Prediction of runner eccentricity and Alford force of a Kaplan turbine based on variational mode decomposition

The rotor blades of an axial‐flow turbine are cantilever structures, and there is inevitably a gap between them and the casing. Due to factors such as rotor wear and unit vibration, the eccentricity of the impeller will change during the operation of the turbine, resulting in the impeller being affected by additional radial forces, which can even lead to rubbing or biting between the impeller and the casing. To monitor the eccentricity of the impeller and the additional radial forces in real time during the operation of the turbine, this study conducted numerical simulations of the internal flow of the turbine under different eccentricities of the impeller, and analyzed the characteristics of pressure pulsation and impeller radial force in the turbine using the variational mode decomposition method. The results showed that there was a good linear relationship between the eccentricity of the impeller and the amplitude of the frequency corresponding to the rotor in pressure pulsation at the monitoring point and the Alford force acting on the impeller. Based on this finding, we established mathematical formulas for the relationship between the pressure pulsation at the monitoring point and the eccentricity of the impeller, as well as the eccentricity of the impeller and the Alford force acting on it. According to these formulas, we only need to monitor the pressure pulsation during the operation of the turbine to realize the real‐time monitoring of the eccentricity of the impeller and the Alford force, which is of great significance for ensuring the safe and stable operation of the turbine.


| INTRODUCTION
Kaplan turbine is a type of hydraulic power generation machinery widely used in large-flow and low-head hydropower stations. 1Due to the cantilever structure of the blade of the Kaplan turbine runner, there must be a clearance between the top of the blade and the casing. 2he fluid near the blade tip clearance is affected by the pressure difference between the blade pressure surface and the suction surface, which flows from the pressure surface to the suction surface through the blade tip clearance, forming a clearance leakage flow. 3The flow near the blade tip clearance generates a complex threedimensional flow structure due to the interaction between the clearance leakage flow and the mainstream. 4he clearance width has a significant impact on the characteristics of clearance leakage flow.Shi and colleagues 5,6 used numerical simulation methods to study the influence of clearance width on the evolution behavior of clearance leakage vortices.The study showed that as the clearance width continued to increase, the position of the tip leakage vortex core gradually moved inward and toward the trailing edge, and the vortex suction strength increased.Wu and colleagues 7,8 studied the performance of Kaplan turbines under different clearance widths.The study showed that within a certain range, as the clearance width increases, the separation vortices gradually strengthen, and the flow inside the runner tends to become chaotic.
During the operation of the hydraulic turbine, due to factors such as runner wear, cavitation, and unit vibration, the runner will form a certain degree of eccentricity, resulting in uneven circumferential distribution of blade tip clearance. 9The uneven distribution of clearance width will reduce the stability of flow field in the blade top area, cause the leakage flow at the blade top to show obvious unsteady characteristics, induce high amplitude pressure pulsation, lead to the increase of runner radial force, and even lead to the rubbing or seizure between the impeller and the housing in serious cases. 10,11At present, the calculation of runner eccentricity is mainly achieved through direct measurement, and it is very difficult to monitor the changes in runner eccentricity in real time during the operation of hydraulic turbines.Because the eccentric operation of the runner inevitably leads to changes in the internal pressure pulsation of the hydraulic turbine, it is possible to predict the degree of eccentricity of the runner through changes in the pressure pulsation signal.Variational mode decomposition (VMD) 12 is an adaptive, completely nonrecursive signal processing method.Compared with other time-frequency analysis directions, the VMD method has a complete theoretical foundation and good noise robustness. 13,14At present, there have been a large number of studies applying the VMD method to early mechanical fault diagnosis and showing good results, [15][16][17] which provides a theoretical basis for us to use the VMD method to predict the degree of runner eccentricity.
The degree of eccentricity has a significant impact on the excitation force of the runner.Thomas and Alford were the first to pay attention to the problem of excitation force (Alford force) caused by uneven gap distribution and proposed the calculation formula for Alford force. 18,19Subsequently, a large number of scholars conducted further research on the shortcomings of this formula in practical applications.Pan et al. 20 used computational fluid dynamics (CFD) numerical calculation methods to analyze in detail the effects of factors such as rotational frequency, rotational speed, inlet airflow angle, eccentricity, and average tip clearance on the Alford force of turbine impellers.aus der Wiesche et al. 21proposed a semiempirical model for predicting Alford forces in steam turbines, and the predicted results of this model better match existing experimental data.In the field of hydraulic machinery, Brennen 22 theoretically analyzed the mechanism of fluid excitation force in the tip clearance of axial-flow water pumps and provided a theoretical calculation formula for Alford force, discussing the influence of vortex direction on Alford force.Wang et al. 23 conducted in-depth research on Alford effect and Alford force of tubular turbine under different runner eccentricities through the method of combining experiment and numerical simulation and found that with the increase of runner eccentricity, the average value of runner radial force gradually increases, and the frequency may be offset.
Overall, in the field of aerodynamic machinery, a large number of researchers have conducted in-depth research on the impact of runner eccentricity on Alford force, revealing the characteristics of Alford force under different eccentricities, and establishing Alford's prediction model.However, in hydraulic machinery, there is limited research on Alford force, and further research is needed on the characteristics and prediction formulas of Alford force under different eccentricities.In addition, the calculation of runner eccentricity in the above research is mainly through direct measurement, which is very difficult to calculate when the hydraulic turbine is running.
This study takes the model Kaplan turbine as the research object and uses numerical simulation methods to simulate the internal flow of the turbine under different head and eccentricity conditions.The effectiveness of numerical simulation was verified by comparing the results of numerical simulation with experimental results.Then, we used the VMD method to analyze the internal flow pressure pulsation and radial force characteristics of the runner and found the features in the pressure pulsation signal that can effectively reflect the eccentricity of the runner.The relationship between this feature and the eccentricity of the runner, the eccentricity of the runner, and the average Alford force are established.It is able to conduct real-time monitoring of the eccentricity status and Alford of the runner during the operation of the hydraulic turbine, ensuring the operation safety and stability of the unit, which has important engineering significance.

| Kaplan turbine unit
The research object of this paper is the model Kaplan turbine, which is composed of five parts: volute, stay vane, guide vane, runner, and draft tube.Among them, the number of blades for stay vane and guide vane are both 24, and the number of blades for the runner is 5.The schematic diagram of the model scale turbine (fluid domain) is shown in Figure 1, and the geometric parameters are shown in Table 1.The runner speed of the model hydraulic turbine is 1770 rpm, and the runner diameter is 0.365 m.There is a gap between the runner and the casing, and when the runner is not eccentric, the gap width is 3.2 mm.This article mainly considers the operation of the model hydraulic turbine under two operating conditions, Condition-01 and Condition-02, with different guide vane opening angle and runner opening angle.Under Condition-01, the rated head of the water turbine is 20 m, the flow rate is 1.02 m 3 /s, the opening of the guide vane is 46°, and the angle of the runner blades is 25°.Under Condition-02, the water head is 15 m, the flow rate of the hydraulic turbine is 0.55 m 3 /s, the opening of the guide vane is 35°, and the angle of the runner blades is 14°.

| Runner eccentricity definition
The main content of this study is the prediction of runner eccentricity and Alford force during the operation of hydraulic turbines.The eccentricity of the runner is expressed as the eccentricity level φ, and its calculation formula is as follows: where h tc is the gap width when the runner is not eccentric, which is 3.2 mm.E r is the eccentricity of the runner, and the definition of E r is shown in Figure 2.
From the figure, it can be seen that the turbine casing and runner have their respective centers.We define the linear distance between the casing center and the runner rotation center as the runner eccentricity E r .When the runner is not eccentric, the center of the casing overlaps with the center of the runner rotation, and the eccentricity of the runner E r = 0.When the runner is eccentric, there is a certain difference between the center of the casing and the rotation center of the runner, and the eccentricity of the runner E r > 0. This study considered four cases of runner eccentricity φ = 0, 0.2, 0.4, and 0.6.When φ = 0.6, the eccentricity of the runner E r = 1.92 mm.This is a very small eccentricity of the runner, as the ratio of eccentricity to runner diameter D r is only 0.5%, which is almost imperceptible in actual production.

| Turbulence model
To ensure computational accuracy and save computational resources, the detached Eddy simulation (DES) 24 turbulence model was selected for numerical calculations in this study.The DES turbulence model is a hybrid model of Reynolds-averaged Navier-ransStokes (RANS) 25 and large Eddy simulation (LES). 26In the DES turbulence model, the RANS model is used to solve the region near the wall, while the LES simulation method is used in other regions.Through this approach, the DES turbulence model can accurately simulate the three-dimensional vortex structure in the flow field, while avoiding the high requirement of near-wall mesh in the LES model.It can improve the simulation accuracy of vortex structures without significantly increasing the consumption of computational resources, making it the most effective turbulence model in current and future turbulence simulations.Whether DES uses the RANS turbulence model or the LES turbulence model depends on the size of the dissipation term, and its formula for calculation 27 is x y z Δ = min{Δ , Δ , Δ }. (5)

| VMD
VMD is a nonrecursive decomposition method based on variational problems, which can effectively decompose nonstationary and nonlinear sequences and avoid the appearance of mode mixing phenomenon. 28VMD assumes that the signal sequence can be decomposed into k intrinsic mode function (IMF) components {u k (t)}, each with its own central frequency and limited bandwidth.The variational problem 29 can be described as follows: minimize the sum of estimated bandwidths of each IMF component {u k (t)}, subject to the constraint that the sum of these k mode functions equals the original signal x(t), which is mathematically represented as 30 where {u k } = {u 1 , u 2 , …, u k } represents the k IMF components obtained by decomposing the original signal; To solve the constrained variational problem, we introduce Lagrange multiplier λ (t) and quadratic penalty term α to convert the constrained optimization into an unconstrained one.
Subsequently, the saddle point of Equation ( 8) is obtained through the alternate direction method of multipliers, and the optimal solution is obtained by iteratively updating {u k }, {ω k } and λ in the frequency domain, 31 which gives the IMFs {u k } and their corresponding central frequencies {ω k }.

| Setup of simulation
In this case, CFD is used for numerical simulation of axial-flow turbine, and DES turbulence model is selected In steady state simulation, the maximum number of iteration steps is set to 1000, and the convergence index of the continuity equation and momentum equation is 1 × 10 −6 .Unsteady state calculations are based on the results of steady state calculations.A total of 20 runner rotations were calculated for unsteady state simulations.Each runner rotation is set to 180 timesteps, and the maximum number of iterations per step is set to 10.The convergence index of the continuity equation and momentum equation is also 1 × 10 −6 .

| Grid preparation and check
In CFD simulation, the quality and quantity of grid partitioning have a significant impact on the accuracy of numerical simulation results and the speed of computer operation.In this study, considering the complex geometry of Kaplan turbine, we adopted unstructured grids with strong geometric adaptability to discretize the entire fluid domain.To balance the number of grids and simulation quality, it is essential to conduct grid independence checks.This article uses the grid convergence index (GCI) proposed by Celik et al. 32 to examine three grid schemes (M1, M2, M3).The number of grids for M1, M2, and M3 grid schemes is 8,974,415, 4,311,985, and 2,022,024, respectively.Figure 3 shows the changes in the discharge (flow rate) of turbine under different grid schemes and the extrapolated values based on Richard's extrapolation method.As the number of grids increases, the calculation accuracy continues to improve.The GCI value for the M1 and M2 grid schemes is 0.05%, and the GCI value for the M2 and M3 grid schemes is 0.2%.The GCI of the three grid schemes is less than 3%, meeting the grid convergence criteria.Taking into account both computational accuracy and speed, the M2 grid scheme was ultimately selected for the grid division of the hydraulic turbine.The schematic diagram of the grid division is shown in Figure 4.

| Validation of simulation
To verify the effectiveness of numerical simulation, Figure 5 compares the simulation results and experimental results of the efficiency of the model turbine under two operating conditions when the runner is not eccentric.The experiment was completed on a hydraulic machinery closed test rig following the IEC 60193 standard 33 (see Figure 5A).The power P is measured through torque meter, the head H is measured through differential pressure gauge, and the flow rate Q is | 1573 measured through electromagnetic flowmeter.Efficiency η according to the formula η = P/ρgQH calculation can be obtained.From Figure 5B, it can be seen that under two operating conditions, the efficiency of the hydraulic turbine obtained from numerical simulation is close to the experimental results, and the difference between the two is within the error bar range of the experimental values.CFD simulation can effectively simulate the internal flow of hydraulic turbines, and analyzing the internal flow of hydraulic turbines based on numerical simulation results is feasible.

| COMPARISON OF PERFORMANCE
Figure 6 shows the influence of eccentricity of Kaplan turbine runner on turbine flow rate, shaft power, and hydraulic efficiency under different operating conditions.Figure 6 (a) is for condition 01 and (b) is for condition 02.
From Figure 6, it can be seen that under Condition-02, as the eccentricity level of the runner φ increases, the overflow flow of the hydraulic turbine continues to increase.The shaft power and hydraulic efficiency of the hydraulic turbine both increase when the eccentricity of the runner is between 0 and 0.2.When the eccentricity level of the runner φ is 0.2, the shaft power and efficiency of the hydraulic turbine reach their maximum value.This means that a smaller eccentricity of the runner can to some extent improve the shaft power and hydraulic efficiency of the hydraulic turbine.However, as φ continues to increase, when it is between 0.2 and 0.4, the shaft power and efficiency of the hydraulic turbine change from an increase to a slow decrease; When φ is greater than 0.4, the shaft power and efficiency of the turbine rapidly decrease.From Figure 6, it can be seen that under Condition-01, the impact of runner eccentricity on flow rate is different from that under Condition-02.As φ increases, the flow rate of the hydraulic turbine first decreases and then increases.The trend of changes in turbine shaft power and efficiency is consistent with Condition-02, but the magnitude of their changes is different.Specifically, under Condition-01, the eccentricity of the runner has a greater impact on the power and efficiency of the turbine shaft.Overall, the eccentricity of the runner has almost no effect on the performance of the turbine.Under Condition-01, compared to the noneccentric condition, when φ is 0.6, the difference in turbine flow rate, shaft power, and hydraulic efficiency is only 0.007%, 0.112%, and 0.131%, respectively.Therefore, it is difficult to determine the eccentricity status of Kaplan turbine runner based on the change in performance.

| PREDICTION OF ECCENTRICITY OF RUNNER 6.1 | Layout of monitoring point
The eccentricity of runner seriously threatens the safe and stable operation of the turbine unit.During the operation of the turbine, the state of the eccentricity may change due to blade wear, cavitation, and vibration.Therefore, real-time monitoring of the eccentricity is of great significance to the stable operation of the turbine.
Since the eccentricity of runner has little effect on the performance of the turbine, but it will inevitably lead to the change of the pressure in the turbine, the eccentricity of runner can be monitored by the change of the pressure pulsation inside the turbine.Considering the convenience of implementation, only one monitoring point is arranged in the draft tube of the turbine to monitor the pressure pulsation.The location of the monitoring point is shown in Figure 1.

| Analysis of pressure pulsation signal
VMD method can decompose the nonstationary signal into several IMFs, and each IMF has different characteristic scales.By analyzing the characteristic of each IMF, the characteristics that can effectively reflect the eccentricity are found, so as to accurately reflect the degree of the eccentricity of runner of the original signal.2. In this paper, the original pressure pulsation signal is decomposed into five IMFs.From IMF5 to IMF1, the center frequency increases gradually.The center frequency of IMF5 is the lowest among the five IMFs.When the runner is not eccentric, the dominant frequency in IMF5 is 147.8Hz, which is five times the rotating frequency of the turbine f n = 29.6Hz and equals to the runner blade passing frequency f b = 147.8Hz.This indicates that IMF5 is most affected by the runner blades when the runner is not eccentric.However, this situation changes when the runner is eccentric.When the runner has a certain degree of eccentricity, the main frequency of IMF5 is reduced to 29.6 Hz, which equals to f n .As the eccentricity of the runner increases, the peak-to-peak value of IMF5 also increases.This indicates that when the runner is eccentric, the rotation of the runner has a greater impact on the internal pressure pulsation of the turbine, and the greater the eccentricity, the greater the impact.Compared with the time-domain diagram of IMF5 under different operating conditions, when the runner is not eccentric, the period of IMF5 is shorter and the pressure pulsation is denser.When the runner is eccentric, the waveforms of IMF5 are similar under different eccentricities, but the peak-to-peak values are significantly difference.As the eccentricity increases, the peak-to-peak value increases.IMF4 is the component whose center frequency is only higher than IMF5 decomposed by VMD.When the runner is not eccentric, IMF4 is composed of multiple signals with similar amplitude, and there is no signal with a particularly prominent amplitude.The frequency range is 500-650 Hz, and the center frequency is 602 Hz.When the runner is eccentric, the frequency range of IMF4 reduces, and there is a peak amplitude whose frequency is equal to f b = 147.8Hz.By comparing the time-domain diagram, it can be found that the timedomain diagram of IMF4 when the runner is eccentric and the one of IMF5 when the runner is not eccentric are similar.It can be seen that when the runner is eccentric, the modes obtained by VMD changes due to the increase of the amplitude with the runner frequency.By comparing the amplitude of runner blade passing frequency with different runner eccentricity, we can find that the amplitude pulses within a certain range with the increase of runner eccentricity.It can be seen that the eccentricity of the runner has little effect on the amplitude of runner blade passing frequency.
When the runner is not eccentric, similar to IMF4, the frequency-domain diagram of IMF3 is also composed of multiple signals with similar amplitude, but the frequency range is different.The frequency range of IMF3 is 650-800 Hz, and the center frequency is 717 Hz.When φ = 0.2 and φ = 0.4, the frequency range is 500-750 Hz, and the center frequency is near 620 Hz.When φ = 0.6, the frequency range is 500-700 Hz, and the center frequency is 652 Hz.The frequency range of IMF3 is approximately around 709.6 Hz, which is equal | 1577 to 24 times of runner frequency.That is mainly due to the impact of guide vane.Because the amplitudes of the modes are small, it shows that the guide vane has little impact on the pressure pulsation.
Except for the operating condition of φ = 0.6, IMF2 has the highest energy among all modes, with a frequency range of about 800-1000 Hz and a center frequency of about 830 Hz.There is a peak amplitude in the frequency-domain diagram of IMF2, with a frequency of 827.9 Hz, which is 28 times that of the runner rotating frequency f n .This is very close to the sum of the number of guide vanes and the number of runner blades, indicating that the interaction between guide vanes and runner has a greater impact on the pressure pulsation in the draft tube.There are significant differences in the time-domain diagrams of IMF2 under these three operating conditions.As the eccentricity increases, the pulsation trend of the time-domain diagrams becomes steeper.When the eccentricity of the runner φ = 0.6, the frequency range of IMF2 is 700-800 Hz, and the center frequency is 745 Hz, which is still affected by guide vanes.
IMF1 is the component with the highest center frequency obtained by VMD.Except for the operation condition of φ = 0.6, the frequency range is 1500-1700 Hz, which is about 48-58 times that of the turbine runner frequency f n , mainly due to the impact of double frequency (48-58 times the runner frequency).The pulsation amplitude of IMF1 is small.Compared with other components, the time-domain diagram of IMF1 has a shorter period and more intensive pulsations.When φ = 0.6, there is a peak amplitude in the frequency-domain diagram of IMF1, whose frequency is equal to 28f n = 827.9Hz, similar to the IMF2 under other operation conditions.Similar to the variation trend of the amplitude corresponding to the blade frequency, the amplitude corresponding to 28f n does not show a significant trend with the change in the eccentricity of the runner.When the eccentricity of the runner is 0.2, the amplitude corresponding to 28f n is the largest, equal to 1923.2.
Figure 8 shows the time-domain diagram and its corresponding frequency-domain diagram of each component obtained by decomposing the pressure pulsation signal at the monitoring point under Condition-01 using VMD. Figure 8 (a) is for φ = 0, (b) is for φ = 0.2, (c) is for φ = 0.4, and (d) is for φ = 0.6.The center frequency of each IMF is shown in Table 3.It can be seen from Figure 8 that when φ = 0 and φ = 0.2, the center frequency of IMF5 is larger, and its frequency-domain diagram contains two peak amplitudes, whose corresponding frequencies are f n and f b , respectively.Under these two conditions, the period and density of timedomain diagram of IMF5 are substantially same, but the amplitude is larger when φ = 0.2.When φ = 0.4 and φ = 0.6, the center frequency of IMF5 is small, and there is only one peak amplitude in its frequency-domain diagram, and the frequency is f n .Comparing the amplitude corresponding to runner frequency under different eccentricities, it can be seen that as the eccentricity of the runner increases, the amplitude corresponding to runner frequency under Condition-01 increases.
When φ = 0 under Condition-01, the frequency range of IMF4 is 500-650 Hz, with a center frequency of 592 Hz; the frequency range of IMF3 is 650-800 Hz, with a center frequency of 732 Hz.When φ = 0.2, the frequency range of IMF4 is 500-750 Hz, with a center frequency of 623 Hz.This frequency range is basically consistent with the sum of the frequency ranges of IMF4 and IMF3 under noneccentric conditions because the difference in VMD makes the range divided into two parts.The frequency range is around 24f n , mainly due to the impact of guide vanes on flow.When φ = 0.4 and φ = 0.6, IMF4 is mainly affected by the passing frequency of the runner blades, and there is a peak amplitude in the frequency-domain diagram of IMF4, with the frequency equal to f b = 147.8Hz.Comparing the amplitude of f b under different eccentricity conditions, it can be found that as the eccentricity of the runner increases, the amplitude of f b first rapidly increases and then remains variating around 1650.
When φ = 0.2, there are two peak amplitudes in the frequency-domain diagram of imf3, and their corresponding frequencies are 27f n = 798.3Hz and 28f n = 827.9Hz, which are mainly caused by the impact of runner blades and guide vanes.When φ = 0.4 and φ = 0.6, the frequency range of IMF3 is 500-750 Hz, and the center frequency is about 600 Hz.It is similar to that of IMF4 under φ = 0.2, which is mainly caused by the impact of the guide vane.From the above analysis, it can be seen that due to the impact of guide vane blades, a specific frequency range around 24f n will be formed in the frequency- domain diagram of pressure pulsation, and the amplitude of pressure pulsation is small.When φ = 0 and φ = 0.4, the energy of IMF2 is the largest, and there is an obvious peak amplitude in its frequency-domain diagram, whose corresponding frequency is 28f n = 827.9Hz, which is caused by the joint impact of the runner and guide vane.When φ = 0, its amplitude is 2527.7;when φ = 0.4, its amplitude is 2943.4.When φ = 0.2, the signal generated under the joint impact of runner and guide vane is decomposed into IMF3, and its amplitude is 1221.1.when φ = 0.2, the frequency range of IMF2 is 1000-1450 Hz, and the center frequency is 1217 Hz.The frequency-domain diagram of IMF2 is composed of a series of signals with low amplitude.When φ = 0.6, the dominant frequency of IMF2 is 26f n = 768.7 Hz.
Except for φ = 0.6, the frequency range of IMF1 under other operation conditions is 1450-1750 Hz, which is mainly affected by double frequency (48-58 times that of runner frequency), and its frequencydomain diagram is composed of a series of signals with low amplitude.When φ = 0.6, there is a peak amplitude in the frequency-domain diagram of IMF1, which is 3628.3, and its corresponding frequency is 28f n = 827.9Hz.Comparing the amplitude of 28f n under different eccentricities, it can be seen that as the eccentricity increases, the amplitude first decreases and then increases, and reaches the minimum value of 1221.1 when φ = 0.2.

| Establishment of prediction formula for eccentricity of runner
Through the analysis of the pressure pulsation signal decomposed by VMD, it can be seen that there are five obvious characteristic frequencies of the draft tube pressure pulsation signal, which are runner frequency f n , runner blade passing frequency f b , the frequency generated by the impact of guide vane (24f n ), the frequency generated by the joint impact of guide vane and runner (29f n ) and double frequency (48-58f n ).The amplitude of runner frequency increases with the increase of eccentricity, while the amplitude of runner blade passing frequency and the frequency generated by the joint impact of runner and guide vane does not show an obvious change trend with the change of eccentricity.The signals of the frequency generated by guide vane and double frequency are composed of a series of signals with lower amplitude.Therefore, to better quantify the relationship between the pressure pulsation in the draft tube and the eccentricity of the runner, this paper selects the amplitude corresponding to the runner frequency to represent the extent of the draft tube pressure pulsation.
Figure 9 shows the amplitude of pressure pulsation in the turbine under different operation conditions corresponding to the runner frequency A fn and its trend line.Figure 9 (a) is for condition 01 and (b) is for condition 02.

(A) (B)
F I G U R E 9 amplitude of pressure pulsation corresponding to the runner frequency A fn and its trend line.
HU ET AL.
| 1581 The trend line of A fn is based on linear function fitting, and its basic equation is as follows: where c 0 and c 1 are constants.It can be seen from Figure 9 that good fitting results can be obtained by fitting the relationship between the amplitude of the pressure pulsation corresponding to runner frequency and the eccentricity of the runner through the linear relationship.Under Condition-02, the slope of the trend line is 2903.6, the intercept is −3.9, and the fitting correlation coefficient is 0.993.Under Condition-01, the slope of the trend line is 4219.8, the intercept is 84.6, and the fitting correlation coefficient is 0.989.Under Condition-02, the slope and intercept of the trend line are smaller than those under Condition-01, and the fitting result is consistent with the larger amplitude of pressure pulsation under Condition-01.The equation clearly shows the relationship between A fn and φ.Using this equation, we can monitor the eccentricity of the runner in real time during the operation of the turbine by pressure pulsation data monitoring by the sensor.

| Comparison of intensity and amplitude
Through above research, we can monitor the eccentricity of the impeller in real time.However, the specific characteristics of the radial force acting on the impeller under different eccentricities are still unknown.In this section, we studied the characteristics of the radial force acting on the impeller under different eccentricities.Figure 10 shows the influence of the impeller eccentricity on the average intensity and amplitude of the radial force F r of the turbine under Condition-02.Figure 10 (a) is for condition 01 and (b) is for condition 02.As shown in the figure, under Condition-02, when the impeller eccentricity is zero, the average intensity of F r is 22.8 N and the amplitude of F r is 48.3 N.With the increase of the impeller eccentricity, the average intensity and amplitude acting on the impeller increases rapidly.When the impeller eccentricity is 0.6, the average intensity is 57.7 N, which is 2.5 times that of the noneccentric case.
The amplitude shows an increasing trend when the impeller eccentricity is between 0 and 0.4.When the impeller eccentricity exceeds 0.4, the amplitude decreases slightly.When the impeller eccentricity is 0.4, the amplitude reaches its maximum, which is 1.6 times that of the noneccentric case.
Figure 11 shows the influence of impeller eccentricity on the average intensity and amplitude of F r of the turbine under Condition-01.Figure 11 (a) is for condition 01 and (b) is for condition 02.It can be seen from the figure that when the impeller eccentricity is zero, the average intensity and amplitude of F r under Condition-01 are 55.5 and 100.7 N, respectively.With the increase of the impeller eccentricity, the average intensity of F r gradually increases.When the impeller eccentricity is 0.6, the average intensity of F r is 73.5 N, which is 1.3 times the average intensity of F r of the noneccentric case.With the increase of the impeller eccentricity, the amplitude of F r continues to increase.When the impeller eccentricity is 0.6, the amplitude of F r reaches its maximum at 155.0 N, which is 1.5 times that of the noneccentric case.| 1585 IMF2 is 1450-2000 Hz, and the central frequency is around 1640 Hz.When the eccentricity is 0.4 and 0.6, the frequency range of IMF2 is 1000-2000 Hz, and the central frequency is around 1610 Hz.This range is mainly influenced by the second harmonic frequency (48-58 times f n ), and the amplitude in the frequencydomain plot is relatively small.Under different eccentricities, the frequency range of the IMF1 component is 2000-2650 Hz, which is mainly influenced by the third harmonic frequency (72-87 times f n ).When the impeller is not eccentric, there is a clear peak in the IMF1 component at the frequency of 2483.6 Hz, which corresponds to 84 times f n , indicating that the radial force acting on the impeller is greatly influenced by the third harmonic frequency.
Figure 13 shows the time-and frequency-domain plots of the radial force fluctuation signal based on VMD decomposition under Condition-01.Figure 13 (a) is for φ = 0, (b) is for φ = 0.2, (c) is for φ = 0.4, and (d) is for φ = 0.6.The central frequencies of each IMF component are shown in Table 5.The frequency range of IMF5 is the same as that under Condition-02, which is 0-500 Hz, but the central frequency differs slightly.When the impeller is not eccentric, there is a peak in the frequency-domain plot of IMF5 at the blade passing frequency f b , indicating that the blade passing frequency has a significant influence on the radial force acting on the impeller.When the impeller is eccentric, the influence of the blade passing frequency on the radial force decreases, and the corresponding amplitude also decreases significantly.At the same time, the frequency corresponding to the amplitude of IMF5 becomes the runner rotational frequency f n , indicating that the influence of the runner rotational frequency on the radial force increases when the impeller is eccentric, and the amplitude corresponding to the runner rotational frequency increases with the increase of the impeller eccentricity.
Except for the condition with an eccentricity of 0.4, the frequency range of IMF4 is 500-1000 Hz, and the central frequency is around 800 Hz.When the eccentricity is 0.4, the range of 500-1000 Hz is divided into two components, IMF4 and IMF3.This frequency range is mainly due to the combined effect of the guide vanes and impeller blades.As the eccentricity increases, the amplitude corresponding to 24 times f n first decreases and then increases.When the impeller eccentricity is 0.4, the amplitude corresponding to 24 times f n is the smallest.The amplitude corresponding to 29 times f n first decreases and then increases.When the impeller eccentricity is 0.2, the amplitude corresponding to 29 times f n is the smallest.From the changes in amplitude, it can be seen that the influence of 24 times f n continues to decrease as the eccentricity increases from 0.2 to 0.4, while the influence of 29 times f n begins to increase, which leads to the separation of IMF4 and IMF3 components during VMD decomposition.
When the eccentricity is 0.4, the frequency range of IMF2 is 1000-2000 Hz, and the central frequency is around 1706 Hz.Except for the condition with an eccentricity of 0.4, the frequency range of IMF3 is 1000-1450 Hz, and the frequency range of IMF2 is 1450-2000 Hz.The superimposed frequency range of these two components coincides with the frequency range of the IMF2 component (1000-2000 Hz) under the eccentricity of 0.4, which is mainly influenced by the second harmonic frequency (48-59 times f n ).From the frequency-domain plot, it can be seen that the amplitude of the radial force fluctuation is generally low in this range, indicating a weak influence of the second harmonic frequency on the radial force.
Under Condition-01, the frequency range of IMF1 is the same as that under Condition-02, which is 2000-2650 Hz, and is mainly influenced by the third harmonic frequency (72-87 times f n ).Under Condition-01, there is a peak in the IMF1 component at a frequency of 2513.2Hz (85 times f n ).As the eccentricity increases, the corresponding amplitude first increases and then decreases, reaching its maximum value when the eccentricity is 0.2.
From the above analysis of the radial force fluctuation signal, it can be concluded that the fluctuation of the radial force acting on the impeller is mainly influenced by five characteristic frequencies, which are the runner rotational frequency f n , the frequency due to the influence of the guide vanes (24 times f n ), the frequency due to the combined effect of the guide vanes and impeller blades (29 times f n ), the second harmonic frequency (48-58 times f n ), and the third harmonic frequency (72-87 times f n ).Compared with the main frequencies of the pressure fluctuation in the draft tube, the influence of the blade passing frequency on the radial  force is relatively weak, while the influence of the third harmonic frequency on the radial force increases significantly.In some operating conditions, its influence may even be higher than that of the second harmonic frequency on the radial force.

| Prediction of Alford force
Alford force F A is a radial excitation force caused by eccentricity of the runner, and its calculation formula is as follows: In the formula, F r0 is the average intensity of F r when the eccentricity of the runner is zero; F ¯r is the average intensity of F r under different eccentricities.The variation of Alford force with eccentricity under different conditions is shown in Figure 14.The trend line of F A is fitted based on linear function, and its basic equation is as follows: where F ATL is the Alford force of the impeller evaluated by the trend line, and C 0 and C 1 are constants.The trend lines of the Alford force based on linear function fitting are shown in Figure 13.It can be seen from Figure 14 that the variation of the impeller Alford force with the impeller eccentricity under different head is basically linear, with a fitting correlation coefficient of 0.974 and 0.988, respectively.The slope C 1 of F ATL under Condition-02 is 57.76, while the slope of F ATL under Condition-01 is 29.15.This indicates that the additional radial force induced by impeller eccentricity is smaller under Condition-01.This may be because the internal flow of the Kaplan turbine is more stable under Condition-01, resulting in a smaller influence of impeller eccentricity on the Alford force.However, as the Alford force grows rapidly under Condition-02, we should pay more attention to the impact of impeller eccentricity induced Alford force on the operation of the unit and avoid accidents (Table 6).

F I G U R E 2
Definition of runner eccentricity.(A) Non-eccentric (overlapped center).(B) Eccentric (offset of center) eccentricity E r as the turbulence model.The fluid medium is 25°C water, with a density of ρ = 1 × 10 3 kg/m 3 , kinematic viscosity ν = 1 × 10 −6 m 2 /s, and the reference pressure is set to one standard atmospheric (1 atm) pressure.The boundary conditions are set as follows: 1.The inlet of the spiral case is set as the inlet of the entire calculation domain, and the boundary condition of the inlet is set as the pressure inlet.The specific pressure value is determined by the head under different conditions.Under Condition-01, the inlet pressure is set to total pressure of 1.96 × 10 5 Pa.Under Condition-02, the inlet pressure is set to total pressure of 1.47 × 10 5 Pa. 2. The outlet of draft tube is set as the outlet of the entire calculation domain, the outlet boundary condition is set as the pressure outlet, and the relative pressure is 0 Pa. 3.There are multiple interfaces used to transmit data throughout the entire fluid domain.Among them, the interfaces between the fixed guide vane and runner, runner and draft tube are dynamic interfaces, and the other interfaces are static interfaces.The walls are all no-slip wall boundaries.

FF I G U R E 6
I G U R E 5 Comparison of efficiency between numerical and experimental data.(A) Test rig schematic map.(B) Full characteristics with comparison of efficiency.BEP, best efficiency point; CFD, computational fluid dynamics; Exp., experimental.Variation of turbine performance with runner eccentricity level under different operating conditions.

Figure 7
shows the time-domain diagram and its corresponding frequency-domain diagram of each IMF of the pressure pulsation signal of the monitoring point under Condition-02 decomposed by VMD.Figure 7 (a) is for φ = 0, (b) is for φ = 0.2, (c) is for φ = 0.4, and (d) is for φ = 0.6.The center frequency of each IMF is shown in Table HU ET AL.

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I G U R E 8 Time-and frequency-domain diagram of pressure pulsation signal based on variational mode decomposition under Condition-01.IMF, intrinsic mode function.

T A B L E 3
Center frequency of each IMF based on VMD under Condition-02.

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I G U R E 10 Effect of eccentricity of runner on the average intensity and amplitude of radial force under Condition-02.F I G U R E 12 Time domain diagram and corresponding frequency spectrum of radial force pulsation signal based on VMD decomposition under Condition-02.IMF, intrinsic mode function.F I G U R E 12 (Continued).

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I G U R E 13 Time domain diagram and corresponding frequency spectrum of radial force pulsation signal based on variational mode decomposition under Condition-01.IMF, intrinsic mode function.
T A B L E 4 Center frequency of each IMF based on VMD under Condition-01.
Center frequencies of various IMF components based on VMD decomposition under Condition-02.Center frequencies of various IMF components based on VMD decomposition under Condition-01.
T A B L E 5 F I G U R E 14The magnitude and trend of Alford force on the runner under different operating conditions.T A B L E 6Abbreviations: IMF, intrinsic mode function; VMD, variational mode decomposition.