Influence of Load Variation on the Flow Field and Stability of the Francis Turbine

Abstract

With the development of a power system predominantly reliant on new energy sources, turbine generator sets are increasingly required to operate under wide load conditions, resulting in numerous unstable flow phenomena and substantial economic losses for power stations. This study employs the Shear Stress Transport (SST) k-ω turbulence model to combine numerical simulations with experimental methods. It calculates the guide vane opening at the rated head of a Francis turbine and examines the internal flow field characteristics and pressure pulsations under various operating conditions. The findings indicate that the entropy production ratio in the draft tube is the highest among all load conditions, ranging from about 72.7% to 95.9%. Energy dissipation in the vaneless zone and the runner increases with greater opening. At 45% and 100% load conditions, the draft tube is mainly influenced by dynamic and static interference, single and double frequencies induced by runner rotation, and low-frequency fluctuations of the vortex and. Under 60% load conditions, pressure fluctuations in the draft tube are primarily caused by the eccentric vortex band, characterized by higher intensity and a frequency of 0.2 f_n. Numerical results closely align with experimental observations. The findings provide essential guidance for ensuring the stable operation of power plant units.

1. Introduction

The construction and operation of hydropower projects are closely associated with energy supply, flood control safety, and the promotion of sustainable development at both national and regional levels. The water turbine generator set is a crucial component for hydropower energy production, and its stability significantly influences the overall safety and reliability of hydropower projects. With the establishment of new power systems dominated by renewable energy, hydropower generation is required to meet three key criteria: continuous regeneration, flexible regulation, and inertial support. These attributes enable hydropower to perform vital functions such as power grid adjustment and balancing intermittent renewable energy sources. However, the wide-load operation of turbine generator units introduces numerous unstable flow phenomena within the turbine, complicating power station operation and necessitating frequent maintenance. In severe cases, such instability may result in catastrophic damage to equipment and facilities, leading to significant economic losses.

Extensive numerical and experimental research has been conducted by scholars globally to analyze the internal pressure pulsation of hydraulic turbines. Chen et al. [1] discovered that variations in the active guide vane opening of a Francis turbine significantly influence the dynamic and static interactions within the vaneless region. Li et al. [2,3,4,5] conducted unsteady numerical simulations at the rated head for various guide vane openings, investigating the causes and characteristics of pressure pulsations across different flow components. Liang et al. [6,7,8,9,10] analyzed pressure pulsations within the vaneless region, while Ren et al. [11] studied the flow characteristics and pressure pulsations inside the draft tube of a Francis turbine running at its rated head with various guide vane openings. Wang et al. [12] conducted numerical simulations of the unsteady flow field and sound field in a Francis turbine with varying guide blade openings, revealing that pressure pulsations within the runner were primarily influenced by blade passing frequency and low-frequency pressure fluctuations. Zuo et al. [13], through numerical simulations and experimental methods, identified that pressure pulsations of varying frequencies were caused by different types of blade vortices, leading to a weaker pressure fluctuation at the midsection of the runner blade and a stronger fluctuation near the lower ring. Trivedi et al. [14] discovered that the amplitude of pressure pulsation in the vaneless zone and runner decreased when the turbine was shut down under steady-state best efficiency point (BEP) load conditions, compared to normal shutdown scenarios. Hu et al. [15], utilizing computational fluid dynamics (CFD) simulations, investigated the flow characteristics and pressure pulsation distribution of a turbine across five distinct operating conditions. Their analysis provided insights into the variation laws of pressure pulsations. Li et al. [16] employed numerical simulations to evaluate the performance of hydraulic turbines operating at different rotational speeds, concluding that an increase in speed broadened the turbine’s high-efficiency zone and significantly reduced pressure pulsations in the runner. Minakov et al. [17] employed the Detached-Eddy Simulation (DES) turbulence model to simulate unsteady turbulence within the channel of a high-head hydropower station, revealing that low-frequency pulsations in the runner were primarily attributed to the vortex region downstream of the runner. Jamali et al. [18] observed that pressure pulsations are more pronounced at lower power levels. Unterluggauer et al. [19] employed both experimental and numerical approaches to investigate the startup process of a prototype Francis turbine runner. They observed the formation of reverse-rotating vortices in the draft tube as the guide vanes opened. In most instances, four such vortices were found to be the primary cause of high-pressure pulsations. Kumar et al. [20] examined the formation of spiral vortex bands and their influence on hydraulic turbine performance, with a focus on the pressure pulsations caused by these structures. Joy et al. [21] discovered that the adjustable guide vane system in a high-head Francis turbine substantially reduced pressure pulsations and enhanced the operational flexibility of the turbine. Wang et al. [22] determined that pressure fluctuations in the draft tube are a primary factor contributing to hydraulic instability in Francis turbines. Yamamoto et al. [23] found that, under deep load conditions, cavitation eddies formed between the runner blades, causing both pressure pulsations and erosion. Wang et al. [24] employed the shear stress transport (SST) k-ω turbulence model coupled with the pressure implicit with the splitting of operators (PISO) algorithm to more accurately predict runner pressure pulsations. Platonov et al. [25] performed an experimental study to characterize the ground pressure pulsations in a Francis turbine, analyzing the flow structures downstream of the runner.

The hydraulic losses in a turbine are closely associated with factors such as friction losses, vortex motion, and backflow. Traditional assessment methods for hydraulic losses are unable to accurately determine their distribution and specific locations. While the entropy generation theory has been primarily applied to three-dimensional pumps, it remains underutilized in the study of Francis turbines. Compared with the traditional head loss coefficient based on pressure drop calculations, the local entropy generation rate more accurately reflects the physical flow losses. [26] Thus, the proper application of entropy production theory can also serve to assess flow losses in hydraulic machinery, providing a robust theoretical foundation for further enhancing the efficiency of the unit. Based on entropy production theory, Yu et al. [27,28,29] found that the vortex bands inside the draft tube rotate in the same direction as the flow passage, albeit at a lower angular velocity. This results in low-frequency, high-amplitude pressure pulsations within the draft tube. Wang et al. [30] developed a method for identifying the transmission path of fluid pressure pulsations by utilizing the time-delay transfer entropy method combined with wavelet theory. Zhou et al. [31] proposed an improved strategy to mitigate the formation of vortex bands, thereby reducing the associated pressure pulsations in the draft tube of the runner. Luo et al. [32] found that generation at the wall surface of the J-shaped groove and draft tube was significantly lower than that of the non-J-shaped groove. Additionally, the groove reduced the circumferential velocity component, leading to significant weakening of pressure pulsations and eddy current intensity within the conical section. Trived et al. [33] conducted comprehensive experimental and numerical investigations into pressure pulsations within the vaneless space, runner, and draft tube of Francis turbine.

Using the SST k-ω turbulence model, an analysis was performed to examine the flow field characteristics and pressure pulsations in units with varying guide vane openings. The results were compared with experimental data to evaluate the impact of guide vane openings on unit stability, providing insights to support the stable operation and maintenance of Francis turbines.

2. Mathematical Model

2.1. Fundamental Governing Equation

The flowing medium inside the turbine can be regarded as an incompressible fluid, so its basic equation can be expressed as:

Continuum equation:

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

(1)

Momentum equation:

$${\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+\nu {\displaystyle \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}}+g_i$$

(2)

where t is time, u is the fluid velocity, ρ is the fluid density, p is the pressure, g is the gravitational acceleration, ν is the viscosity coefficient of fluid motion, x is the coordinate. The subscripts i and j are coordinate tensors.

2.2. Turbulence Model

The SST k-ω turbulence model is widely employed in fluid machinery due to its robustness. Numerous experts and scholars have utilized this model to investigate the internal flow field and stability of Francis turbines and pumps. By integrating vortex characteristics, entropy production theory, and pressure pulsation analysis, they have gained deeper insights into the performance and reliability of these machines [34,35,36,37].

When simulating turbulent flow, the SST k−ω model automatically applies the standard k−ω model in the near wall region, and automatically switches to the standard k−ω model in the far field. The transport term of turbulent shear stress is reflected in the equation of SST k−ω model, which is widely used in the unsteady flow simulation of rotary hydraulic machinery such as water turbines. Its expression is as follows:

k equation:

$${\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial (k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\nu +{\sigma }_k{\nu }_t\right){\displaystyle \frac{\partial k}{\partial x_i}}\right]+P_k-\beta *k\omega$$

(3)

ω equation:

$${\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial (\omega u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\nu +{\sigma }_{\omega }{\nu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_i}}\right]+2(1-F_1){\displaystyle \frac{{\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\partial k}{\partial x_i}}{\displaystyle \frac{\partial \omega }{\partial x_i}}+P_{\omega }-\beta {\omega }^2$$

(4)

where P_k is the turbulent kinetic energy generation term, P_ω is the turbulent dissipation term, F₁ is a mixed function, ν_t is the turbulent eddy viscosity, and β, β*, σ_k, σ_ω and σ_ω2 are turbulent model constants.

3. Construction of Geometric Model and Division of Mesh

3.1. Three-Dimensional Geometrical Model

The geometric model of the complete flow channel for the Francis turbine at the research power station was developed using Unigraphics NX 12.0 modeling software. The design specifications of the turbine are detailed in

Table 1. Design Parameters of Hydraulic Turbines.

Parameter	Value	Parameter	Value
Rated Head Hr (m)	159.3	Number of runner blades	15
Rated output P (MW)	90	Number of stay vanes	20
Rated Speed nr (r/min)	272.7	Number of guide vanes	20
Rated Discharge Qr (m3/s)	62.7	Runner inlet diameter D1 (mm)	2975
Suction Height Hs (m)	−3	Runner outlet diameter D2 (mm)	2472

, and the three-dimensional model of the water body within the turbine’s full channel is illustrated in Figure 1.

3.2. Grid Division and Independence Verification

Meshing was conducted using Ansys ICEM and Ansys Workbench Mesh. Due to the intricate geometry of the turbine flow components, including the runner blade, which is a three-dimensional twisted surface with an irregular shape, and the draft tube, which exhibits a complex flow pattern prone to vortex formation, a structural grid was applied. For the volute, fixed guide vane, and movable guide vane, an unstructured mesh was used, with local encryption in areas of interest. This approach reduces numerical calculation time while ensuring accuracy. The complexity of the runner structure is depicted in Figure 2. Grid independence verification for water intake head and efficiency demonstrated that when the grid count reached 12.1 million, further increases in the number of grids had minimal impact on improving calculation accuracy, with the water head and efficiency stabilizing, as shown in Figure 3. Therefore, 12.1 million grids were selected for the flow calculations.

3.3. Boundary Condition

The calculation employed velocity inlet and static pressure outlet. All solid walls were set as non-slip boundaries. Transient calculations used the results from steady-state analysis as the initial conditions. Each 2° rotation of the runner was taken as a time step, with 180° corresponding to one period, and the runner’s rotational frequency was f_n = 4.54 Hz. The time step for one revolution was 0.22 s, with a step length of 0.001 s. The total simulation time for 20 revolutions was 4.4 s, and the convergence criterion for the results was set to 10⁻⁵.

4. Analysis of Calculation Results

To evaluate internal flow behavior and stability variations in the turbine under different guide vane openings, three load conditions were selected, corresponding to a rated head of 159.3 m and guide vane openings of 45%, 60%, and 100%. As shown in Figure 4, the calculated results closely match the experimental data, with all errors remaining within 0.5%.

According to the second law of thermodynamics, entropy generation is an inevitable and irreversible process in energy conversion. Due to fluid viscosity, Reynolds stress, and turbulence, mechanical energy is invariably transformed into internal energy [38]. From a thermodynamic standpoint, the energy loss within the internal flow of a Francis turbine can be assessed using the entropy production method. In turbulent flows, the total local entropy production rate includes contributions from both the time-averaged motion and the turbulence-induced entropy production rate, as demonstrated in Equations (5)–(7). It is worth noting that the contribution from the time-averaged motion is minimal and can be disregarded [27,39]. CFX post is used to edit and input the entropy yield formula, and then form a new variable for entropy yield calculation.

$${\dot{S}}_D^{″}={\dot{S}}_{\overline{D}}^{″}+{\dot{S}}_{D^{\prime }}^{″}$$

(5)

$$\begin{array}{l}{\dot{S}}_{\overline{D}}^{″}={\displaystyle \frac{2\mu }{T}}[{({\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_1}})}^2+{({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_2}})}^2+{({\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_3}})}^2]+\\ {\displaystyle \frac{\mu }{T}}[{({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_1}}+{\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_2}})}^2+{({\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_1}}+{\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_3}})}^2+{({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_3}}+{\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_2}})}^2]\end{array}$$

(6)

$$\begin{array}{l}{\dot{S}}_{D^{\prime }}^{″}={\displaystyle \frac{2\mu }{T}}[{({\displaystyle \frac{\partial u_1^{\prime }}{\partial x_1}})}^2+{({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_2}})}^2+{({\displaystyle \frac{\partial u_3^{\prime }}{\partial x_3}})}^2]+\\ {\displaystyle \frac{\mu }{T}}[{({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_1}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial x_2}})}^2+{({\displaystyle \frac{\partial u_3^{\prime }}{\partial x_1}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial x_3}})}^2+{({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_3}}+{\displaystyle \frac{\partial u_3^{\prime }}{\partial x_2}})}^2]\end{array}$$

(7)

where ${\dot{S}}_D^{″}$ is total entropy production rate; ${\dot{S}}_{\overline{D}}^{″}$ is entropy production rate induced by time-averaged movement; ${\dot{S}}_{D^{\prime }}^{″}$ is entropy production induced by velocity fluctuations. T is temperature; u_i’ is turbulent fluctuation velocity; ${\overline{u}}_i$ is time-averaged velocity; i means the three directions in the Cartesian coordinate system (i = 1, 2, 3); μ is effective dynamic viscosity.

Thus, the draft tube exhibits the highest proportion of entropy production across all load conditions, ranging from approximately 72.7% to 95.9%, as shown in Figure 5. Conversely, the entropy production in the volute, stay vanes, and guide vanes regions is minimal, particularly under the 45% load condition.

4.1. Internal Flow Field Analysis

The comparison of energy losses at different loads is shown in

Table 2. Comparison of energy losses at different loads.

	45%	60%	100%
Vaneless	The entropy production at the trailing edge of the guide vane is minimal.	The entropy production around the stay vane is higher than that of the other two operating conditions	The entropy production range of the trailing edge of the guide vane is the largest, and the two ends are higher than the middle
Runner	There is almost no high-entropy production area in the tail of the blades		Span = 0.9 The entropy yield range of the blade tail was the largest
Draft tube	High-entropy production areas occur at both ends of the proximal wall	High entropy production areas occur at both ends of the proximal wall, and the range of elbow segments is the largest	The high-entropy production zones are symmetrically distributed about the centerline of the straight conical section.

4.1.1. Vaneless Space

As the load increases, the region of high entropy production in the vaneless zone progressively expands, as illustrated in Figure 6. Wake loss and flow separation are primary contributors to the increased entropy production in the guide vane region. The high-velocity flow within the vaneless region generates a significant velocity gradient. As the flow rate increases, the vortex motion and backflow, which result from flow separation, become more prominent in specific flow paths of the guide vanes. This leads to flow path blockage. At 60% load, the entropy production around the stay vane is higher than that at 45% and 100%. In the case of 100% loading, the entropy production near the top and bottom of the guide vane is higher than that in the middle area.

4.1.2. Runner

As a primary contributor to hydraulic loss in the turbine unit, entropy generation was analyzed across single rotating surfaces with varying spread directions using blade-to-blade analysis. Three flow surfaces, labeled Span0.1, Span0.5, and Span0.9, are defined from the lower shroud to the upper crown of the flow passage, as illustrated in Figure 7. Across all load conditions, high entropy production on the runner blades was predominantly concentrated at the inlet and outlet of the suction surface. Entropy production near the lower circulation surface was greater than that near the upper crown. The primary cause of entropy generation at the blade inlet was the significant impact of inlet flow on the leading edge of the runner blade, which induced local backflow and substantial hydraulic loss. At the blade outlet, hydraulic losses arose from the high velocity cycle and a notable increase in entropy production.

Flow separation begins at the inlet of the runner blade and extends from the upper crown to the lower ring, as illustrated in Figure 8. Near the blade inlet at the upper crown, flow separation was minor, with limited reflux observed at the blade’s leading edge. However, as the reflux region expanded and moved, flow separation intensified. With increasing guide vane opening, the extent of flow separation in the runner channel decreased and shifted toward the blade’s trailing edge. The yellow squares mainly mark the small swirls at the edge of the blade intake.

4.1.3. Draft Tube

The pressure pulsations induced by the eccentric vortex zone in the draft tube constitute a primary source of unit oscillations. Energy losses arise from the precession of the vortex, velocity-induced diffusion near the walls, and secondary flows within the draft tube. Hydraulic losses can be mitigated through measures such as filling the draft tube, modifying the water flow within it, or controlling the eccentricity of the vortex core.

Figure 9 illustrates the distribution of entropy production in the draft tube. Under partial load conditions (non-rated conditions), areas of high entropy production are primarily concentrated at both ends of the straight cone section and the two ends of the elbow section. In contrast, under rated load conditions, these regions are symmetrically distributed in the central area of the straight conical section, exhibiting lower entropy production compared to partial load conditions.

Figure 10 shows the cross-sectional view of the draft tube. The tangential velocity near the wall is obviously higher than the center velocity in the straight cone section of the draft tube, which results in a significant swirl intensity. This effect is more pronounced as the guide vane opening increases, resulting in the formation of a vortex zone near the wall, as shown in Figure 11a,b. The secondary flow resulting from the vortex zones in the elbow section leads to increased hydraulic losses within the draft tube. Under partial load conditions, the vortex centers at different cross-sections exhibit continuous movement and variation, with the changes in flow lines aligning with regions of high entropy generation. Minimizing the secondary flow near the wall vortex zones in both the vertical cone section and the elbow section can help reduce hydraulic losses within the draft tube during partial load conditions. Under rated load conditions, as the inlet flow velocity of the draft tube increases, the downstream centerline flow velocity remains stable, while the range of maximum flow velocities narrows. The vortex center remains stable, facilitating the formation of a single cylindrical vortex zone, as shown in Figure 11c. Compared to the entropy generation distribution, secondary recirculation in the elbow section is nearly eliminated.

4.2. Pressure Pulsation Analysis

The comparison of pressure pulsations under different loads is shown in

Table 3. Comparison of pressure pulsations under different loads.

	45%	60%	100%
Vaneless		A low-frequency pressure ripple of 0.2 fn is observed, with the largest amplitude of pressure ripple resulting from dynamic and static interference.
Runner		A low-frequency pressure pulsation of 0.8 fn occurs	The amplitude of pressure fluctuation caused by dynamic and static interference is the largest
Draft tube	The amplitude of the pressure ripple and the extent of the vortex zone are clearly observable.	The pressure ripples caused by the vortex belt exhibit the largest amplitude.	The maximum value is the amplitude of the pressure ripple due to the dynamic and static interference

.

4.2.1. Monitoring Point Setting

To monitor transient variations in runner pressure pulsation, monitoring points were strategically arranged along the turbine’s key flow paths. The monitoring points include SG (stay vane to guide vane) and GR (guide vane to runner), representing the vaneless area, R (runner) for the runner flow path located at 50% blade height, and DT (draft tube) for the draft tube, as shown in Figure 12.

The pressure pulsation intensity of the Francis turbine was quantified using the pulsation coefficient C_p, calculated as:

$$C_p={\displaystyle \frac{p_i-\overline{p}}{\rho gH}}\times 100\%$$

(8)

where p_i is the pressure at point i, Pa; $\overline{p}$ is the average pressure, Pa; and H is the water head, m.

In the unsteady computations, one time step was taken for every 2° rotation of the runner. With a rotational speed of 272.7 rpm (4.54 Hz), pressure pulsation data from the last five periods were extracted. Using the fast Fourier transform (FFT), the time-domain signals were converted to the frequency domain, allowing discrimination of pulsation amplitudes across different frequencies and identification of the pulsation sources.

4.2.2. Vaneless Space

Across all load conditions, the pressure pulsation coefficient near the runner monitoring point was observed to be high, with a peak at the primary frequency of 15. This is due to uneven pressure distribution caused by the runner’s periodic rotation through the vaneless area. A more severe uneven pressure distribution results in higher pulsation amplitudes, as depicted in Figure 13. As the guide vane opening (load) increases, the decreasing gap between the guide vanes and the flow passage results in a rise in pulsation amplitude and enhanced dynamic-static interference. Additionally, at the 60% load condition, a low-frequency pressure pulsation at 0.2 f_n was observed, attributed to the strong vortex band formation in the draft tube.

4.2.3. Runner

Figure 14 illustrates the pressure pulsation variations within the runner. The results indicate that the pressure pulsations in this region are primarily caused by dynamic-static interference, with the maximum pulsation amplitude concentrated at the runner inlet. The dominant frequency of these pulsations is 20 f_n. The pressure pulsation decreases progressively downstream. A single frequency generated by the rotation of the runner itself is evident under all three operating conditions. At 60% load, low-frequency pressure pulsations with a frequency of 0.8 f_n are observed, increasing gradually from the inlet to the outlet. It shows that under this working condition, the pressure fluctuation generated by the eccentric vortex belt in the tailpipe propagates upstream and the low-frequency pressure fluctuation generated by the vortex inside the runner is caused by the combined action, resulting in the internal flow disorder and unstable operation of the runner. Under 100% load, the pressure pulsation amplitude is higher compared to the other two operating conditions.

4.2.4. Draft Tube

According to the principle of vortex motion, under partial load, when the $Q/(\mathsf{\Gamma }R)$ (Q is the flow rate, $\mathsf{\Gamma }$ is the potential equal ring, and R is the radius of the tube) value is small, to a certain extent, there will be reflux in the center of the vortex, and the reflux will flow into the runner blade flow channel and then flow out of the flow channel. The main flow and return flow form a velocity triangle in the flow channel of the runner blades. In this way, the sudden jump in velocity and the change of direction in the flow channel will form a flow discontinuity in both the axial and axial directions, which will become the source of vortex formation. Due to the vortices formed by the circumferential velocity being opposite to the runner’s rotation direction, while those formed by the axial velocity are aligned with it, two phenomena occur at the draft tube inlet: on one hand, a forced vortex forms at the axis center; on the other hand, an axially symmetric vortex ring is generated by the backflow at the blade outlet. In actual operation, various factors prevent the vortex rings from being perfectly axially symmetric, leading to induced axial velocities that cause the original central vortex to deviate from its initial position. This deviation marks the onset of helical vortex formation.

Figure 15 illustrates the variations in pressure pulsation in the draft tube under different operating conditions. It is evident that under 45% and 100% load conditions, the draft tube is primarily affected by dynamic and static interference, the double frequency generated by runner rotation, and low-frequency pressure pulsations induced by the vortex band. The pressure pulsation amplitude with a main frequency of 0.2 f_n is notably higher than under other conditions. The amplitude increases progressively from the inlet, reaches a peak at DT06, and then declines at DT13. This behavior occurs because, under these conditions, the vortex band in the draft tube is predominantly concentrated in the straight cone section, with only a portion extending into the elbow section. As a result of the interaction between the runner and the draft tube, the pressure pulsation amplitude is relatively high in the inlet section of the draft tube. The vortex zone entering this region emerges as a key factor contributing to the instability within the draft tube.

To mitigate or eliminate the effects of pressure pulsation, several measures can be taken: installing a cross in the draft tube, extending the conical section of the draft tube, increasing the cone angle, adding water barriers and diversion partitions, or improving the design of the discharge cone.

5. Analysis of the Test Results

This evaluation employs a range of sensing elements to transform the non-electrical signals from various sections of the unit into electrical signals. These signals are then processed through electrical measurement techniques. The electrical signals are converted into data displayed on a computer, enabling analysis of the transformed signals. The principle underlying the stability test measurement is illustrated in Figure 16, while the field installation of the sensors is shown in Figure 17.

5.1. Pressure Pulsation Within the Vaneless Region

The relative amplitude of pressure pulsation in the vaneless region ranges from 2% to 4.5%, as depicted in Figure 18. Within the output range of 40–60 MW (44%~67% load), the relative amplitude of pressure pulsation in this region initially increases and then decreases, aligning with the numerical simulation results. This trend indicates that during variations in opening, the force exerted by the runner on the guide vane becomes more pronounced, enhancing the radial and axial propagation of pressure pulsations and increasing the likelihood of hydraulic vibration. When the output exceeds 90 MW (100% load), the relative amplitude of pressure pulsation shows a tendency to rise.

5.2. Pressure Pulsation of the Draft Tube

Figure 19 illustrates the test outcomes regarding the relative amplitude variations of draft tube pressure pulsation across different operating conditions. It is noted that the relative amplitude of these pressure pulsations varies between 0.5% and 5.5%. Overall, the pressure pulsation demonstrates a pattern of initially decreasing followed by an increase. Within the output range of 0–35 MW (below 40% load), the relative amplitude of pressure pulsation in the draft tube is influenced by the reduced flow rate at the runner outlet, which causes a reduction in the axial surface velocity at the runner outlet. Furthermore, a circular, tree-shaped vortex zone predominantly develops within the draft tube, leading to a significant relative amplitude of turbine pressure pulsation. In the output range of 35–70 MW (40%–78% load), the draft tube is mainly influenced by the eccentric vortex zone, which gradually diminishes as the guide vane opening increases. This behavior aligns closely with the results obtained from numerical simulations. At the output range of 80–90 MW (89%–100% load), the amplitude of pressure fluctuation in the draft tube remains relatively moderate, with no evidence of abnormal fluctuations. After 90 MW (100% load), the pressure pulsation demonstrates an upward trend.

6. Conclusions

The effect of guide vane opening on turbine pressure pulsation was evaluated by comparing computational results across different load conditions. These assessments were supported by pressure pulsation measurements taken on the model turbine testing platform. The primary findings and conclusions are as follows:

• The draft tube accounts for the majority of the turbine’s energy dissipation, ranging from about 72.7% to 95.9%. The peak dissipation occurs at the 45% load condition. The energy loss between the vaneless region and the runner is mainly due to backflow and eddy currents caused by flow separation. At full load (100%), the energy loss in the vaneless region and the runner exceeds that observed at 45% and 60% loads.
• The pressure fluctuations caused by the vortex belt are the primary contributors to the unstable operation of the unit. At 45% and 100% load conditions, the draft tube is mainly affected by dynamic and static interference, rotor-generated single and double frequencies, and low-frequency vortex belt oscillations. Under 60% load, the pressure pulsation in the draft tube is predominantly driven by the vortex band at a frequency of 0.2 f_n, with its amplitude notably higher than that observed at 45% and 100% load conditions. Measures can be taken to reduce or eliminate hydraulic losses, such as introducing air into the draft tube, modifying the water flow within it, or controlling the eccentricity of the vortex belt.
• The relative amplitude of pressure pulsation in the draft tube ranged from 0.5% to 5.5%, markedly higher than that observed in the vaneless region. Additionally, the amplitude was more pronounced under partial load conditions, consistent with the findings from numerical simulations.