Stability Analysis of Vaneless Space in High-Head Pump-Turbine under Turbine Mode: Computational Fluid Dynamics Simulation and Particle Imaging Velocimetry Measurement

When the Francis-type reversible pump-turbine runs under partial load, the pressure pulsation amplitude and frequency in vaneless space are high, posing a serious threat to the stability of unit operation. Water presents weak compressibility in a high-head pump-turbine, thereby affecting the amplitude–frequency characteristics of pressure pulsation. This study used numerical simulations in a model and prototype pump-turbine and particle image velocimetry (PIV) in a model pump-turbine to examine the internal flow field and pressure pulsation characteristics and determine the effect of the flow in the vaneless space on the amplitude–frequency characteristics of the pressure pulsation. The pressure pulsation amplitude–frequency characteristics were verified through prototype tests. The effects of the weak compressibility of the water on the propagation law of pressure pulsation throughout the flow passage of the prototype and model pump-turbine were roughly similar but exhibited certain differences. Considering the weak compressibility of water, the pressure pulsation fluctuations in each flow passage of the prototype and model pump-turbine exhibit varying degrees of improvement, which is more obvious at the prototype scale. Therefore, the pressure wave disturbance caused by the weak compressibility of the water has different effects on the prototype scale and model scale of the high-head Francis pump-turbine.


Introduction
Currently, pumped-storage power stations are globally used as energy storage devices with huge capacities. As of 2020, the total installed capacity of China's pumped-storage power plants was 40 million kilowatts. A pump-turbine is the core component of a pumpedstorage power station, and its operational stability directly affects the benefit of the power station and the stability of the power grid. Due to the increasing head of pumped-storage units, the water embodies weak compressibility. This characteristic has an impact on both the flow and pressure pulsation characteristics inside the unit and is of high research value.
In pump-turbines, the vaneless space between the guide vane and the runner blades is the most unstable region of flow [1][2][3]. The rotating stalls in the vaneless space [4,5] and the rotor-stator interaction (RSI) [6,7] threaten the stable operation of the pump-turbine. The direct manifestation of the hydraulic stability is the pressure pulsation phenomenon,  22 22 mesh. The grid is shown in Figure 2. The grid independence was also verified. When the number of grids reached 4.9 million, the efficiency and head fluctuation were less than 0.05%. This value met the accuracy required by the numerical simulation. The number of grids in each part is shown in Table 2. The average value of y+ in the near-wall region of the runner and guide vane was less than 15, which met the requirements of the turbulence model.   Using ICEM and Turbogrid, the model pump-turbine was divided into a hexahedral mesh. The grid is shown in Figure 2. The grid independence was also verified. When the number of grids reached 4.9 million, the efficiency and head fluctuation were less than 0.05%. This value met the accuracy required by the numerical simulation. The number of grids in each part is shown in Table 2. The average value of y+ in the near-wall region of the runner and guide vane was less than 15, which met the requirements of the turbulence model.
Using ICEM and Turbogrid, the model pump-turbine was divided into a hexahedral mesh. The grid is shown in Figure 2. The grid independence was also verified. When the number of grids reached 4.9 million, the efficiency and head fluctuation were less than 0.05%. This value met the accuracy required by the numerical simulation. The number of grids in each part is shown in Table 2. The average value of y+ in the near-wall region of the runner and guide vane was less than 15, which met the requirements of the turbulence model.

Boundary Conditions and Operating Points
SST k-ω turbulence model [33][34][35] was used in the CFD simulation by ANSYS CFX. The operating point parameters are listed in Table 3. The flow inlet and opening outlet are adopted. The conditions of the inlet and speed of rotation of the runner were selected according to the results of the model test. For interface processing, the general grid interface was used to connect the parts. In the interface type in the stable calculation, type None was used for the static interface, and the frozen rotor approach was applied for the dynamic and static interfaces. In the unstable calculation, the static-static interface still adopted the type None, whereas the dynamic-static interface adopted a transient rotor-stator interaction. The calculation time step was 1/360 of the runner rotation period, 0.000340977 s for the prototype and 0.000138899 s for the model. A root mean square of residuals of 10 -5 was used as the convergence condition for the iterative calculation. Further, 20 runner rotation periods were calculated, and the last 5 periods were selected for analysis. The unit flow and unit speed are expressed as follows: where n 11 is the unit speed (rpm), Q 11 is the unit flow (L/s), n is the speed (rev/min), Q is the flow (L/s), H is the working head (m), and D 2 is the nominal diameter of the runner (m).  Figure 3 shows the location of pressure pulsation monitoring points during the transient calculation, where 108 of these are located in the vaneless space, with three layers of 36 points each. The monitoring points in the vaneless space are named in a clockwise direction, and owing to space limitations, some monitoring points such as VL36 and VL10 are marked in.

Weakly Compressible Equation
The equation expressing the state of the water is based on the relationship between the water's density and pressure, determined from the acoustic pressure Equation (3) [11,36].
where r is the actual density of water; 0 r is the reference density of water, 998 kg/m 3 ; 0 P is the reference pressure, 101,325 Pa; and 0 c is the speed of sound, 1400 m/s.

Computational Fluid Dynamics Results
The dimensionless pressure coefficient, Cp, and ∆H/H are reference values for quantifying the intensity of pressure pulsations in pump-turbines, as shown in Equation (4).
where Cp is the dimensionless pressure pulsation coefficient (%), i p is the corresponding pressure at point i (Pa), p is the time-averaged pressure, H is the head (m), ρ is the density of water, and g is the gravitational acceleration. ∆H is the pressure pulsation peakto-peak value, m;

Weakly Compressible Equation
The equation expressing the state of the water is based on the relationship between the water's density and pressure, determined from the acoustic pressure Equation (3) [11,36].
where ρ is the actual density of water; ρ 0 is the reference density of water, 998 kg/m 3 ; P 0 is the reference pressure, 101,325 Pa; and c 0 is the speed of sound, 1400 m/s.

Computational Fluid Dynamics Results
The dimensionless pressure coefficient, C p , and ∆H/H are reference values for quantifying the intensity of pressure pulsations in pump-turbines, as shown in Equation (4).
where Cp is the dimensionless pressure pulsation coefficient (%), p i is the corresponding pressure at point i (Pa), p is the time-averaged pressure, H is the head (m), ρ is the density of water, and g is the gravitational acceleration. ∆H is the pressure pulsation peak-to-peak value, m; p imax , p imin is the corresponding maximum and minimum pressure values at point i, Pa. Figures 4 and 5 show the time-domain and frequency-domain plots of the pressure pulsation monitoring point, VL6 of the prototype test, respectively, where f 1 is the bladepassing frequency (BPF), 9f n . As seen in the figure, the primary frequency of the pressure pulsation monitoring point, VL6, in the vaneless space of the prototype test was BPF 9f n (i.e., 73.32 Hz), and the relative pressure pulsation amplitude was 1.97%. The secondary frequency was 18 f n , that is, 146.91 Hz, and the relative pressure pulsation amplitude was Machines 2022, 10, 143 6 of 20 0.94%. Additionally, the relative pressure pulsation amplitude, 4f n , was relatively small (32.58 Hz) at a relative pressure pulsation amplitude of 0.28%.    Figure 6 shows the frequency-domain and time-domain diagrams of pulsation considering the compressible and non-compressible cases of water. E both the compressible and non-compressible simulations can better reflect t frequencies in the prototype test, but the difference in their prediction of t frequency amplitude is relatively large. The main frequency f1 relative pressure p amplitude is 1.32% in the incompressible simulation, while the f1 relative pulsation amplitude is 1.86% in the compressible simulation, which is close prototype test results. In addition, the incompressible simulation fails to ob characteristic frequency, f3, while the relative pressure pulsation amplitud characteristic frequency f3 in the compressible simulation is 0.25%. Figure 6b sh time-domain diagram of the pressure pulsation of the pressure monitoring poin the vaneless space within one rotating cycle of the runner considering the com and non-compressible cases of the water, and the pressure pulsation in the vanele within one rotating cycle of the runner in both the compressible and non-com simulations shows periodic changes, but the relative pressure pulsation amplitu weakly compressible condition is larger than that in the non-compressible co Therefore, the results yielded when the weak compressibility of the water in the n simulation is considered are closer to those of the prototype test than the results when the compressibility of the water is not considered.    Figure 6 shows the frequency-domain and time-domain diagrams of pulsation considering the compressible and non-compressible cases of water. E both the compressible and non-compressible simulations can better reflect t frequencies in the prototype test, but the difference in their prediction of t frequency amplitude is relatively large. The main frequency f1 relative pressure p amplitude is 1.32% in the incompressible simulation, while the f1 relative pulsation amplitude is 1.86% in the compressible simulation, which is close prototype test results. In addition, the incompressible simulation fails to ob characteristic frequency, f3, while the relative pressure pulsation amplitud characteristic frequency f3 in the compressible simulation is 0.25%. Figure 6b sh time-domain diagram of the pressure pulsation of the pressure monitoring poin the vaneless space within one rotating cycle of the runner considering the com and non-compressible cases of the water, and the pressure pulsation in the vanel within one rotating cycle of the runner in both the compressible and non-com simulations shows periodic changes, but the relative pressure pulsation amplitu weakly compressible condition is larger than that in the non-compressible c Therefore, the results yielded when the weak compressibility of the water in the n simulation is considered are closer to those of the prototype test than the result when the compressibility of the water is not considered.  Figure 6 shows the frequency-domain and time-domain diagrams of pressure pulsation considering the compressible and non-compressible cases of water. Evidently, both the compressible and non-compressible simulations can better reflect the main frequencies in the prototype test, but the difference in their prediction of the main frequency amplitude is relatively large. The main frequency f 1 relative pressure pulsation amplitude is 1.32% in the incompressible simulation, while the f 1 relative pressure pulsation amplitude is 1.86% in the compressible simulation, which is closer to the prototype test results. In addition, the incompressible simulation fails to obtain the characteristic frequency, f 3 , while the relative pressure pulsation amplitude of the characteristic frequency f 3 in the compressible simulation is 0.25%. Figure 6b shows the time-domain diagram of the pressure pulsation of the pressure monitoring point VL6 in the vaneless space within one rotating cycle of the runner considering the compressible and non-compressible cases of the water, and the pressure pulsation in the vaneless space within one rotating cycle of the runner in both the compressible and non-compressible simulations shows periodic changes, but the relative pressure pulsation amplitude in the weakly compressible condition is larger than that in the non-compressible condition. Therefore, the results yielded when the weak compressibility of the water in the numerical simulation is considered are closer to those of the prototype test than the results yielded when the compressibility of the water is not considered.  Table 3).
The internal flow field characteristics of the prototype and the model were compared and analyzed, and Figure 7 shows the relative velocity distribution of the annulus within the vaneless space.   The internal flow field characteristics of the prototype and the model were compared and analyzed, and Figure 7 shows the relative velocity distribution of the annulus within the vaneless space.  Table 3).
The internal flow field characteristics of the prototype and the model were compared and analyzed, and Figure 7 shows the relative velocity distribution of the annulus within the vaneless space.     Comparison of the relative pressure pulsation amplitude of the four monitoring points in the vaneless space of the prototype and the model machine indicates that the distribution pattern of the relative pressure pulsation amplitude in the vaneless space of the prototype pump-turbine is consistent with that of the model pump-turbine, but the relative pressure pulsation amplitude in the vaneless space of the prototype pump-turbine is larger than that of the model pump-turbine, and the compressible simulation is larger than the incompressible simulation. This difference in the relative pressure pulsation amplitude caused by the compressibility of water is more obvious in the prototype pumpturbine.
The pressure pulsation time-frequency characteristics of the prototype and model pump-turbine vaneless space monitoring point VL6 are further analyzed, as shown in Figure 9, where the blade rotation frequency fn is 8.15 Hz and 19.99 Hz for the prototype pump-turbine and model pump-turbine, respectively, under the rated operating condition of the turbine, and its BPF is 9fn. Comparison of the relative pressure pulsation amplitude of the four monitoring points in the vaneless space of the prototype and the model machine indicates that the distribution pattern of the relative pressure pulsation amplitude in the vaneless space of the prototype pump-turbine is consistent with that of the model pump-turbine, but the relative pressure pulsation amplitude in the vaneless space of the prototype pump-turbine is larger than that of the model pump-turbine, and the compressible simulation is larger than the incompressible simulation. This difference in the relative pressure pulsation amplitude caused by the compressibility of water is more obvious in the prototype pump-turbine.
The pressure pulsation time-frequency characteristics of the prototype and model pump-turbine vaneless space monitoring point VL6 are further analyzed, as shown in Figure 9, where the blade rotation frequency f n is 8.15 Hz and 19.99 Hz for the prototype pump-turbine and model pump-turbine, respectively, under the rated operating condition of the turbine, and its BPF is 9f n .
The main frequencies of pressure pulsation in the vaneless space of both the prototype and the model pump-turbine are BPF, and its harmonic frequencies are 9f n and 18f n . In addition, there is a relatively small pressure pulsation amplitude of 4f n . The blade-passing frequency and its harmonic frequencies 9f n and 18f n are RSI frequencies. 4f n is obtained from the upward propagation of the runner channel.
Whether compressibility is considered or not, the numerical simulation results are consistent with the main frequency of pressure pulsation of the test results, but there is still a certain difference in amplitude. Under the incompressible condition, the main frequency amplitude of pressure pulsation in the vaneless space of the prototype pump-turbine is significantly smaller than that of the prototype test results, and the main frequency amplitude of pressure pulsation in the vaneless space of the model pump-turbine is lower than that of the model test results, but the difference is smaller. Under the weakly compressible condition, the main frequency amplitude of pressure pulsation in the vaneless space of the prototype and model pump-turbine is increased, and the main frequency amplitude of pressure pulsation in the vaneless space of the prototype pump-turbine is higher than that of the model pump-turbine when the weak compressibility of the water is considered. Generally, when considering the weak compressibility of water, the main frequency amplitude of pressure pulsation is closer to the experimental value, and it is more obvious in the numerical simulation of the prototype pump-turbine. This is because the speed of sound in water is approximately 1400 m/s, the wavelength is approximately 5-30 m, the RSI frequency of the pump-turbine is approximately 50-200 Hz, and its corresponding acoustic wave wavelength is similar to the scale of the prototype pump-turbine, so this pressure wave perturbation caused by the compressibility of the water causes acoustic resonance in the flow field and enhances its pressure pulsation amplitude. amplitude caused by the compressibility of water is more obvious in the prototype pumpturbine.
The pressure pulsation time-frequency characteristics of the prototype and model pump-turbine vaneless space monitoring point VL6 are further analyzed, as shown in Figure 9, where the blade rotation frequency fn is 8.15 Hz and 19.99 Hz for the prototype pump-turbine and model pump-turbine, respectively, under the rated operating condition of the turbine, and its BPF is 9fn. The main frequencies of pressure pulsation in the vaneless space of both the prototype and the model pump-turbine are BPF, and its harmonic frequencies are 9fn and 18fn. In addition, there is a relatively small pressure pulsation amplitude of 4fn. The bladepassing frequency and its harmonic frequencies 9fn and 18fn are RSI frequencies. 4fn is obtained from the upward propagation of the runner channel.
Whether compressibility is considered or not, the numerical simulation results are consistent with the main frequency of pressure pulsation of the test results, but there is still a certain difference in amplitude. Under the incompressible condition, the main frequency amplitude of pressure pulsation in the vaneless space of the prototype pumpturbine is significantly smaller than that of the prototype test results, and the main frequency amplitude of pressure pulsation in the vaneless space of the model pumpturbine is lower than that of the model test results, but the difference is smaller. Under the weakly compressible condition, the main frequency amplitude of pressure pulsation in the vaneless space of the prototype and model pump-turbine is increased, and the main frequency amplitude of pressure pulsation in the vaneless space of the prototype pumpturbine is higher than that of the model pump-turbine when the weak compressibility of the water is considered. Generally, when considering the weak compressibility of water, the main frequency amplitude of pressure pulsation is closer to the experimental value, and it is more obvious in the numerical simulation of the prototype pump-turbine. This is because the speed of sound in water is approximately 1400 m/s, the wavelength is approximately 5-30 m, the RSI frequency of the pump-turbine is approximately 50-200 Hz, and its corresponding acoustic wave wavelength is similar to the scale of the prototype pump-turbine, so this pressure wave perturbation caused by the compressibility of the water causes acoustic resonance in the flow field and enhances its pressure pulsation amplitude.

Particle Image Velocimetry Methods
The pressure pulsation amplitude in the vaneless space is affected by compressibility and the vortex motion in the vaneless space, which is examined using the PIV test.

Particle Image Velocimetry System and Model Pump-Turbine Parameters
The PIV test system is composed of tracer particles, a pulsed laser and sheet light source lens group, a CCD camera, a synchronizer, and a special workstation. The main technical indicators of the PIV test system are listed in Table 4. The parameters of the model pump-turbine used in the test are consistent with those of the CFD model pump-turbine.

Test Site
The structural system of the test bench includes a high-pressure water tank, a model water pump-turbine test section (including a dynamometer), a low-pressure water tank, and a vacuum pump system, which is a closed-loop test system. The system was supplied by two 24SA-10 single-stage double-suction centrifugal water pumps. The pumps were connected to two 850 kW bi-directionally rotating DC motors. The DC motors were powered by two sets of fully digital DC speed-regulating devices. The pump unit can realize stable operation within a range of 400-1000 r/min. The maximum lift of the single pump was 85 m. The maximum flow rate was 1.0 m 3 /s, and the maximum speed was 1000 r/min. The maximum water head of the test bench was 170 m, and the speed stabilization accuracy was 0.1%. The diameter of the high-pressure pipeline of the system was 500 mm, and that of the low-pressure pipeline was 700 mm. A bi-directional electromagnetic flowmeter was used for flow calibration. The diameter of the high-pressure water tank and the lowpressure water tank of the test bench was 2.4 m. These large containers stabilize the water flow before and after the model unit, and the water flows through the electromagnetic flowmeter. The parameters of the test bench are listed in Table 5. The position and arrangement of the camera are shown in Figure 10. The imaging area is the position of the vaneless space between the runner and guide vane, and the imaging area is determined by the position of the two screws on the end shaft of the guide vane. The pulsed laser was directed to the experimental observation area from the transparent window at the volute. The test operating points are listed in Table 6.     Figure 11 shows the internal flow field in the vaneless space under different operating conditions. Machines 2022, 10, x FOR PEER REVIEW 12 of 20 the vortex structure changes from a single vortex structure to a double vortex structure, and the vortices are distributed in pairs and rotate in opposite directions. With the progressive increase in load, the unsteady vortex structure in the vaneless space is further reduced at 100% load condition, and the vortex structure is changed to a single vortex structure with a smaller vortex structure. At the same load, the vortex structure increases with the increase in unit speed, and the position moves toward the guide vane exit runner. Analysis of continuous images of OP1, 2, and 3 test results. Figures 12-14 show the variation of the internal flow field in the vaneless space over time at different speed operating points, OP1, OP2, and OP3, of the unit under 50% load condition of the turbine, respectively. The rotor rotation times for one revolution at working points OP1, OP2, and OP3 are 0.0525 s, 0.0510 s, and 0.0502 s, respectively, and the sampling frequency is 15 Hz. To analyze the variation of the internal flow field in the vaneless space over time, six different moments, T1, T2, T3, T4, T5, and T6, of the same time interval ΔT are selected. Under 50% load conditions, the vortex structure in the vaneless space is relatively large and is distributed singly in the observation area, and the vortex rotation is clockwise. At the operating point, OP1, the vortex structure is relatively small, and the vortex structure appears close to the runner outlet, while at OP2 and OP3, where the unit speed is relatively high, this vortex structure is very large, and the pressure at the center of the vortex is extremely low owing to the very high vortex center flow velocity at this time. This local low pressure caused by the vortex causes periodic disturbance to the high-pressure flow field in the vaneless space where the rotation of the runner significantly contributes to the high amplitude of pressure pulsation in the vaneless space under partial load.

Test Results
Under the 75% load condition, the flow field in the vaneless space is relatively stable and smooth at the lower speed, OP4, but as the unit speed increases, a clockwise rotating vortex and a counterclockwise rotating vortex are distributed in pairs at OP5 and OP6, and the vortex structure is reduced to a certain extent compared with the 50% load vortex structure. At 100% load condition, the flow field in the vaneless space is relatively stable, the vortex structure in the vaneless space is very small at OP8 and OP9, and the distribution of vortex is not even observed at OP7.
At 50% load condition, the unsteady vortex structure in the vaneless space is single vortex distribution, and the vortex structure is larger. When the unit load increases to 75%, the vortex structure changes from a single vortex structure to a double vortex structure, and the vortices are distributed in pairs and rotate in opposite directions. With the progressive increase in load, the unsteady vortex structure in the vaneless space is further reduced at 100% load condition, and the vortex structure is changed to a single vortex structure with a smaller vortex structure. At the same load, the vortex structure increases with the increase in unit speed, and the position moves toward the guide vane exit runner.
Analysis of continuous images of OP1, 2, and 3 test results.  At 50% load, the flow regime inside the vaneless space is more turbulent; large-scale unsteady vortex structures appear frequently in the observation cycle and are observed once every ∆T apart. At operating point OP1, the unsteady vortex structure in the vaneless space is located near the runner inlet side, and the vortex structure is relatively small. As the rotational speed increases, the unsteady vortex structure in the vaneless space becomes larger at operating points OP2 and OP3, and the flow regime in the vaneless space deteriorates. The large-scale unsteady flow structure appears in the vaneless space between the guide vane outlet and the runner inlet, and this unsteady vortex structure with high frequency and large-scale structure disturbs the flow field in the vaneless space, thereby decreasing the stability of the flow field inside the vaneless space. At 50% load, the flow regime inside the vaneless space is more turbulent; large-scale unsteady vortex structures appear frequently in the observation cycle and are observed once every ΔT apart. At operating point OP1, the unsteady vortex structure in the vaneless space is located near the runner inlet side, and the vortex structure is relatively small. As the rotational speed increases, the unsteady vortex structure in the vaneless space becomes larger at operating points OP2 and OP3, and the flow regime in the vaneless  At 50% load, the flow regime inside the vaneless space is more turbulent; large-scale unsteady vortex structures appear frequently in the observation cycle and are observed once every ΔT apart. At operating point OP1, the unsteady vortex structure in the vaneless space is located near the runner inlet side, and the vortex structure is relatively small. As the rotational speed increases, the unsteady vortex structure in the vaneless space becomes larger at operating points OP2 and OP3, and the flow regime in the vaneless This unstable vortex structure is generated in the runner channel, so its characteristic frequency is related to the rotating frequency of the runner blades, and because of its development to the vaneless space, it intensifies the RSI effect, so this unstable vortex structure causes the main frequency amplitude of the pressure pulsation in the vaneless space to increase significantly, i.e., the BPF 9f n . With the change of working conditions, in the operating range of the unit, the unit load is attenuated as it increases and is enhanced as the unit speed increases in the same load condition.

Propagation Law of Pressure Pulsation in the Vaneless Space
The source of high-frequency pressure pulsation in the vaneless space, which is the location of the strongest pressure pulsation in a pump-turbine, was analyzed in CFD and PIV studies; therefore, the propagation law was analyzed. Figure 15 shows the relative pressure pulsation amplitude distribution in the full-flow channel of the prototype and model pump-turbine under compressible and incompressible conditions, and the pressure pulsation monitoring points were selected as VO1, ST2, GV2, VL6, RV1, RV2, RV3, DT1, and DT2. A total of nine monitoring point locations were compared and analyzed. structure causes the main frequency amplitude of the pressure pulsation in the vaneless space to increase significantly, i.e., the BPF 9fn. With the change of working conditions, in the operating range of the unit, the unit load is attenuated as it increases and is enhanced as the unit speed increases in the same load condition.

Propagation Law of Pressure Pulsation in the Vaneless Space
The source of high-frequency pressure pulsation in the vaneless space, which is the location of the strongest pressure pulsation in a pump-turbine, was analyzed in CFD and PIV studies; therefore, the propagation law was analyzed. Figure 15 shows the relative pressure pulsation amplitude distribution in the fullflow channel of the prototype and model pump-turbine under compressible and incompressible conditions, and the pressure pulsation monitoring points were selected as VO1, ST2, GV2, VL6, RV1, RV2, RV3, DT1, and DT2. A total of nine monitoring point locations were compared and analyzed.  Figure 15 shows that the relative pressure pulsation amplitude distribution pattern is the same for the prototype and model pump under compressible and incompressible conditions, the strongest pressure pulsation area is located in the vaneless space, and the relative pressure pulsation amplitude gradually decreases upstream and downstream along the vaneless space. Regardless of whether the weak compressibility of the water is considered, the pressure pulsation intensity of the prototype pump-turbine in the fullflow channel exceeds that of the model pump-turbine. The compressible simulation results are higher than the incompressible simulation results, and this difference in numerical calculation owing to the consideration of the weak compressibility of the water is more obvious in the prototype pump-turbine. Figure 16a,b shows that the pressure pulsation at monitoring point, VO1, in the inlet runner of the prototype pump-turbine spiral casing is mainly affected by the vortex rope of the draft tube under the incompressible condition, showing low-frequency vibration with low amplitude, while the pressure pulsation under the compressible condition is mainly affected by the RSI, showing amplitude fluctuations corresponding to the number of runner blades. The pressure pulsation at monitoring point VO1 of the model pump-  Figure 15 shows that the relative pressure pulsation amplitude distribution pattern is the same for the prototype and model pump under compressible and incompressible conditions, the strongest pressure pulsation area is located in the vaneless space, and the relative pressure pulsation amplitude gradually decreases upstream and downstream along the vaneless space. Regardless of whether the weak compressibility of the water is considered, the pressure pulsation intensity of the prototype pump-turbine in the full-flow channel exceeds that of the model pump-turbine. The compressible simulation results are higher than the incompressible simulation results, and this difference in numerical calculation owing to the consideration of the weak compressibility of the water is more obvious in the prototype pump-turbine. Figure 16a,b shows that the pressure pulsation at monitoring point, VO1, in the inlet runner of the prototype pump-turbine spiral casing is mainly affected by the vortex rope of the draft tube under the incompressible condition, showing low-frequency vibration with low amplitude, while the pressure pulsation under the compressible condition is mainly affected by the RSI, showing amplitude fluctuations corresponding to the number of runner blades. The pressure pulsation at monitoring point VO1 of the model pumpturbine shows the characteristics of high-frequency vibration, but the fluctuation pattern of pressure pulsation under incompressible conditions is related to the number of runner blades, and there are nine fluctuations in the runner cycle, which is influenced by RSI, and four fluctuations of pressure pulsation under compressible conditions are four times in a runner cycle, which is related to the flow separation in the runner.

Full-Flow Channel Pressure Pulsation Characteristics
In Figure 16c In Figure 16c,d, the pressure pulsation at the stay vane monitoring point ST2 of t prototype pump-turbine is affected by both the vortex rope of the draft tube and R under incompressible conditions, and the pressure pulsation shows the characteristics both low-frequency vibration and high-frequency vibration. Further, a significant increa is observed in the amplitude of the pressure pulsation under compressible condition where one of the high-frequency components is significantly enhanced. The time-doma characteristics of the pressure pulsation at the stay vane runner monitoring point ST2 the model pump-turbine are relatively consistent, both being high-frequency vibratio related to the number of blades.
The pressure pulsation fluctuation at the guide vane monitoring point GV2 of t prototype and model pump-turbine in Figure 16e,f is affected by the RSI under bo compressible and incompressible conditions, showing a regular fluctuation related to t number of blades, but the pressure pulsation amplitude fluctuation of the prototy pump-turbine is more obvious under compressible conditions. In Figure 16g,h, t fluctuation of the pressure pulsation amplitude at the prototype and model pump-turbi runner monitoring point RV2 is mainly influenced by the RSI, which is related to t number of guide vanes, and the pressure pulsation fluctuation of the prototype pum turbine is more obvious than that of the model pump-turbine.
In Figure 16i,j, the straight cone section of the draft tube of the prototype pum turbine, DT1, mainly vibrates at low frequencies under incompressible conditions, wi small pulsation amplitudes, while under compressible conditions, the pressure pulsati fluctuations are significantly enhanced, and the frequency of amplitude fluctuations  Figure 16g,h, the fluctuation of the pressure pulsation amplitude at the prototype and model pump-turbine runner monitoring point RV2 is mainly influenced by the RSI, which is related to the number of guide vanes, and the pressure pulsation fluctuation of the prototype pump-turbine is more obvious than that of the model pump-turbine.
In Figure 16i,j, the straight cone section of the draft tube of the prototype pump-turbine, DT1, mainly vibrates at low frequencies under incompressible conditions, with small pulsation amplitudes, while under compressible conditions, the pressure pulsation fluctuations are significantly enhanced, and the frequency of amplitude fluctuations is significantly increased, with fluctuation patterns related to the phenomenon of flow separation in the runner. The pressure pulsation fluctuations of the model pump-turbine draft tube straight cone section monitoring point DT1 in the incompressible conditions are small and almost tend to be stable, while the pulsation amplitude of low-frequency fluctuations appears in the compressible conditions. The frequency-domain characteristics of the full-flow channel pressure pulsation for the prototype and model pump-turbine under compressible and incompressible conditions are shown in Figure 17. The similarities in the propagation pattern of the characteristic frequency of pressure pulsations along the flow channel in the prototype and model pump-turbine under compressible and incompressible conditions are displayed in Figure 17. In the prototype pump-turbine, the pressure pulsation special sign frequencies are 1fn, 4fn, 9fn, 18fn, 22fn, and 36fn. 1fn is the RSI frequency between the runner and the draft tube. 1fn increases upstream along the runner outlet to the spiral casing inlet, decreases rapidly downstream through the vaneless space at the runner outlet, decreases downstream along the runner outlet to the elbow section of the draft tube, and decreases during the propagation downstream along the runner outlet to the elbow section of the draft tube. The compressible simulation is consistent with the incompressible simulation. 4fn is associated with a deliquescence in the runner inlet, a characteristic frequency that is not captured in the incompressible simulation. 4fn is strongest in the compressible simulation at the runner inlet position and decreases upstream and downstream along the flow channel from the runner inlet.
9fn, 18fn, and 36fn are the RSI frequencies and their harmonic frequencies in the vaneless space between the runner and the guide vane. In the incompressible simulation, the RSI frequencies and their harmonic frequencies in the vaneless space decrease from the vaneless space upstream and downstream along the flow channel. 9fn and 18fn in the compressible simulation maintain the same propagation pattern as the incompressible, but 36fn increases at the stay vane during propagation upstream along the flow channel, probably owing to the pressure wave disturbance caused by the weak compressibility of the water amplifying this characteristic frequency. 22fn is subject to the RSI frequency between the guide vane and the runner, i.e., the guide vane passing frequency, which is highest in amplitude in the runner inlet channel and decreases along the upstream and downstream from the runner inlet channel.
The characteristic frequencies of pressure pulsation in the model pump-turbine are consistent with those of the prototype pump-turbine, and their main characteristic frequencies are 1fn, 4fn, 9fn, 18fn, and 22fn, and the compressible simulation and The similarities in the propagation pattern of the characteristic frequency of pressure pulsations along the flow channel in the prototype and model pump-turbine under compressible and incompressible conditions are displayed in Figure 17. In the prototype pump-turbine, the pressure pulsation special sign frequencies are 1f n , 4f n , 9f n , 18f n , 22f n , and 36f n . 1f n is the RSI frequency between the runner and the draft tube. 1f n increases upstream along the runner outlet to the spiral casing inlet, decreases rapidly downstream through the vaneless space at the runner outlet, decreases downstream along the runner outlet to the elbow section of the draft tube, and decreases during the propagation downstream along the runner outlet to the elbow section of the draft tube. The compressible simulation is consistent with the incompressible simulation. 4f n is associated with a deliquescence in the runner inlet, a characteristic frequency that is not captured in the incompressible simulation. 4f n is strongest in the compressible simulation at the runner inlet position and decreases upstream and downstream along the flow channel from the runner inlet. 9f n , 18f n , and 36f n are the RSI frequencies and their harmonic frequencies in the vaneless space between the runner and the guide vane. In the incompressible simulation, the RSI frequencies and their harmonic frequencies in the vaneless space decrease from the vaneless space upstream and downstream along the flow channel. 9f n and 18f n in the compressible simulation maintain the same propagation pattern as the incompressible, but 36f n increases at the stay vane during propagation upstream along the flow channel, probably owing to the pressure wave disturbance caused by the weak compressibility of the water amplifying this characteristic frequency. 22f n is subject to the RSI frequency between the guide vane and the runner, i.e., the guide vane passing frequency, which is highest in amplitude in the runner inlet channel and decreases along the upstream and downstream from the runner inlet channel.
The characteristic frequencies of pressure pulsation in the model pump-turbine are consistent with those of the prototype pump-turbine, and their main characteristic frequencies are 1f n , 4f n , 9f n , 18f n , and 22f n , and the compressible simulation and incompressible simulation propagation laws are more similar. The 1f n increases upstream from the runner outlet channel to the runner inlet channel and decreases rapidly after passing through the vaneless space. 9f n and 18f n decrease upstream and downstream from the vaneless space. 22f n from the runner inlet channel along the upstream and downstream also maintains a decreasing trend. The difference between the compressible and incompressible simulations comes from the different regularity of 4f n propagation upstream. In the incompressible simulation, 4f n gradually decreases along the upstream and downstream from the runner inlet channel, while in the compressible simulation, 4f n gradually decreases along the downstream and gradually increases along the upstream to the spiral casing inlet.

Conclusions
In this paper, the pressure pulsation characteristics of the vaneless space of the prototype and the model pump-turbine are compared and analyzed based on the weak compressibility of the water, and the vortex flow in the vaneless space of the model pump-turbine is measured using the PIV test.

1.
Under compressible conditions, the pressure pulsation value calculated by CFD is closer to the test measured value of the prototype pump-turbine, and the increased unit instability caused by the compressibility of water should be considered in the design of high-head pump-turbines.

2.
Important causes of instability in the vaneless space of pump-turbines include RSI and vortex flow; RSI is the main source of instability, while the vortex motion in the vaneless space enhances the pressure pulsation caused by RSI. The main frequencies of pressure pulsation in the vaneless space of the prototype and the model pump-turbine are all RSI frequencies, and their harmonic frequencies, in addition to the relative pressure pulsation amplitude, are low in relation to the runner inlet runner pressure side flow separation 4f n .

3.
This unsteady vortex structure in the vaneless space owing to the operation of the unit in a non-optimal operating zone, has been identified by PIV tests as exacerbating flow instability in the vaneless space. This is caused by the negative attack angle resulting from the mismatch between the runner inlet water flow angle and the runner inlet blade placement angle, which generates a stall vortex on the pressure side of the runner inlet and a backflow structure near the runner inlet. With the change of operating conditions, the disturbance decreases with the increase in the unit load in the unit operating range and increases with the increase in the speed under the same load condition.

4.
The influence of the weak compressibility of water on the propagation law of pressure pulsation in the full-flow channel of the prototype and model pump-turbine is generally similar but with some variability. Considering the weak compressibility of water, the fluctuation of pressure pulsation in each flow channel of the prototype and model pump-turbine is increased to different degrees, but this effect is more obvious at the prototype scale, and the harmonic frequency 36f n of RSI at the prototype scale is amplified during the propagation along the upstream. This indicates that the pressure wave disturbance due to the weak compressibility of water is different in the prototype and model scales of the high-head Francis-type pump-turbine. This difference is attributed to acoustic resonance.