Effect of Geometric Parameters in the Seal Clearance on the Modal Characteristics of Pump-Turbine Runner

Abstract

The runner of a pump turbine features a relatively flat structural configuration. The clearance cavities formed between the upper crown, lower band, and surrounding stationary components play a critical role in its dynamic behavior and operational stability. Consequently, a detailed modal analysis of the runner is essential to ensure safe and stable operation. In this study, an acoustic–structure coupling method is adopted to investigate the wet modal characteristics of the pump-turbine runner, explicitly accounting for the added mass effect induced by the fluid in the external flow passages. By systematically varying the geometric parameters of the upper crown clearance cavity, the influence of seal clearance dimensions on the runner’s modal characteristics is examined. The results demonstrate that the radial clearance and the axial height of the seal cavity are the most influential parameters, reducing natural frequencies by up to 15.85% and 16.93%, respectively. The pitch of the seal teeth shows a secondary yet notable effect, inducing a frequency variation of 13.73%. In contrast, local labyrinth seal parameters, such as the number of teeth and tooth width, have a comparatively limited effect. This study provides practical guidance for vibration risk prediction, anti-resonance design, and operational stability assessment of high-head, large-capacity turbine runners by revealing the quantitative relationship between geometric parameters and modal frequencies.

1. Introduction

To address the global energy crisis, pumped-storage power plants are vital for energy storage and grid regulation, with the reversible pump turbine serving as their core component [1,2,3]. Compared to conventional hydraulic turbines, the pump-turbine runner features a flatter geometry. The resulting intense fluid–structure interaction in a water-immersed environment significantly reduces its natural frequencies [4,5]. Moreover, these units frequently undergo transient processes such as start-stop cycles and load variations, making them more susceptible to flow-induced resonance and posing a serious threat to operational safety and stability [6]. Structural vibration can be decomposed into multiple modes, each characterized by specific natural frequencies, damping ratios, and mode shapes. These parameters are essential for predicting dynamic responses and avoiding resonance [7,8]. However, the modal characteristics of hydraulic machinery are significantly influenced by the surrounding fluid. The added mass induced by the fluid is the primary cause of the observed natural frequency shift, making the accurate prediction of the runner’s wet modal characteristics crucial for resonance avoidance and ensuring safe, stable operation.

The analysis of wet modal characteristics primarily employs three approaches: analytical, experimental, and numerical simulation. In theoretical analysis, Vogel and Skinner [9] pioneered the development of early mathematical models under various boundary conditions, which were subsequently improved and extended by researchers such as Southwell and Leissa [10,11]. Further analytical work has extended to more complex structures. For instance, Mao et al. [12] proposed an analytical model for the added mass and damping of marine propellers. Concurrently, semi-empirical models based on experimental data have also been developed [13,14,15]. Zeng’s study [16] systematically summarized and refined theoretical prediction models for hydrofoil added damping. In experimental methods, vibration measurement for fully submerged structures relies on various sensors. Common contact devices include accelerometers [17] and PZTs [18,19], while optical components like distributed optical fiber sensors [20] and fiber Bragg grating sensors [21] are also applied. For non-contact measurement, the laser Doppler vibrometer [22] is the primary tool. The measured data from these devices enable the identification of the system’s natural frequencies, mode shapes, and damping through methods like the Fast Fourier Transform and the Short-Time Fourier Transform [23,24,25]. Given the high cost and complexity of testing intricate hydraulic machinery, numerical simulation provides a practical alternative. Liang et al. [26], for example, validated the effectiveness of numerical methods for wet modal analysis by comparing calculated results with experimental rotor modal characteristics.

Subsequent research has increasingly focused on identifying key parameters governing wet modal behavior, encompassing structural characteristics, fluid conditions, and boundary conditions. Regarding structural factors, the runner’s operating conditions and design are dominant. Studies show that stress-stiffening from operational loading can increase its natural frequencies [27], while these frequencies decrease with a reduction in the speed of sound within the fluid medium [28]. Under off-design conditions, inflow is prone to inducing resonance, often manifesting as bending-torsional deformation that may lead to blade fracture [29]. Concerning fluid conditions, both cavitation and immersion depth directly influence added mass. Cavitation reduces the average fluid density, thereby decreasing the added mass [30]. Furthermore, the variation in added mass with immersion depth is closely related to the mode shape; its rate of change decreases significantly when the liquid level approaches the modal nodal lines [31]. In terms of boundary conditions, proximity to solid walls is a critical factor. The closer a submerged structure is to a wall, the more pronounced the added mass effect becomes [32,33]. For disk-like structures, added mass is particularly sensitive to variations in axial clearance, though its change also becomes significant when the radial clearance is sufficiently small [34]. Related studies [35,36] have further investigated clearance effects. Among these, the acoustic-structure coupling method has proven effective for accurately predicting the natural frequencies of submerged structures near walls [35]. The influence of clearance size on the modal characteristics of complex runner structures has also been extensively analyzed [37,38,39].

In practical engineering, labyrinth seals are critical sealing components at the clearance between the runner crown and band. Their performance is vital for high-head units. Studies indicate that the clearance size is a key parameter: reducing the labyrinth clearance width effectively decreases leakage under both turbine and pump operating conditions [40]. Even under transient loading, the presence of this gap, though transmitting minimal flow, significantly improves the accuracy of flow field simulations [41]. Adjusting the lower-ring clearance of a Francis turbine can optimize draft tube flow, thereby reducing leakage losses [42]. However, seal wear during operation increases clearance, triggering a nonlinear rise in leakage, a process closely tied to the geometric parameters of the seal teeth [43]. To address this, research focuses on developing new seal structures like adaptive labyrinth seals [44] and studying the leakage flow in configurations such as spiral labyrinth brush seals [45], highlighting their indispensable role in ensuring efficient and stable unit operation.

While existing studies have primarily focused on the overall dimensions of the sealing clearance, this work further investigates the specific structural features of the labyrinth seal ring inside the clearance. By integrating both the macroscopic clearance dimensions and the internal labyrinth geometry, the influence of key geometric parameters of the sealing clearance on the wet modal characteristics of the runner is systematically examined. A quantitative correlation between these parameters and the modal frequencies is established, providing a theoretical basis and practical reference for vibration risk prediction, anti-resonance design, and operational stability assessment of the runner in high-head, large-capacity pump-turbine units.

2. Research Objects and Methods

2.1. Research Object

2.1.1. Pump-Turbine Model

This study considers a full-scale, high-head pump turbine from an operating power station. The runner consists of six pairs of alternating long and short blades and is fully submerged in water under normal operating conditions. The surrounding water within the actual flow domain is explicitly taken into account and is divided into internal and external flow passages. The external passage is further classified into the upper crown chamber and the lower band chamber. Figure 1 illustrates the runner geometry and the corresponding water domain model.

2.1.2. Labyrinth Seal Structure

As the present study is dedicated to a comparative assessment of upper crown seal configurations, a comb-type labyrinth seal widely adopted in high-head pumped-storage power stations is selected for analysis. A schematic illustration of the seal geometry is shown in Figure 2, A schematic diagram illustrating this labyrinth seal structure and its key parameters is presented in Figure 3. The key geometric parameters investigated include the number of comb teeth (N), tooth height (l₁), tooth pitch (l₂), tooth width (l₃), pitch circle radius of the teeth (R), axial height of the seal chamber (H), and the radial clearance in the vaneless space (L). These fundamental dimensions are systematically varied and summarized in (

Table 1. Geometric parameters of the seal clearance.

Parameter	Value	Unit
Number of seal teeth N	2, 3, 4	[—]
Height of the seal tooth l1	40, 80, 120	[mm]
Pitch of the seal teeth l2	8, 16, 24	[mm]
Width of the seal tooth l3	5, 10, 15	[mm]
Pitch circle radius of the seal teeth R	1100, 1160, 1220	[mm]
Axial height of the seal chamber H	25, 45, 65	[mm]
Radial clearance in the vaneless space L	20, 80, 160	[mm]

).

2.2. Numerical Simulation Method

The acoustic-structure coupling method is employed to investigate the wet modal characteristics of submerged structures. The surrounding water is modeled as a stationary, inviscid, and incompressible fluid, with constant mean density and pressure throughout the entire fluid domain. Under these assumptions, the fluid momentum equation can be simplified to the acoustic wave equation, namely the Helmholtz equation.

$${\nabla }^{2}p={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\alpha }^{2}p}{\mathsf{\partial }{t}^2}}$$

(1)

where p is the fluid pressure, $c=\sqrt{K/{\rho }_0}$. For the speed of sound in water. t is time; ${\nabla }^2$ For the Laplace operator.

Further application of the Galerk in method yields the governing equations in integral form within the fluid domain. Expressed in matrix form, this is:

$$\left[M_f\right]\left\{\ddot{p}\right\}+\left[C_f\right]\left\{\dot{p}\right\}+\left[K_f\right]\left\{p\right\}=\left\{F_{sf}\right\}$$

(2)

where $\left[M_f\right]$ is the fluid equivalent “mass” matrix, $\left[C_f\right]$ Equivalent “damping” matrix for fluids, $\left[K_f\right]$ Equivalent “stiffness” matrix for fluids, $\left\{F_{sf}\right\}$ Fluid “loads” generated by structural displacement at the coupling interface.

The finite element form of the structural dynamics equation for submerged structures is:

$$\left[M_s\right]\left\{\ddot{a}\right\}+\left[C_s\right]\left\{\dot{a}\right\}+\left[K_s\right]\left\{a\right\}=\left\{F_{sf}\right\}+\left\{F_s\right\}$$

(3)

where $\left[M_s\right]$ is the structural quality matrix, $\left[C_s\right]$ is the structural damping matrix, $\left[K_s\right]$ is the structural stiffness matrix, $\left\{F_s\right\}$ is the applied load vector, $\left\{a\right\}$ is the node displacement vector.

Combining the above two equations yields the following directly coupled fluid–structure interaction equations:

$$\left[\begin{array}{cc}\left[M_s\right]& 0\\ \left[M_{fs}\right]& \left[M_f\right]\end{array}\right]\left\{\begin{array}{l}a\\ \left\{\ddot{p}\right\}\end{array}\right\}+\left[\begin{array}{cc}\left[C_s\right]& 0\\ 0& \left[C_f\right]\end{array}\right]\left\{\begin{array}{l}\left\{\dot{a}\right\}\\ \left\{\dot{p}\right\}\end{array}\right\}+\left[\begin{array}{cc}\left[K_s\right]& \left[K_{sf}\right]\\ 0& \left[K_f\right]\end{array}\right]\left\{\begin{array}{l}a\\ p\end{array}\right\}=\left\{\begin{array}{l}{F}_s\\ 0\end{array}\right\}$$

(4)

where $\left[M_{fs}\right]={\rho }_0{\left[R\right]}^T$ is the equivalent coupling “mass” matrix, $\left[K_{sf}\right]=-\left[R\right]$ is the equivalent coupling “stiffness” matrix.

Further simplifying the above equation and considering that the structural external load is zero in modal analysis while neglecting damping, the equation for the free vibration of the immersed structure in the fluid domain is obtained as:

$$(\left[M_s\right]+\left[M_a\right])\left\{\ddot{a}\right\}+(\left[K_s\right]+\left[K_a\right])\left\{a\right\}=0$$

(5)

where $\left[M_a\right]$ For the additional quality matrix, $\left[K_a\right]$ is the additional stiffness matrix.

Since this paper does not consider pressure changes caused by fluid motion and only addresses the pressure effects generated by the fluid itself, the above method can be applied for analysis.

By solving the above equations, the natural frequencies of the structure in air and within the working flow channel can be obtained respectively:

$$f_a=\sqrt{{\displaystyle \frac{\left[K_s\right]}{\left[M_s\right]}}}$$

(6)

$$f_w=\sqrt{{\displaystyle \frac{\left[K_s\right]+\left[K_a\right]}{\left[M_s\right]+\left[M_a\right]}}}$$

(7)

The inherent stiffness of the structure is far greater than the added stiffness from the fluid, and is therefore generally neglected. Equation (7) can be written as:

$$f_w=\sqrt{{\displaystyle \frac{\left[K_s\right]}{\left[M_s\right]+\left[M_a\right]}}}$$

(8)

This study introduces the frequency roll-off rate (FRR) and the added mass coefficient to describe the added mass effect:

$$FRR={\displaystyle \frac{f_a-f_w}{f_a}}$$

(9)

$$\lambda ={\displaystyle \frac{\left[M_a\right]}{\left[M_s\right]}}={\left({\displaystyle \frac{f_a}{f_w}}\right)}^2-1$$

(10)

2.3. Mesh Generation

In the finite element model developed in this study, the computational domain includes the runner structural domain, the internal flow passage, and the fluid domains within the upper crown and lower band seal clearances, as shown in Figure 4. The fluid–structure interaction domain was discretized using the meshing tools integrated in the ANSYS Workbench 2022 platform, specifically within the Modal Acoustics module. To minimize the influence of mesh discretization on the numerical results, a mesh sensitivity analysis was performed. Five finite element models with different element sizes were constructed and analyzed, and the corresponding numbers of elements and nodes are summarized in (

Table 2. Number of elements and nodes for different grid models.

	Mesh 1	Mesh 2	Mesh 3	Mesh 4	Mesh 5
Number of nodes	258,961	371,894	732,598	1,275,856	2,130,317
Number of elements	170,751	246,632	491,808	860,382	1,441,546

). Figure 5 presents the convergence behavior of the natural frequencies for the selected modes across the different mesh models. The results indicate that, with progressive mesh refinement, the natural frequencies of all considered modes converge to stable values. Considering both result convergence and computational efficiency, Mesh Model 4 was selected for subsequent analyses, comprising 1,275,856 nodes and 860,382 elements.

2.4. Computational Setup

Modal analysis was performed using the Modal Acoustics module in ANSYS Workbench. For in-vacuo modal analysis of the runner, the fluid domains were deactivated. The acoustic fluid was assigned a density of 1000 kg/m³ and a speed of sound of 1483.2 m/s, while the structural material was defined as structural steel with a density of 7850 kg/m³, a Young’s modulus of 201 GPa, and a Poisson’s ratio of 0.3. To approximate realistic boundary conditions, a fixed support was applied to the runner crown in the vicinity of the central shaft, as illustrated in Figure 6. Fluid–structure interaction was enabled by defining a Fluid–Structure Interface (FSI) on the wetted surfaces between the runner blades and the acoustic domain. In addition, the bolted connection at the center of the runner shaft was modeled using a fixed constraint to ensure numerical stability. The interface between the external fluid domain and the surrounding infinite field was treated using a perfectly matched layer (PML), serving as an absorbing boundary to simulate the non-reflective propagation of acoustic waves. This treatment effectively suppresses spurious wave reflections, thereby avoiding artificial resonance and providing a more realistic representation of the acoustic response under actual operating conditions.

3. Discussion of Results

3.1. Influence of Fluid Medium on Runner Modal Characteristics

3.1.1. Comparative Analysis of Wet and Dry Modal Characteristics of the Runner

Owing to the geometric characteristics of pump-turbine runners, the axial stiffness of the upper crown and lower band is relatively low compared with that of the blades, particularly near the runner inlet. As a result, the upper crown and lower band regions at the inlet, which are not directly supported by the blades, are prone to deformation. The structural configuration of the upper crown and lower band resembles that of band disks, giving rise to modal characteristics similar to those of a disk. For such a centrally constrained disk-like structure, the mode shapes primarily exhibit axial deformations, characterized by different numbers of nodal diameters (ND) and nodal circles (NC). Accordingly, the natural modes can be classified using the notation (ND, NC) [46], with the values of ND and NC determined directly through visual inspection of the mode shapes. Modes of different orders are distinctly identified by their unique combinations of ND and NC counts. Figure 7 illustrates typical runner mode shapes. All modes considered in this study display predominantly axial vibration characteristics.

Finite element analysis yielded the first 12 modal orders of the runner in air and the first 10 modal orders when submerged in water (Figure 8). All nodal-diameter modes with n ≥ 1 appear as degenerate mode pairs, with the first 12 modes exhibiting typical nodal-diameter or nodal-circle patterns. This indicates that, within the low-order mode range, the vibration is dominated by global bending deformations. When the runner is immersed in water, the added mass effect of the surrounding fluid substantially reduces the natural frequencies, while the mode shapes remain highly similar to those observed in air.

Table 3. Modal parameters of the runner.

Modal	In Air fa/Hz	In Water fw/Hz	Frequency Reduction Rate FRR	Added Mass Factor λ
1 ND	128.43	48.77	0.620	5.936
0 NC	162.57	50.57	0.689	9.334
2 ND	186.46	82.17	0.559	4.147
3 ND	249.29	114.03	0.543	4.779
4 ND	348.29	160.18	0.540	3.728

compares the natural frequencies of the rotor in the submerged condition (within the operational flow passage) and in vacuo, and lists the frequency reduction ratio (FRR) and the added mass factor (λ) for each vibrational mode.

Owing to the relatively flat structural profile of pump-turbine runners and the corresponding sealing design, which produces similarly flat geometries for the upper crown chamber and lower band chamber, the entire external surfaces of the runner crown and band are fully wetted. Moreover, these surfaces are located in close proximity to the rigid walls of the stationary components. Compared with conventional hydraulic turbines, pump-turbine runners are therefore more sensitive to the added mass effect of the surrounding water. For ND modes, the reduction in natural frequency decreases with increasing modal order, accompanied by a corresponding decline in the added mass factor (λ), indicating a diminishing influence of the added mass effect. In contrast, the added mass factor for the 0 NC (zero nodal circle) mode is significantly higher than that for ND modes, reflecting the distinctive characteristics of the mode shape. In the 0 NC mode, all cross-sections of the crown and band oscillate in phase, moving in the same direction. By comparison, in an ND mode, when one sector of the crown or band moves upward, an adjacent sector moves downward. The fluid displaced by the upward motion partially compensates for the “cavity” generated by the downward motion of the opposing sector, resulting in a lower added mass factor for ND modes.

3.1.2. Influence of Fluid Domain on Runner Modal Characteristics

In high-head pump turbines, the runner operates with extremely narrow clearances relative to solid boundaries, such as the head cover and bottom ring, within the actual flow passages. These confined gaps cause the rigid walls to exert strong constraints on the adjacent fluid, resulting in pronounced added mass effects. To accurately identify the key fluid regions governing the wet modes of the runner and to delineate their effective influence zones, fluid–structure interaction (FSI) models were systematically constructed at multiple scales. Initially, the near-field fluid surrounding the runner was divided into three primary subregions: the upper crown chamber, the lower band chamber, and the runner’s internal flow passage. The runner’s modal characteristics under the isolated influence of each subregion were computed (

Table 4. Runner near-field flow calculation.

Number	Description
A	In air
WI	Internal flow passage only
WII	Upper crown chamber only
WIII	Lower band chamber only
WIV	Entire flow field

) to quantitatively assess their respective contributions to the added mass effect. Based on the influence identified in the near-field subregions, the analysis was subsequently extended to the entire distal fluid domain including the moveable guide vanes, stay vanes, and volute to define the boundaries of the effective influence region (

Table 5. Runner far-field flow calculation.

Number	Description
A	In air
YI	Near-runner flow field
YII	Guide vane domain included
YIII	Stay vane domain included
YIV	Volute domain included

).

This study focuses on the analysis of five vibrational modes of the runner. The results show that, under different fluid domain configurations, the mode shapes remain largely unchanged, while the natural frequencies vary due to the influence of the surrounding fluid. Figure 9 compares the natural frequencies of the runner for four distinct fluid configurations, with the in-vacuo frequencies provided as a reference. When considering only the internal flow passage, the runner’s natural frequencies experience the smallest reduction, indicating that the added mass effect from this region is relatively weak. Inclusion of the fluid within the lower band chamber results in a more pronounced frequency decrease. The most significant reduction occurs when only the upper crown chamber is considered, yielding frequencies closest to those obtained from the fully coupled model incorporating all fluid domains. This finding demonstrates that the upper crown chamber is the dominant contributor to the added mass effect in this fluid–structure interaction system. Interestingly, for both the 1 ND and 2 ND modes, the natural frequencies calculated using only the upper crown chamber or the lower band chamber are lower than those from the fully coupled model. This observation indicates that, when multiple fluid domains coexist, the added mass contributions are not simply additive but are subject to mutual constraints.

The added mass factors shown in Figure 10 further quantify the influence of each flow passage. The analysis indicates that the wet modal characteristics of the runner are dominated by a distinct region of influence, primarily the near-field fluid domain especially the upper crown chamber. The internal flow passage exhibits the smallest added mass factor, as the fluid there moves readily in phase with the runner vibration. The lower band chamber has a more pronounced effect, while the upper crown chamber contributes most significantly due to strong confinement by adjacent rigid walls. Notably, for modes such as 1 ND and 2 ND, the added mass factors for models considering only the upper crown chamber or the lower band chamber are higher than those obtained from the fully coupled model. This indicates that when multiple fluid domains are connected, the fluid displaced by runner vibration can transfer between passages, partially relieving local fluid squeezing. This phenomenon, referred to as coupling-induced attenuation, demonstrates that the added mass effect is not simply additive. Meanwhile, the added mass factors associated with distal fluid domains, including the volute, show negligible variation compared with those of the near-field fluids. This confirms that fluid located far from the runner, due to its larger inertia and reduced excitation efficiency, contributes only marginally to the total system added mass. Overall, these results indicate that the upper crown chamber is the critical region governing the runner’s dynamic behavior. Accordingly, subsequent parametric studies focus on its geometric parameters, while the influence of distal fluid domains on the runner’s modal characteristics can be safely neglected.

3.2. Study on the Influence of Seal Clearance Geometry on the Wet Modal Frequency of the Runner

3.2.1. Influence of Labyrinth Seal Comb Structure on Runner Modal Characteristics

To systematically examine the influence of seal clearance geometry on the runner’s wet modal characteristics, five key design variables were selected: the number of labyrinth seal teeth, tooth width, tooth height, tooth pitch, and the pitch circle radius of the teeth. The analysis indicates that the wet modal behavior of the runner exhibits pronounced nonlinear sensitivity to these geometric parameters, with the degree of sensitivity depending on both the parameter type and the modal order. As shown in Figure 11, two general trends emerge regarding the effect of each parameter on the natural frequencies. Increasing the comb tooth width leads to a monotonic decrease in frequency, whereas variations in the number of teeth, tooth height, tooth pitch, and pitch circle radius produce a frequency response that initially increases and then decreases. This behavior suggests the existence of an optimal parameter combination that maximizes the system stiffness. Further examination reveals mode-dependent sensitivities. Nodal circle (NC) modes are more sensitive to changes in the number of teeth, tooth width, and pitch circle radius, with maximum frequency variation rates reaching 11.45%. In contrast, ND modes respond more strongly to variations in tooth height and tooth pitch, achieving peak variation rates of up to 13.73%.

3.2.2. Influence of Seal Clearance Geometry on Runner Modal Characteristics

(1) Axial Height of the Seal Chamber

Based on the baseline seal chamber geometry, three distinct runner configurations with varying axial heights of the seal chamber were modeled, while all other structural parameters were kept constant. Figure 12 illustrates the natural frequencies of typical vibrational modes for each runner configuration. As the axial height of the seal chamber increased from 25 mm to 65 mm, the natural frequencies of all five monitored modes exhibited a consistent upward trend. A comparative analysis of the frequency increase rates, using the configuration with a 65 mm axial height as the reference (

Table 6. Natural frequency reduction relative to the 25 mm seal cavity height baseline.

	H = 45 mm	H = 65 mm
1 ND	−13.69%	−13.85%
0 NC	−14.57%	−15.85%
2 ND	−12.30%	−14.12%
3 ND	−10.30%	−11.47%
4 ND	−0.37%	−0.58%

), reveals a nonlinear relationship: the increase in natural frequency is more pronounced when the axial height is raised from 25 mm to 45 mm, whereas a further increase from 45 mm to 65 mm results in a noticeably smaller and more gradual frequency change. Classified by modal type, variations in the axial height of the seal chamber exerted a more significant influence on the first four ND modes and the NC mode, with observed shifts exceeding 10% and reaching a maximum of 15.85%. In contrast, the impact on the fifth ND mode was relatively minor, exhibiting a maximum variation of only 0.58%.

(2) Radial Clearance in the Vaneless Space

Based on the baseline seal chamber geometry, three distinct runner configurations with varying radial clearances in the vaneless space were modeled by altering this specific clearance while keeping all other structural parameters constant. Figure 13 illustrates the variation in the natural frequencies of typical vibrational modes for each runner. As the radial clearance in the vaneless space increased from 20 mm to 160 mm, the natural frequency of the ND mode initially decreased and then increased. A comparative analysis of the frequency change rates, using the configuration with a 20 mm radial clearance as the baseline (

Table 7. Natural frequency reduction relative to the 20 mm vaneless space radial clearance baseline.

	L = 80 mm	L = 160 mm
1 ND	14.73%	11.90%
0 NC	10.65%	16.93%
2 ND	12.73%	10.84%
3 ND	9.87%	8.49%
4 ND	0.90%	0.40%

), indicates that the frequency shift is most pronounced when the radial clearance increases from 20 mm to 80 mm, exhibiting a maximum variation of 14.73%. In contrast, a further increase in radial clearance from 80 mm to 160 mm results in a more gradual frequency change. Conversely, the natural frequency of the NC mode exhibits a monotonically decreasing trend with increasing radial clearance. Regarding the magnitude of influence, variations in the radial clearance of the vaneless space exert the most significant effect on the NC mode, with a maximum variation reaching 16.93%. The effect is also considerable for the first four ND modes, whereas the impact on the fifth ND mode is relatively minor.

3.2.3. Sensitivity Analysis of Wet Modes to Seal Clearance Geometry

We conducted a sensitivity analysis across a seven-dimensional parameter space, encompassing the number of teeth, tooth width, tooth pitch, pitch circle radius, and tooth height of the labyrinth seal, together with the axial height of the seal chamber and the radial clearance in the vaneless space. The ratio of the change in natural frequency to its initial value was defined as the normalized frequency shift. This approach systematically quantified the sensitivity indices of each geometric parameter on the ND and NC mode frequencies (

Table 8. Sensitivity analysis of runner wet modal parameter.

Parameter	Variation Range	1 ND Luffing	0 NC Luffing	2 ND Luffing	3 ND Luffing	4 ND Luffing
Number of seal teeth (N)	3 → 4	−3.32%	−11.45%	−1.65%	−0.92%	−0.24%
Height of the seal tooth (l1)	80 → 120	−9.09%	−0.51%	−3.60%	−0.38%	+4.33%
Pitch of the seal teeth (l2)	16 → 24	−13.73%	−5.02%	−6.06%	−6.85%	−0.33%
Width of the seal tooth (l3)	5 → 15	−5.74%	−13.98%	−2.63%	−1.39%	−0.19%
Pitch circle radius of the seal teeth (R)	1160 → 1220	−2.89%	−10.42%	−1.57%	−0.73%	−0.33%
Axial height of the seal chamber (H)	65 → 25	−13.85%	−15.85%	−14.12%	−11.47%	−0.58%
Radial clearance in the vaneless space (L)	20 → 80	−14.73%	−10.65%	−12.73%	−9.87%	−0.90%

). The results indicate that the parameters defining the seal chamber geometry generally exhibit higher sensitivity on the runner natural frequencies than those defining the labyrinth seal comb structure. For the NC mode, parameters such as the number of seal teeth, tooth width, pitch circle radius, and tooth height show lower sensitivity, resulting in limited frequency variation. In contrast, the tooth pitch, axial height of the seal chamber, and radial clearance in the vaneless space display high sensitivity to the NC mode, causing considerably larger frequency shifts. For the ND mode, most parameters exhibit strong sensitivity, except for the tooth pitch and tooth height. Among these, the radial clearance in the vaneless space and the axial height of the seal chamber exert the most pronounced influence, producing the largest variations in natural frequency.

3.3. Analysis of the Influence of Labyrinth Seal Parameters on Harmonic Response Characteristics

To thoroughly investigate the influence of key geometric parameters specifically, the comb tooth pitch, the axial height of the seal chamber, and the radial clearance in the vaneless space on the dynamic response of the runner and the associated fluid damping effects, a harmonic response analysis was conducted based on the validated acoustic–structure interaction model. The analysis was performed using the Harmonic Acoustics module in ANSYS Workbench 2020 R1. The computational domain included the runner structure and the surrounding fluid volume, with boundary conditions and constraints configured identically to those applied in the preceding modal analysis. A harmonic pressure excitation with an amplitude of 1 Pa was applied at the inlet boundary of the fluid domain to simulate flow passage pressure pulsations. The frequency sweep ranged from 20 Hz to 250 Hz, covering the dominant modal frequencies of the runner. Response probes were positioned at the blade tips, and displacement responses were recorded assuming a structural damping ratio of 0.003. The results of the harmonic response analysis are presented in Figure 14.

The results reveal significant differences in how the various geometric parameters influence the dynamic characteristics of the runner. Variations in the tooth pitch exert a relatively limited effect on the frequency response. Within the investigated range of 8 mm to 24 mm, the response curves display highly similar shapes, with primary resonance peaks occurring near 50 Hz and 200 Hz. Neither the peak amplitudes nor the corresponding frequencies show significant shifts. In contrast, the axial height (H) of the seal chamber has a more pronounced effect. The response curves for H = 45 mm and H = 65 mm are relatively close, exhibiting stable resonance characteristics. However, at H = 25 mm, the response exhibits marked differences: the amplitude of all resonance peaks decreases significantly, and the smoothness of the curve deteriorates at high frequencies, resulting in a more complex response. The radial clearance (L) in the vaneless space also substantially affects the response amplitudes. The responses stabilize at L = 80 mm and L = 160 mm, whereas at L = 20 mm, the primary low-frequency resonance peak shifts to approximately 50 Hz. Furthermore, the high-frequency response amplitude near 200 Hz is significantly higher than in the other cases, indicating an increased risk of resonance. In summary, the axial height of the seal chamber and the radial clearance in the vaneless space are the primary parameters governing runner vibration, whereas variations in the tooth pitch have a comparatively minor effect on the vibration characteristics.

4. Conclusions

Based on a quantitative analysis of seven key geometric parameters of the seal clearance and the fluid regions around the runner, this study systematically clarifies the influence of these parameters on the wet modal characteristics of a pump-turbine runner. The results show that the effects of different geometric parameters on the runner’s dynamic behavior vary significantly.

(1)

The wet modal characteristics are mainly governed by the fluid in the immediate vicinity of the runner. The upper crown chamber plays the most critical role, whereas the influence of the far-field fluid domain is negligible. This finding supports the use of a simplified near-field model for efficient modal analysis.

(2)

Among the seal clearance geometric parameters, the axial height of the seal chamber and the radial clearance in the vaneless space exhibit the most pronounced influence on wet modal frequencies. In particular, the radial clearance in the vaneless space strongly affects the NC modes, with a maximum frequency shift of 16.93%.

(3)

For the labyrinth seal teeth, the tooth pitch is the dominant geometric parameter influencing the modal behavior. It can cause a frequency variation of up to 13.73% in the 1ND mode.

(4)

Harmonic response analysis further confirms that the axial height of the seal chamber and the radial clearance in the vaneless space are key geometric factors governing the runner’s vibration response and resonance risk. Variations in these parameters notably shift resonant amplitudes and frequency distributions. For example, reducing the radial clearance in the vaneless space to 20 mm leads to a pronounced resonance peak with substantially amplified amplitude near 200 Hz.