Mechanism analysis of the abnormal head-cover vibration of an ultra-high-head pump-turbine operating in pump mode by CFD and FEM

ABSTRACT

Hydraulic excitation causes abnormal head-cover vibration in some pump-turbines, particularly as design heads continue to rise. Pump-turbines with heads exceeding 600 m have reported such vibrations in pump mode, and it is urgent to reveal the reasons. A study was conducted on a head-cover of a pump-turbine with a maximum head of 660 m by 3D computational fluid dynamics (CFD) and finite element method (FEM) numerical simulations to investigate the hydraulic excitation source and vibration mechanism. Results show that the intensive vibration is related to the coincidence of frequency and mode between abnormal pressure pulsations with frequency 6–8f_n and the low-order wet mode of the head-cover. A new flow pattern termed the rolling vortex was discovered in the vaneless space under high-head pumping conditions. The 6–8f_n pressure pulsations result from the interaction between rolling vortices and guide-vanes, similar to the rotor-stator interaction. The flow mechanism of the rolling vortices is that the outflow velocity near the hub at the runner outlet is higher than that near the shroud, due to the positive blade lean angle and the X-shape distorted blades. Suppressing rolling vortices may help eliminate the abnormal head-cover vibration, which will be attempted in further study.

KEYWORDS:

• Pump-turbine
• head-cover
• pump mode
• pressure pulsations
• flow-induced vibration
• numerical simulation

Nomenclature

$H_{max}$	=	Maximum pump head (m)
$n_r$	=	Rated rotational speed (rpm)
$f_n$	=	Rated rotational frequency (Hz)
$P_r$	=	Rated power in pumping mode (MW)
$Z_r$	=	Number of runner blades
$Z_g$	=	Number of guide-vanes
$Z_s$	=	Number of stay-vanes
$\gamma$	=	Guide-vane opening (degree)
$p$	=	Static pressure (Pa)
$p_g$	=	Gauge pressure (Pa)
$a$	=	Acoustic velocity (m/s)
${\rho }_0$	=	Density of water at 1 atm (kg/m3)
${\rho }_l$	=	Density of water at local pressure (kg/m3)
$k$	=	Oil film stiffness of water guide bearings (N/m)
$y^{+}$	=	Y plus value near the wall
$C_p$	=	Pressure coefficient
$C_v$	=	Velocity coefficient
$H^{\mathrm{\prime }}$	=	Normalized head
${\gamma }^{\mathrm{\prime }}$	=	Normalized guide-vane opening
$D^{\mathrm{\prime }}$	=	Normalized displacement
${\sigma }^{\mathrm{\prime }}$	=	Normalized equivalent stress
$\nu$	=	Number of diametrical nodes

1. Introduction

Due to its flexible and fast peak-shaving ability, pumped storage can fully compensate for the randomness, fluctuation, and intermittency of new energy sources such as wind and solar energy, thereby improving the reliability of the power grid (Barbour et al., 2016). Applying pump-turbine motor-generator units with higher head and larger capacity is the main way to increase the capacity of pumped storage power stations and reduce construction costs. Currently, the highest rated head of operating pumped storage power stations in China exceeds 700 m, and the largest capacity is 400 MW. Such high parameters pose challenges to the safe and stable operation of pump-turbine units. The head-cover is a crucial sealing component that experiences complex stress conditions (Zhi & Ma, 2012), particularly directly bearing the pressure in the flow passage. The rise in head leads to a direct increase in hydrostatic pressure, and, significantly, it results in the runner design favouring a radial flow pattern. The long, flat, and distorted runner passage, along with the narrow vaneless space, combined with multiple operating modes and frequent transient processes, lead to highly noticeable pressure pulsations (Zuo et al., 2015). Complex and intense pressure pulsations may cause obvious vibration and strong noise. An ultra-high-head pump-turbine recently put into operation in China has experienced a vertical vibration alarm on the head-cover in pump mode, and subsequently commissioned pump-turbines at this power station have all encountered this phenomenon. When the head exceeds 640 m, the vertical vibration amplitude of the head-cover increases as the head rises. The frequency components with increased amplitude are concentrated in the range of 6–8f_n (where f_n is the rotational frequency of the runner). However, this phenomenon does not occur in turbine mode. Prolonged operation under vibration conditions will increase the risk of failure and damage to structural components such as the head-cover and bolts (Egusquiza et al., 2012; Frunzaˇverde et al., 2010), posing a threat to the safe and stable operation of pumped storage power plants. Therefore, it is urgent to identify the causes of the abnormal vibration of the head-cover.

The unstable pressure pulsations are considered to be the main factor causing vibration in hydraulic machinery. Identifying the hydraulic excitation is the first step in studying flow-induced vibration issues. The pressure pulsations in the pump-turbine are usually caused by unstable flow patterns, such as vortex rope in draft-tube (Favrel et al., 2017; Valentín et al., 2017), interblade vortices (Magnoli et al., 2019; Sun et al., 2020), rotational stall (Hasmatuchi et al., 2011; Y. Liu et al., 2021), cavitation (Al-Obaidi, 2019, 2023), and rotor-stator interaction (Al-Obaidi & Alhamid, 2023; Ye et al., 2022). These flow patterns appear in different operating conditions, generating different pressure pulsation frequencies. However, the abnormal vibration phenomenon studied in our research cannot be linked to these pressure pulsations. Vortex rope in draft-tube, interblade vortices, and rotating stalls usually result in pressure pulsations with frequencies lower than 3f_n. These phenomena generally do not occur near the rated pumping condition. Unstable flow phenomena such as cavitation and Karman vortex shedding (Dörfler et al., 2013) typically result in high-frequency pressure pulsations, which were also ruled out in field tests. From a frequency characteristic perspective, rotor-stator interaction (RSI) and phase resonance (PR) are more likely to cause abnormal pressure pulsations in the frequency band 6–8f_n. Whether operating in pump or turbine modes, the high-amplitude pressure pulsations caused by RSI in the vaneless space are the main vibration source of pump-turbines. It is usually considered that RSI is caused by the interaction between the stationary guide-vanes and the rotating runner blades, and its frequencies are related to the number of runner blades and guide-vanes as well as the rotational frequency (Tanaka, 2011). In addition, improper matching of these three parameters can result in phase resonance (Fang et al., 2021; Lv et al., 2022), wherein pressure pulsations induced by RSI propagate to the spiral-case, causing amplitude superposition due to the simultaneous phase. Regardless of whether it is pressure pulsation caused by RSI or PR, the frequency of these pulsations remains constant during stable operation, and does not change with the increase of head as observed in abnormal vibration frequencies. Given that the abnormal vibration frequency has not been previously reported, the abnormal vibration we encountered may be attributed to a special flow phenomenon.

The natural properties of the structure play a crucial role in determining its response to specific frequency excitations. Considering the contact between the lower surface of the head-cover and water, the fluid-added mass effect cannot be ignored. Acoustic fluid-structure interaction technology is widely used to assess the wet modal characteristics of hydraulic machinery, and many experiments have confirmed its accuracy. However, the modal analysis only provides information on individual modes, while the actual dynamic response may involve a combination of multiple modes. Therefore, a more comprehensive understanding of vibration mechanisms can be achieved by analyzing the dynamic response under hydraulic excitation. With the advancement of computational capabilities and coupling algorithms, the fluid-structure interaction (FSI) method can accurately simulate and analyze fluid-induced vibration problems (Dompierre & Sabourin, 2010; Khalfaoui et al., 2022). Chen et al. (2021) studied the dynamic stress of the runner during the startup process of a pump-turbine and found that the dynamic stress of the rotor increases rapidly with the increase of the rotational speed. Li et al. (2022) found that the additional mass and damping of water can greatly reduce the high-order natural frequency of the runner; the pressure pulsations with a small amplitude but a frequency coincidence with the natural frequency of the runner can cause resonance and enlarge the amplitude of the specific vibration mode. At present, many researchers have studied the distributions of stress and strain of the head-cover, as well as the strength of the bolts (Luo et al., 2019; Wang et al., 2022). However, there are few studies on the abnormal vibration response of the head-cover.

It is generally believed that when the frequency of hydraulic excitation is close to the natural frequency of the structure, resonance may occur, leading to a significant increase in vibration amplitude. Research by He et al. (2019) suggests that evaluating resonance only based on the similarity between excitation frequency and natural frequency is not comprehensive; it is also necessary to compare the mode consistency between the two. The pressure pulsation characteristics in the pump-turbine exhibit distinct spatial features, making it challenging to reflect modal information by analyzing the spectral information of only a few monitoring points. The dynamic mode decomposition (DMD) technique can extract the primary modes of pressure pulsations. Each mode has a singular frequency and can be visually represented in 2D or 3D forms (Li et al., 2024). By comparing the modes of pressure pulsations with the structural modes, a more comprehensive explanation of flow-induced vibration mechanisms can be achieved.

Despite extensive research on pressure pulsations and flow patterns, the hydraulic excitation sources and the mechanism causing abnormal head-cover vibrations under high-head pumping conditions remain unclear. To address engineering challenges and fill the gap, our study utilized a unidirectional fluid-structure interaction numerical method to investigate the abnormal vibration phenomenon of the head-cover of an ultra-high head pump-turbine. Additionally, we integrated the dynamic mode decomposition technique to facilitate the analysis of pressure pulsations. The subsequent section presents the research object and the numerical methods. The third section verifies the accuracy of the numerical methods. The fourth section focuses on the propagation law and generation mechanism of potential hydraulic excitation sources and confirms the cause of the abnormal head-cover vibration through flow-induced vibration simulations. The sources of abnormal pressure pulsations and their frequency calculation method, as well as abnormal vibration mechanism, are discussed in the fifth section, and the conclusions are provided in the final section.

2. Numerical methods

2.1. Object and 3D computational model

A prototype pump-turbine of an ultra-high head pumped storage power station was selected as the research object. The maximum head of the pump-turbine in pump mode is 660 m. Figure 1 shows the 3D model of the pump-turbine and head-cover, and Table 1 lists the basic parameters. The entire 3D flow passage of the pump-turbine includes the main flow passage and the clearance flow passage. The main flow passage includes the spiral-case, stay-vanes, guide-vanes, runner, and draft-tube. Extending the upstream and downstream pipelines to reduce the influence of boundaries (Al-Obaidi, 2024a). The clearance flow passage includes the hub clearance, shroud clearance, and pressure equalizing pipes. The hub clearance refers to the thin layer of water between the lower surface of the head-cover (stationary surface) and the upper surface of the runner hub (rotating surface). The shroud clearance refers to the thin layer of water between the upper surface of the bottom ring (stationary surface) and the lower surface of the runner shroud (rotating surface). The pressure equalizing pipes connect the hub clearance and the draft-tube to balance the pressure. During the pumping operation, the water enters the draft-tube and flows out of the spiral-case through the runner and vane zone. The structural model includes the head-cover and a part of the stay ring. The head-cover is welded from the Q345 steel plate, and divided into two pieces for installation. In the simulation, the stay ring and the two pieces of head-cover are integrated into a body, and the components installed on it such as the water guide bearing and main shaft seal, are ignored.

Figure 1. 3D model of prototype pump-turbine and head-cover.

Table 1. Basic parameters of the prototype pump-turbine.

Parameters	Value
Maximum pump head Hmax (m)	660
Rated rotational speed nr (rpm)	500
Rated power in pumping mode Pr (MW)	350
Number of runner blades	5 (long) + 5 (short)
Number of guide-vanes	16
Number of stay-vanes	16
High-pressure side diameter of blade at shroud side (m)	4.41
Low-pressure side diameter of blade at shroud side (m)	2.12
Blade lean angle at runner inlet (degree)	4.50

2.2. Simulation mesh and grid independence study

The entire flow passage was divided into five computational domains for grid generation, namely the spiral-case zone, vane zone, runner zone, draft-tube zone, and clearance zone. Commercial software ICEM CFD was used to discretize the 3D computational domains. The vane zone was discretized by wedge-shaped grids, and all the remaining domains were discretized by hexahedral grids. The y⁺ value in the near-wall region of the runner and guide-vanes is less than 30, while for other components such as spiral-case, stay-vanes, and draft-tube, the y⁺ value is kept below 100 (Guo et al., 2023; Liu et al., 2020). And the expansion factor of the grids near the wall is controlled to be 1.2. The pumping condition with a head of 640 m was selected to verify the independence of the fluid grid. Five sets of grids with different levels of refinement were generated, and the number of elements was approximately 5, 7, 9, 11, and 13 million, respectively. The result of grid independence analysis is shown in Figure 2(a), using the amplitudes of typical pressure pulsations obtained from monitoring point VS01 (marked in Figure 6) as the convergence criteria. The dimensionless coefficients ${\text{C}}_{\text{p}}$, normalized values ${\mathrm{D}}^{\mathrm{\prime }}$, and ${\sigma }^{\mathrm{\prime }}$ are defined in Equation (1(1) $\begin{array}{r}{C}_p=\frac{p-\overline{p}}{{\rho }_l{v_r}^2/2},D^{\mathrm{\prime }}=\frac{D}{D_{max}},{\sigma }^{\mathrm{\prime }}=\frac{\sigma }{{\sigma }_{max}}\end{array}$(1) ). When the total number of elements exceeds 9 million, the changes in the amplitude of typical frequencies such as 10f_n and 5f_n produced by RSI, and the abnormal pulsation frequencies 7.35f_n, and 7.84f_n are less than 4%. Considering computational accuracy and resources, a total of 9.13 million grids were ultimately selected for the simulations and analysis, with 1.14, 1.92, 2.51, 1.07, and 2.49 million grids for the spiral-case zone, vane zone, runner zone, draft-tube zone, and clearance zone, respectively. The local details of the selected grid are shown in Figure 3, where mesh refinement for local areas such as the runner, vane (Figure 3(a)), and clearance passage (Figure 3(b)) is shown. (1) $\begin{array}{r}{C}_p=\frac{p-\overline{p}}{{\rho }_l{v_r}^2/2},D^{\mathrm{\prime }}=\frac{D}{D_{max}},{\sigma }^{\mathrm{\prime }}=\frac{\sigma }{{\sigma }_{max}}\end{array}$(1) where p and $\overline{p}$ are the transient pressure and average pressure, respectively, Pa; ${\rho }_{\text{l}}$ is the density of water, kg/m³; v_r is the runner circumferential velocity at its maximum radius, m/s; D and σ are the displacement and equivalent stress of structure; D_max and σ_max are their maximum values.

Figure 2. Grid independence analysis. (a) Fluid grid; (b) Solid grid.

Figure 3. Schematic of local mesh. (a) spiral-case, vanes, and runner; (b) clearance; (c) head-cover.

The head-cover is relatively regular and has rotational symmetry, therefore, the solid domain was discretized by hexahedral elements with the element type Solid 186. Five sets of grids with different levels of refinement were generated, and the numbers of nodes were about 0.2, 0.3, 0.4, 0.61, and 0.82 million, respectively. The displacements and equivalent stresses at points S1 and S2 (marked in Figure 4) under the same operating condition as the CFD simulations are shown in Figure 2(b). When the number of nodes exceeds 0.4 million, the differences in displacement and equivalent stress are less than 1%. Therefore, it is appropriate to select a grid with 0.4 million nodes for subsequent flow-induced vibration analysis. The solid grid is shown in Figure 3(c).

Figure 4. Boundary conditions for structure analysis. (a) wet modal analysis; (b) transient dynamics analysis.

2.3. Numerical schemes and boundary conditions for CFD

The commercial software ANSYS CFX, which uses the finite volume method to solve the governing equations of fluid dynamics, was employed for simulating 3D flow fields. The SST $k-\omega$ turbulence model, which helps to capture the unsteady flow phenomena in the pump-turbine (Al-Obaidi, 2019; Zhang et al., 2020), was used in both steady and transient simulations. The weak compressibility of water can affect the amplitude–frequency and propagation characteristics of pressure pulsations, especially in high-head power stations. Therefore, the density of water was defined as a function of pressure (Zhang & Cheng, 2012), and the elasticity effect of the pipe material was considered through the acoustic velocity in the fluid, as shown in equation (2(2) $\begin{array}{r}{\rho }_l={\rho }_0{e}^{(p-p_0)/({\rho }_0{a}^2)}\end{array}$(2) ), which was realized in CFX through the CEL language. (2) $\begin{array}{r}{\rho }_l={\rho }_0{e}^{(p-p_0)/({\rho }_0{a}^2)}\end{array}$(2) where ${\rho }_0$ is the density of water at standard atmospheric pressure (p₀) of 101325 Pa, 998.2 kg/m³; ${\rho }_l$ is the density of water at local pressure (p), kg/m³; e is the natural logarithm; a is the acoustic velocity in the water, adopted as 1200 m/s based on the statistics of some power stations.

The boundary conditions were set as follows. The draft-tube inlet and the spiral-case outlet (defined in the pumping flow direction) were set as the pressure inlet and pressure outlet, respectively. The hydrostatic pressure distribution was considered and given by Equation (3(3) $\begin{array}{r}{p}_{\mathrm{inlet}/\mathrm{outlet}}=p_g-{\rho }_l\mathrm{gh}\end{array}$(3) ), which was also realized through the CEL language. The non-reflective boundary was activated to eliminate the influence of the boundary on the propagation of pressure waves (Fang et al., 2021). (3) $\begin{array}{r}{p}_{\mathrm{inlet}/\mathrm{outlet}}=p_g-{\rho }_l\mathrm{gh}\end{array}$(3) where $p_{\mathrm{inlet}/\mathrm{outlet}}$ is the hydrostatic pressure at a point on the inlet or outlet section, Pa; $p_g$ is the gauge pressure at the centre of the inlet or outlet section, Pa; h is the vertical distance between the point and the centre of the section, m; g is the gravitational acceleration, 9.81 m/s².

All computational domains were connected by interfaces. The runner domain is set as rotational, while the other domains were set as stationary. The frozen rotor model was used to deal with the interface between the stationary domains and the rotating domain in the steady-state simulations, whereas the transient rotor-stator model was used in the transient simulations (Al-Obaidi et al., 2023). The lower surface of the runner shroud and the upper surface of the runner hub were given a rotational speed consistent with the runner, while the remaining surfaces of the runner were set as non-slip stationary walls.

The transient simulations adopted the convergent and stable steady-state results as the initial flow field. The timestep was 0.001 s, which corresponds to 3 degrees of runner rotation at the rated speed (Al-Obaidi, 2024b). The subsequent analysis is based on the results of transient simulations when parameters converge.

2.4. Structural analysis settings

2.4.1. Acoustic modal analysis

Modal analysis can be used to determine the natural frequencies and mode-shapes of the structure. The lower surface of the head-cover directly contacts the water in the hub clearance and vane zones. The vibration of the head-cover will drive the vibration of the surrounding fluid, resulting in a certain amount of added mass. And the added mass will change the modal characteristics of the head-cover. Wet modal analysis can more accurately evaluate the modal frequencies and mode-shapes of the head-cover under real operating conditions by considering the influence of fluid. Wet modal analysis was completed through the Modal Acoustic module in ANSYS Workbench, and Figure 4(a) shows the computational domains and boundary conditions. The computational domains include the solid zones such as the head-cover and the upper ring plate of the stay ring, as well as the fluid zones such as the hub clearance and the guide-vane zone. The bottom surface of the upper ring plate of the stay ring was set as fixed support, and the surfaces of the head-cover in contact with the water were set as the fluid-structure interface (red-coloured lines in Figure 4(a)). The density, elastic modulus, and Poisson's ratio of the head-cover are 7850 kg/m³, 210 GPa, and 0.3, respectively. The acoustic velocity in the fluid is set at 1200 m/s.

2.4.2. Transient dynamics analysis

The transient dynamics simulation was performed by the ANSYS Mechanical module, and material properties were consistent with the wet modal analysis. Figure 4(b) shows the boundary conditions. The head-cover and stay ring were regarded as a whole and fixed support was imposed at the bottom surface of the stay ring. The radial constraint of the water guide bearing during normal operation was simulated using the bearing element (cyan-coloured surface in Figure 4(b)), and the oil film stiffness was given as 1 × 10⁹ N/m. The influence of gravity was also taken into account. The water pressure obtained through 3D CFD was applied to the lower surface of the head-cover (red-coloured lines in Figure 4(b)). The same time step as that of the CFD simulation was used for the transient dynamics simulations.

3. Verification of simulation accuracy

In the field test, abnormal pressure pulsations and vibrations of the prototype unit were measured using pressure transmitters (sampling frequency: 1000 Hz) and vibration velocity sensors (frequency response range: 0.5–200 Hz) (Zheng et al., 2023). Figure 5 presents a comparison between the numerical simulation results and the experimental results of the prototype. The condition with significant vibration during the actual operation of the pump-turbine was selected, and the head is 640 m. The 3D flow field simulation can capture the frequency band abnormal pressure pulsations, and the amplitude is similar to the field test value. The frequencies of pressure pulsations are not completely consistent, with an absolute error of 3 Hz and a relative error of 4.81%. But they are all in the frequency band 6–8f_n, which is consistent with the vibration characteristics of the pump-turbines. In addition to the possible measurement errors observed in the field test and the machining and installation errors of pump-turbine, we believe that the errors originate from the errors at the operating points. The head of the pump-turbine obtained from the numerical simulation was slightly higher than that observed in the field test, although the boundary conditions for the numerical simulation were derived from the field test. The field test indicates that the frequency of abnormal pressure pulsations varies with changes in the head rather than being a fixed value. Therefore, a slightly higher frequency of pressure pulsations obtained from numerical simulation is acceptable.

Figure 5. Comparison of pressure pulsations (a) and vibration velocity (b) in field test and simulation.

Due to the lack of field test data on the vibration velocity of the head-cover under the condition of a head of 640 m, a comparison is made between the field test data of a head of 620 m and the simulation data of a head of 640 m. The results of structural simulation reflect vibration characteristics similar to those of the field test, with similar amplitudes but higher frequencies. This also conforms to the characteristic of abnormal vibration frequency increasing with the head in the real power station. Therefore, it can be considered that the above numerical simulation models and methods can be used to analyze the mechanism of hydraulic excitation sources and the abnormal vibration of the head-cover.

4. Results of flow-induced vibration simulation

In this section, all variables are presented with dimensionless coefficients or normalized values. The pressure coefficient ${\text{C}}_{\text{p}}$, velocity coefficient ${\text{C}}_{\text{v}}$, normalized head ${\mathrm{H}}^{\mathrm{\prime }}$, normalized guide-vane opening ${\gamma }^{\mathrm{\prime }}$, normalized displacement ${\mathrm{D}}^{\mathrm{\prime }}$, and normalized equivalent stress ${\sigma }^{\mathrm{\prime }}$ are defined as follows: (4) $\begin{array}{r}{C}_v=\frac{v}{v_r},H^{\mathrm{\prime }}=\frac{H}{H_{max}},{\gamma }^{\mathrm{\prime }}=\frac{\gamma }{{\gamma }_{max}}\end{array}$(4) where v is the water velocity, m/s; H, γ are the head, and guide-vane opening of structure, respectively; H_max, γ_max are their maximum values, respectively.

4.1. Analysis of pressure pulsations

To obtain the characteristics and propagation law of the pressure pulsations in the flow passage of the pump-turbine, pressure monitoring points were arranged in each computational domain (Figure 6). Figure 6(a) shows the distribution of monitoring points in the spiral-case and the vane zone. The monitoring points marked as SC01 – SC23, SV01 – SC16, and VS01 – VS16 were arranged along the circumference in the spiral-case, stay-vane, and vaneless space, respectively. Figure 6(b) shows the distribution of monitoring points in the draft-tube, in which the 7 monitoring points are marked as DT01 – DT07. To analyze the source and propagation law of pressure pulsations in the clearance passage, a total of 12 monitoring points marked as CH01 – CH06 and CS01 – CS06 were arranged in the hub clearance and shroud clearance, respectively, as shown in Figure 6(c).

Figure 6. Layout of pressure monitoring points. (a) spiral-case and vane zone; (b) draft-tube; (c) clearance passages.

The head-cover of the pump-turbine vibrated relatively severely when the head was 0.97H_max in pump mode, and the corresponding guide-vane opening was 0.69γ_max. Therefore, this operating point was selected to analyze the source and propagation law of the abnormal pressure pulsations in the frequency band 6–8f_n in the flow passage. Due to the strong pulsating characteristics of pressure, the fast Fourier transform was used to process the signal, and the characteristic components of the pressure pulsations at each monitoring point were obtained, as shown in Figure 7. The abnormal pressure pulsations in the frequency band 6–8f_n were monitored in all flow passages, with the main frequencies being 7.35f_n and 7.84f_n but different in amplitude. Overall, the amplitudes of the abnormal pressure pulsations in the vane zone and clearance passage are greater than those in the spiral-case and draft-tube. In the spiral-case, the amplitude changes with the position of the monitoring points. At point SC01 near the spiral-case nose, the amplitude with frequency 7.35f_n is larger than that with frequency 7.84f_n, while at point SC23 near the spiral-case outlet the amplitude with frequency 7.84f_n is larger. The amplitude of abnormal pressure pulsations in the draft-tube gradually decreases from the runner (DT01) to the draft-tube inlet (DT07), and the amplitude with frequency 7.84f_n is larger than that with frequency 7.35f_n. The pressure pulsations in the vaneless space (VS01) are abundant in frequency and high in amplitude, and attenuate gradually from the vaneless space to the stay-vane (SV01). The amplitudes of the abnormal pressure pulsations in the external cavities of the hub clearance (CH03) and shroud clearance (CS03) are almost equal and close to those in the vaneless space, with the main frequencies being 7.35f_n and 7.84f_n, while the pressure pulsations with frequencies 10f_n and 5 f_n in these cavities caused by RSI attenuate rapidly.

Figure 7. Frequency spectrum of pressure pulsations. (a) Monitoring points in spiral-case. (b) Monitoring points in draft-tube. (c) Monitoring points in vane zone. (d) Monitoring points in clearance passage.

The amplitudes of pressure pulsations with frequencies 7.35f_n and 7.84f_n at each monitoring point were extracted, and arranged in the order of the point numbers, as shown in Figure 8. Along the circumference of the spiral-case from the nose to the spiral-case outlet (SC01-SC23), the amplitudes with frequencies 7.35f_n and 7.84f_n show an attenuation trend with rise and fall. The amplitudes reach their maximum near the nose, with several extreme values along the circumference of the spiral-case. The amplitude with frequency 7.84f_n in the draft-tube is higher than that with frequency 7.35f_n, and both decrease from the runner to the draft-tube inlet (DT01-DT07) (Figure 8(b)). In the vaneless space, the amplitude with frequency 7.84f_n is higher than that with frequency 7.35f_n, while they are almost equivalent in the stay-vane zone. From the vaneless space to the stay-vane, the amplitudes with frequencies 7.35f_n and 7.84f_n decrease (Figure 8(c)). The variation trends of the two amplitudes in the hub clearance and shroud clearance are similar. Radially inward along the clearances, the amplitudes are almost constant in the external cavities (CH02-CH03 and CS02-CS03) but decrease suddenly at the sealing rings (CH03-CH04 and CS03-CS04) (Figure 8(d)). Therefore, it can be inferred from the amplitude distributions that the abnormal pressure pulsations in the frequency band 6–8 f_n originate from the vaneless space and propagate to the spiral-case, draft-tube, and clearance passages.

Figure 8. Amplitude variation of pressure pulsations with frequencies 7.35f_n and 7.84f_n. (a) Monitoring points in spiral-case.(b) Monitoring points in draft-tube. (c) Monitoring points in vane zone. (d) Monitoring points in clearance channel.

4.2. Evolution of flow patterns in the pump-turbine

4.2.1. Dynamic mode decomposition of the pressure field

The DMD method was adopted to distinguish the flow structure at characteristic frequencies. More than 3000 snapshots (25 rotation cycles) of the flow field were extracted from the cross-section of the spiral-case and vane zone, and the pressure data with the mean removed was decomposed by the DMD method. Figure 9 shows the eigenvalues distribution and energy proportion of the DMD mode, respectively. Almost all eigenvalues are distributed on the unit circle, indicating that the main flow structures are stable. The mode frequencies with a high energy proportion are consistent with the main frequencies obtained by FFT of the pressure at each monitoring point in section 4.1, which are the abnormal pressure pulsation frequency of 7.84f_n, 7.35f_n, and the blade passing frequencies caused by RSI.

Figure 9. Eigenvalue distribution (a) and energy proportion (b) of DMD modes.

Figure 10 and Figure 11 show the evolution of the pressure mode of the abnormal pressure pulsations. The pressure mode with frequency 7.84f_n is characterized by pulsations in pressure in the vaneless space and circumferential movement of the relative high/low-pressure area in the spiral-case. At the same time, a concentrated area of pressure can be clearly observed near each guide-vane, and simultaneously all high-pressure or all low-pressure, which is a nodal circle mode. The pressure changes simultaneously with time, and its frequency is 7.84f_n. The pressure level in the spiral-case is lower than that of the vaneless space, and its change near the nose lags behind that in the vaneless space. When the pressure in the vaneless space is high, low pressure appears near the nose, while when the pressure in the vaneless space is low, high pressure appears. This is because the propagation of pressure waves takes time. The blocking and reflection effects of the nose force the high/low-pressure area to move clockwise along the spiral-case. The pressure mode with frequency 7.35f_n is characterized by the concentration and rotation of pressure in the vaneless space and the spiral-case. The high and low-pressure areas in the vaneless space each account for half and rotate counterclockwise simultaneously (opposite to the rotation direction of the runner in pump mode), which is a rotating 1 nodal diameter mode. The rotation of the high and low-pressure areas in the vaneless space also drives the rotation of pressure in the spiral-case, and at the same time, the pressure change near the nose is slightly ahead of that in the vaneless space due to the time required for pressure wave propagation.

Figure 10. Pressure mode extracted by DMD method with frequency 7.84f_n.

Figure 11. Pressure mode extracted by DMD method with frequency 7.35f_n.

The DMD decomposition indicates that the abnormal pressure pulsations in the flow field originate from the vaneless space, where the pressure pulsation with frequency 7.84f_n exhibits a nodal circle mode and the other exhibits a rotating 1 nodal diameter mode.

4.2.2. The motion law and generation mechanism of rolling vertices in the vaneless space

Pressure pulsations are closely related to flow patterns. Abnormal flow patterns are the source of abnormal pressure pulsations and vibration. It was found that the abnormal pressure pulsations with frequency around 8f_n are generated by the rolling vortices in the vaneless space. Figure 12 shows the distribution of the vortices identified by the Q criterion at typical times. The vortices are displayed through an isosurface with Q = 5000 s⁻², and coloured by the absolute velocity. Abundant vortices are distributed in the vaneless space and the transitional area between the guide-vanes and the stay-vanes. Several distinct vortex clusters can be identified by the absolute velocity, and the velocity around the vortex clusters in the vaneless space is high and concentrated. The vortex clusters almost fill the entire vaneless space and move around the main shaft like the runner. At t = 0.0 s, vortex cluster 1 is close to guide-vane 1 (GV1). Then vortex cluster 1 rotates clockwise and moves to the guide-vane 2 (GV2) at 0.015 s. Subsequently, vortex cluster 1 continues to rotate and approaches guide-vane 3 (GV3) at t = 0.030 s. The movement law of the remaining vortex clusters is similar to the above description, i.e. all vortex clusters rotate together in the narrow vaneless space. The sizes of the vortex clusters do not remain constant but develop and change during their motion. Figure 13 shows the development of vortex cluster 1, and displays the velocity vectors on the surface of the vortex cluster and the pressure on the guide-vane. At t = 0.0 s, vortex cluster 1 is located at the tail of GV1. The velocity vectors on the upper surface of the vortex cluster 1 (top view) are oriented towards the passage between GV1 and GV2, while the velocity vectors on the lower surface have components towards the runner (bottom view), meaning there are small backflows near the shroud. It indicates that the vortex cluster itself rotates, and rolls forward. This is a new and unreported flow pattern, we termed the ‘rolling vortex’ based on its shape and motion characteristics. Then the rolling vortex moves towards GV2 and is elongated at t = 0.005 s. At t = 0.010 s, a part of the rolling vortex is about to enter the passage between GV1 and GV2. At t = 0.013 s, the rolling vortex collides with GV2. Then at t = 0.015 s, it is cut into two parts by the tail of GV2, with one entering the stay-vane zone and the rest remaining in the vaneless space to repeat the above movement law. The motion of a rolling vortex from GV1 to GV2 caused a change in pressure, resulting in a decrease in pressure at the tail of GV1 and an increase at the tail of GV2. In other words, the rotation of the rolling vortices around the main shaft in the vaneless space caused the pulsations in pressure.

Figure 12. Distribution of vortex in the vaneless space (Q = 5000 s⁻²).

Figure 13. Development of vortex in the vaneless space.

The formation of the rolling vortices in the vaneless space is related to the distortion degree of the runner blades. The blade lean angle of the runner outlet causes the vertically uneven distribution of the outflow velocity, leading to the secondary flow in the vaneless space. Figure 14 shows the geometry of the runner blades. The black and the red lines represent the intersection of the blade with the hub and the shroud, respectively. The long blade and the short blade both have a small positive blade lean angle at the trailing edge, that is, the intersection line of the blade with the hub is inclined towards the pump rotation direction. The intersection of the leading edge and the shroud is taken as the reference, the intersection point between the leading edge and the hub is closer to the centre of rotation. From the leading edge of the blade to the trailing edge, the two lines form an X-shape visually. The distortion degree of the blade affects the flow and velocity distribution in the runner passage. Analyzing based on the simulation result at t = 0.0 s in Figure 12. During the pumping operation, the water enters the runner from the draft-tube and flows along the blade surface. The distortion of the blade's leading edge forces the water to flow into the blade passage more concentrated near the shroud side. Figure 15 shows the contour and vector of the relative velocity near the suction surface of a long blade. Notably, a distinct high-velocity area is observed near the shroud adjacent to the blade's leading edge. With the change of the blade’s distorted direction, water becomes more concentrated near the blade's hub at the trailing edge, with a noticeable flow of water from the shroud towards the hub. Consequently, the velocity near the hub at the blade's trailing edge is larger than that near the shroud. Figure 16 shows the distribution of the radial velocity on different spans. In each blade passage, there is an uneven distribution of radial velocity, with a notable concentration on the suction side of the blade. This concentration is more obvious in the long blade compared to the short one. Additionally, the radial velocity is higher near the hub (Span 0.1) than near the shroud (Span 0.9). Due to the lack of blade constraint, the water rotates in the vaneless space after exiting the runner, leading to the formation of rolling vortices.

Figure 14. Geometry of runner blade.

Figure 15. Contour and vector of relative velocity near the wall on the suction side of a long blade.

Figure 16. Distribution of radial velocity on different flow surfaces.

4.2.3. The influence of head on rolling vortices and abnormal pressure pulsations

The frequencies of abnormal pressure pulsations at different operating points vary, which are related to the correlation between head and guide-vane opening. The characteristics of simulated pressure pulsation are consistent with those obtained from the field test. The obvious frequency component of the head-cover vibration is mainly in the frequency band 6–8f_n, and the amplitude increases with the rise of the head. Figure 17 shows the rolling vortices in vaneless space at different operating points, also identified by Q = 5000 s⁻² and coloured by the absolute velocity. Figure 18 displays the frequency spectrum of pressure pulsations at corresponding operating points in the vaneless space (monitoring point VS01). There are several rolling vortices in the vaneless space at different operating points, and their volume increases with the rise of the head. The larger guide-vane opening at the lower head makes it easier for the rolling vortices to enter the stay-vane zone. Meanwhile, the spatial reduction of the vaneless space hinders the rotation of the rolling vortices around the main shaft. Therefore, the abnormal pressure pulsations are not significant at the lower head operations. As the head rises, the guide-vane opening correspondingly decreases and the vaneless space becomes larger, which makes the rolling vortices easier to grow and rotate around the main shaft, resulting in an abnormal pressure pulsation of 7.2f_n at a head of 0.96H_max. As the guide-vane opening further decreases, the rolling vortices become larger and their rotational frequency increases, resulting in stronger pressure pulsations of 7.84f_n at a head of 0.97H_max.

Figure 17. Rolling vortices in vaneless space at different operating points.

Figure 18. Frequency spectrum of pressure pulsations at different operating points.

4.2.4. The propagation of pressure waves in spiral-case

The pressure pulsations in the vaneless space propagate along the passage between the guide-vanes, causing the pressure in the spiral-case to change periodically. Figure 19 shows the distribution of pressure in the horizontal cross-section of the pump-turbine. The pressure in the spiral-case passage is not evenly distributed at the same moment, and the pressure at the same location also changes over time. The position of local high/low-pressure changes over time, and the propagation of the pressure pulsation manifests as a travelling wave. At t = 0.0 s, a local high-pressure area appears near the nose, while the relatively low-pressure area appears in the rest of the spiral-case. The low-pressure area extends from the guide-vane area to the spiral-case, forming an elliptical shape. The high-pressure area then reduces at the nose and gradually moves towards the outlet along the circumference of the spiral-case, while the elliptical low-pressure area rotates clockwise simultaneously. At t = 0.006–0.012 s, a local low-pressure area appears near the nose, and the high-pressure area continues to move toward the outlet. At t = 0.015 s, the nose pressure increases, and the rest of the area has a pressure distribution similar to t = 0.0 s. The pressure distribution in the spiral-case then repeats the above periodic changes.

Figure 19. Distribution of pressure in the cross-section of pump-turbine.

The pressure pulsations in the spiral-case originate from the vaneless space, with frequencies consistent with those produced by the rolling vortices, albeit with reduced amplitude. The characteristic frequencies are consistent with the frequency generated by the rolling vortices, but the amplitude is smaller. Similar to the phase resonance phenomenon caused by the RSI, the pressure pulsations caused by the interaction between the rolling vortices and guide-vanes propagate along different vane passages to the spiral-case, resulting in a complex superposition of the pressure waves. The reflection at the nose causes the propagation of pressure pulsations along the spiral-case's circumferential direction towards the outlet, where the superimposition of amplitudes results in wave nodes during transmission. Due to the given non-reflective boundary, there is no reflection of pressure pulsations at the spiral-case outlet. Consequently, the pressure distribution in the spiral-case demonstrates a form of travelling wave with oscillating amplitude attenuation.

4.3. Analysis of flow-induced vibration

4.3.1. Wet modal analysis of head-cover

Hydraulic excitation is the main source causing the vibration of pump-turbines, but only a few cases can lead to strong vibration. Previous studies have shown that the vibration response will be very obvious when the frequency of the excitation force is close to the natural frequency of the structure. Therefore, wet modal analysis was conducted to calculate the natural frequency of the head-cover and further confirm the reason of the abnormal vibration.

Figure 20 shows the first 6 mode-shapes of the wet mode, which are coloured by the normalized displacement. The closer the colour to red, the greater the displacement. The head-cover can be regarded as a structure that consists of two circular plates and several connecting ribs, and its mode-shapes are similar to those of a disc. Thus the mode shapes of the head-cover can be described by the nodal circle (NC) and nodal diameter (ND) (Liu et al., 2021). The first-order wet modal frequency (5.94f_n) of the head-cover is near the abnormal vibration frequency band 6–8f_n, and the mode-shape is 0 NC mode, which is manifested as an axial vibration (the z-direction). The displacement reaches its maximum at the main shaft seal and becomes small near the stay ring. The mode-shapes of the second and third-order are similar, with 1 ND mode and a 90° angle difference in the xy plane. The second-order modal frequency (8.66f_n) matches well with the field test values (8.52–8.82f_n). With the number of nodal diameters being 1, the second-order mode exhibits shaking in the x-direction, with the maximum displacement in the centre of the head-cover. The third-order mode exhibits shaking in the y-direction, with a slightly higher frequency than that of the second-order. This may be due to the large total thickness and high stiffness of the rib plate at the head-cover joint. The fourth-order mode is still 0 NC, showing axial vibration (the z-direction). The fifth and sixth-order modes are both 2 ND modes but with a 45° angle difference in the xy plane.

Figure 20. First 6 mode-shapes of wet mode.

The lubricating oil film formed between the water guide bearing and the main shaft not only supports the main shaft but also exerts radial constraints on the head-cover, influencing the modal frequency. And the acoustic velocity in the water also changes the modal frequencies. Figure 21 shows the variation of the wet modal frequencies of the head-cover with oil film stiffness and acoustic velocity. The oil film radial stiffness of the water guide bearing increases the wet modal frequencies, especially for higher-order modes. Due to the additional radial support of the head-cover, the increase in the nodal diameter modal frequency is higher than that in the nodal circle, but the increase is almost zero as the radial stiffness continues to increase. The increase of the acoustic velocity also raises the wet modal frequencies, but the increase in the nodal circle mode is more obvious. When the oil film radial stiffness of the water guide bearing ranges from 0 and 2 × 10⁹ N/m and the acoustic velocity ranges from 1000 to 1400 m/s, the first-order wet modal frequency of the head-cover is in the range of 5.28–6.96f_n and the second-order is in the range of 8.52–9.96f_n, which has a small vibration margin with the frequencies of the abnormal pressure pulsations. Therefore, the fact that the abnormal pressure pulsation frequencies in the flow passage are close to the low-order wet modal frequency of the head-cover, should be the reason for the obvious enhancement of head-cover vibration in the pump mode of this pump-turbine.

Figure 21. Variation of modal frequencies with acoustic velocity and oil film stiffness.

4.3.2. Dynamic response analysis of the head-cover

The unidirectional fluid-structure interaction analysis of the head-cover further proves that the abnormal vibration of the head-cover originates from the pressure pulsations in the flow passage. Figure 22 shows the distribution of axial deformation and equivalent stress of the head-cover under the same operating point as the CFD simulation. The head-cover deforms in the z-direction under the effect of pressure in pump mode. The deformation is symmetrical about the zx plane (the joint plane) and decreases from the centre to the periphery of the head-cover. Due to the lack of axial support, the maximum deformation area is concentrated on the region of the main shaft seal. The distribution of the equivalent stress of the head-cover is also symmetrical. The stress at the rib plate and the flange are relatively high, which is caused by the large bending moment under water pressure.

Figure 22. Distribution of normalized axial deformation (a) and normalized equivalent stress (b).

Figure 23 shows the frequency spectrum of axial vibration displacement and velocity at each monitoring point. Under the pressure in the guide-vane zone and the hub clearance, the head-cover vibrates in the z-direction, and the frequency is in the band 6–8f_n. The main frequencies are consistent with those of the abnormal pressure pulsations, namely 7.35f_n and 7.84f_n. The amplitude of the abnormal axial vibration displacement is small, not exceeding 30% of the limit value specified in ISO 10816-5:2000, and decreases gradually from the centre to the periphery. The variation of vibration velocity also follows the same laws. The amplitude of two abnormal frequencies is larger and almost twice the limit value in the centre of the head-cover.

Figure 23. Frequency spectrum of axial vibration displacement (a) and velocity (b).

5. Discussion of abnormal pressure pulsations and vibration

5.1. Generation mechanism of pressure pulsations in the frequency band 6–8f_n

The uneven distribution of vertical outflow velocity of the runner results in the occurrence of rolling vortices in the vaneless space. The rotating rolling vortices spin themselves and move in the vaneless space around the main shaft, and periodically collide with the guide-vane, causing pressure pulsations, which is similar to the rotor-stator interaction between the runner blade and the guide-vane. Due to the rotation of the runner, the relative positions of the runner blades and the guide-vanes change periodically. The period of RSI depends on the number of rotor blades and stator blades as well as the rotational speed, and its frequency can be calculated by Equation (5(5) $\begin{array}{rl}{f}_s& =Z_r\cdot f_n\end{array}$(5) ) (Tanaka, 2011). The rolling vortices rotating around the main shaft in the vaneless space are equivalent to the rotating runner blade, and their relative position to the guide-vane also changes periodically. The rotational frequency around the main shaft of a single rolling vortex is about 0.49f_n, and there are 16 rolling vortices. Several rolling vortices rotate and pass through each guide-vane, causing periodic pressure pulsations. Standing at a fixed position on a guide-vane located on a stationary coordinate system, we experience 16 pressure pulsations during the time for a rolling vortex to rotate once in the vaneless space. Therefore, the frequency of the pressure pulsations we feel can be calculated through Equation (6(6) $\begin{array}{rl}{f_s}^{\mathrm{\prime }}& =Z_v\cdot {f_n}^{\mathrm{\prime }}\end{array}$(6) ). Similarly to the RSI, the frequency of the interaction between the rolling vortices and the guide-vanes is 16 × 0.49f_n = 7.84f_n. From the perspective of the entire vaneless space, 16 rolling vortices rotate simultaneously and pass through the guide-vanes, resulting in a simultaneous increase or decrease in pressure, which is consistent with the DMD mode with frequency 7.84f_n (Figure 10). Unlike the RSI, the frequency of interaction between rolling vortices and guide-vane changes with operating conditions. This is because the rotation frequency of the rolling vortices around the main shaft denoted as ${\text{f}}_{\text{n}}{}^{\mathrm{\prime }}$ in Equation (6(6) $\begin{array}{rl}{f_s}^{\mathrm{\prime }}& =Z_v\cdot {f_n}^{\mathrm{\prime }}\end{array}$(6) ), is influenced by the guide-vane opening of the pump-turbine. A larger guide-vane opening hinders the rotation of the rolling vortices. Consequently, as the head increases, the guide-vane opening decreases, leading to an increase in the frequency of abnormal pressure pulsations. (5) $\begin{array}{rl}{f}_s& =Z_r\cdot f_n\end{array}$(5) (6) $\begin{array}{rl}{f_s}^{\mathrm{\prime }}& =Z_v\cdot {f_n}^{\mathrm{\prime }}\end{array}$(6) where ${\text{f}}_{\text{s}}$, ${\text{f}}_{\text{s}}{}^{\mathrm{\prime }}$ are the frequency of RSI felt by the guide-vane, and the frequency of interaction between rolling vortices and guide-vanes, respectively, Hz; Z_r, Z_v are the number of runner blades and the number of rolling vortices, respectively; ${\text{f}}_{\text{n}}$, ${\text{f}}_{\text{n}}{}^{\mathrm{\prime }}$ are the rotation frequencies of the runner and the rolling vortex, respectively, Hz.

Similar to the RSI, the interaction between rolling vortices and guide-vanes also causes pressure modes. The equation (7(7) $\begin{array}{rl}m\times Z_g\pm \nu & =n\times Z_r\end{array}$(7) ) (Tanaka, 2011) provides a calculation formula for the modes of pressure pulsations caused by the RSI. Similarly, the pressure mode caused by the interaction between rolling vortices and guide-vanes can be determined by Equation (8(8) $\begin{array}{rl}{m}^{\mathrm{\prime }}\times Z_g\pm {\nu }^{\mathrm{\prime }}& =n^{\mathrm{\prime }}\times Z_v\end{array}$(8) ). Under the combination of 16 rolling vortices and 16 guide-vanes, the equation is $1\times 16\pm 0=1\times 16$. It indicates that the number of the nodal diameter is 0, and the pressure mode generated is of the nodal circle type, as shown in Figure 10. The shape and size of rolling vortices may not be completely consistent, and may also be weakened by the influence of asymmetric spiral-case during their movement. Therefore, there may be less than 16 obvious rolling vortices interacting with 16 guide-vanes. Usually, the pressure modes with fewer nodal diameters are easily excited, and under the combination of 15 rolling vortices and 16 guide-vanes, the equation is $1\times 16-1=1\times 15$. The number of the nodal diameters is 1, and the direction of rotation is opposite to that of the rolling vortices. Standing at a fixed position on a guide-vane located on a stationary coordinate system, the observed rotational speed of the pressure mode is $f_n^{\mathrm{\prime }}\frac{n{Z}_v}{\nu }=0.49f_n\times \frac{1\times 15}{1}=7.35f_n$, and the motion of the pressure mode is the same as the DMD mode shown in Figure 11. (7) $\begin{array}{rl}m\times Z_g\pm \nu & =n\times Z_r\end{array}$(7) (8) $\begin{array}{rl}{m}^{\mathrm{\prime }}\times Z_g\pm {\nu }^{\mathrm{\prime }}& =n^{\mathrm{\prime }}\times Z_v\end{array}$(8) where m and n are positive integers; ν is the number of diametrical nodes; Z_g is the number of guide-vanes.

5.2. Mechanism of abnormal head-cover vibration

The abnormal vibration of the head-cover originates from hydraulic excitation. It is generally believed that the significantly enhanced structural response is due to the coincidence or similarity between the excitation frequency and the natural frequency of the structure. The hydraulic excitation frequencies of the head-cover are complex and have obvious three-dimensional characteristics. Its vibration is the result of multiple basic vibration modes superposition, and simply distinguishing from frequency cannot accurately determine the vibration pattern. The vibration velocity of the head-cover surface in contact with the fluid was extracted, and the vibration mode of the head-cover under the head of 0.97H_max was obtained through the DMD method, as shown in Figure 24. Two basic vibration modes were obtained, with frequencies and modes consistent with the pressure modes of the flow field (as shown in Figure 10 and Figure 11), but different from the wet mode of the head-cover under static water conditions. The vibration mode with frequency 7.84f_n is the same as the mode-shape of the first-order wet mode, exhibiting a nodal circle mode of vertical vibration. The vibration mode with frequency 7.35f_n and the mode-shape of the second-order wet mode both have a nodal diameter, the difference lies in whether it rotates or not. The actual vibration pattern is the superposition of two basic modes, not a simple nodal circle type vibration. Therefore, if there is severe vibration on the head-cover, not only should the frequency be similar, but the excitation mode and mode-shape should also be consistent.

Figure 24. Comparison between vibration mode and mode-shape.

The vibration avoidance margin M adopted widely in engineering is used to characterize the proximity between the natural frequency of the structure and the excitation frequency, thereby evaluating the risk of resonance. M is calculated through Equation (9(9) $\begin{array}{r}M=\left|\frac{f_e-f_s}{f_s}\right|\times 100\mathrm{\%}\end{array}$(9) ) (Li et al., 2022) and is generally required to be greater than 20%. For the first-order wet modal frequency (5.28–6.96f_n), the M is 12.6–48.5%, and for the second-order wet modal frequency (8.52–9.96f_n), the M is 13.7–26.2%. Considering the variation in acoustic velocity and boundary conditions, the vibration avoidance margin of the first – and second-order wet modal frequency is insufficient, and the risk of resonance is relatively high. It can be determined that the intensive vibration is related to the coincidence of frequency and mode between abnormal pressure pulsations in the flow passage and the low-order wet mode of the head-cover. (9) $\begin{array}{r}M=\left|\frac{f_e-f_s}{f_s}\right|\times 100\mathrm{\%}\end{array}$(9) where ${\text{f}}_{\text{e}}$, ${\text{f}}_{\text{s}}$ are the excitation frequency and natural frequency of the structure, respectively.

6. Conclusion

In this study, we revealed the mechanism of the abnormal head-cover vibration of an ultra-high-head pump-turbine operating in pump mode. A new flow pattern that causes abnormal pressure pulsation was identified and its generation mechanism as well as the dynamic characteristics of the head-cover was analyzed. The frequency calculation method of abnormal pressure pulsations and the abnormal vibration mechanism of the head-cover were also discussed. The main conclusions are as follows.

1. The abnormal vibration of the head-cover originates from hydraulic excitation. The insufficient vibration avoidance margin (less than 20%) between the low-order wet modal frequency and the abnormal pressure pulsations (6–8f_n) in the vaneless space is the main reason for the intensive vibration. Despite the abnormal pressure pulsations exhibiting a similar amplitude to those caused by RSI, the similarity in frequency and mode between the hydraulic excitation and the wet mode of the head-cover amplifies the vertical vibration amplitude.

2. A new flow pattern has been discovered in the vaneless space in pump mode, known as rolling vortices, which interact with stationary guide-vanes to generate abnormal pressure pulsations in the frequency band 6–8f_n. The flow mechanism of the rolling vortices is that the outflow velocity near the hub at the runner outlet is obviously higher than that near the shroud, due to the positive blade lean angle and the X-shape distorted blades. Generated by the flow twist, cut by the guide-vanes, and driven by the runner rotation, the rolling vortices change and move in the vaneless space around the main shaft, forming a regular flow pattern that generates regular pressure pulsations. Similar to the rotor-stator interaction, its frequency is the product of the number of rolling vortices and their rotational frequency rotating around the main shaft.

3. The Dynamic Mode Decomposition (DMD)method facilitates the rapid and precise identification of specific flow patterns from the complex internal flow of pump-turbines, enabling the analysis of flow-induced vibration mechanisms. In this study, the pressure modes caused by rolling vortices manifest as the main mode of the nodal circle and the secondary mode of the rotating 1 nodal diameter, which is consistent with the mode-shapes of the low-order wet mode of the head-cover.

This study has identified the hydraulic vibration source that causes abnormal vibration of the head-cover. However, the following limitations still need to be improved. The first is that there is a certain gap between simulation results and field test, which may be improved by considering the influence of head-cover deformation on the flow field and selecting an appropriate turbulence model. The second is that the abnormal head-cover vibration has not been resolved yet. Subsequent research will attempt to improve the stability of the pump-turbine by modifying the runner and optimizing the wet modal frequency of the head-cover.

Acknowledgment

The numerical simulations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University. Thanks to Wenzhe Kang, Zhiyi Yuan,and Xiang Xia for valuable suggestions in writing.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (grant number 51839008) and the Natural Science Foundation of Hubei Province of China (grant number 2022CFB305).