of Planes 6-8 are greater than 3540 Pa, indicating that the vortex rope is located at Planes 1-5. It can be concluded that when there is a vortex rope passing through the section, the average pressure on the draft tube wall is quite different, with an average value of 2.8 × 10 4 Pa to 5 × 10 4 Pa. At the Plane 6 section, the central point pressure is at the critical value of saturated vapor pressure, which is 3540.681 Pa, which means the vortex rope has just disappeared. Thus, the average wall pressure at this section can be seen as the critical value of cavitation. That is to say, the maximum average pressure near the wall of the section to judge cavitation is about 5.1 × 10 4 Pa. Based on the analysis in this paper, it can be seen that the critical average value of the wall monitoring pressure is about 5.25 × 10 4 ± 1000 Pa under this condition. When the actual measured wall pressure is lower than the critical value, the occurrence of cavitation can be preliminarily judged. Further confirmation can be carried out by CFD simulation.

Conclusions
In this paper, the pressure in the draft tube of a pump-turbine are studied and analyzed under the steady-state condition and load rejection transient process.
The average wall pressure calculated and monitored in the in the steady-state condition was first carried out; the results showed the central point pressure being obviously different from that near the wall, and that the center point is the most accurate point to judge cavitation.
Under the transient condition, the moment of cavitation occurrence is identified firstly (for example, the load rejection condition studied in this paper is the point of the maximum speed), and then the instantaneous value of the pressure at that time is calculated and compared. The average pressure at the monitored points in 6.9 s under the transient condition was obtained. Combined with the "critical values" of cavitation present at the central point, the mean wall pressure "critical value" was obtained. For this research, the "critical" wall pressure pulsation is 5.25 × 10 4 ± 1000 Pa.

Conclusions
In this paper, the pressure in the draft tube of a pump-turbine are studied and analyzed under the steady-state condition and load rejection transient process.
The average wall pressure calculated and monitored in the in the steady-state condition was first carried out; the results showed the central point pressure being obviously different from that near the wall, and that the center point is the most accurate point to judge cavitation.
Under the transient condition, the moment of cavitation occurrence is identified firstly (for example, the load rejection condition studied in this paper is the point of the maximum speed), and then the instantaneous value of the pressure at that time is calculated and compared. The average pressure at the monitored points in 6.9 s under the transient condition was obtained. Combined with the "critical values" of cavitation present at the central point, the mean wall pressure "critical value" was obtained. For this research, the "critical" wall pressure pulsation is 5.25 × 10 4 ± 1000 Pa.
For application in engineering, by comparing the average value of the wall pressure pulsation obtained from the transient or steady-state tests with the "critical" value proposed in this paper, the critical conditions of cavitation can be preliminarily predicted. The results of this paper provide a new method for cavitation identification based on the pressure value from the wall monitoring points in engineering.