Application of DMD method to eccentric self-excited vibration of tubular turbine runner

Tubular turbines are widely used in the development and utilization of low-head marine energy. The cantilever type runner inevitably leads to gap between the chamber and blade tip, and non-standard installation can easily lead to eccentric runner, affecting the safe and stable operation of turbine. The overall process of self-excited vibration is an obvious nonlinear process, so the applicability of traditional methods needs to be discussed. In recent years, reduced-order modelling methods based on linear dynamic systems have gradually been introduced into the analysis of hydraulic machinery, and have achieved certain results. Therefore, this study combines computational fluid dynamics (CFD) and dynamic mode decomposition (DMD) methods to conduct low-dimensional mode decomposition and comparative analysis of self-excited vibration caused by different runner eccentricities of tubular turbine. The research shows that as the runner eccentricity continues to increase, the modal frequency distribution of self-excited vibration becomes complex. The DMD method disassembles the low-level representations of self-excited vibration signals, greatly enriching the understanding of eccentric self-excited vibration of the runner, and providing assurance for the safe and stable operation of tubular turbines.


Introduction
In recent years, as the development of high head hydropower resources tends to become saturated, the utilization and development of low head hydropower and marine energy have gradually received attention.Bulb tubular turbine units are a typical way of utilizing low head hydropower resources, which are widely used due to their small civil engineering volume and high efficiency.Due to the unique structural form of the tubular turbine, the runner blades are cantilever type, and there is a gap between the runner chamber and the blade tip.During the installation process of the runner, alignment problems often occur, resulting in eccentric installation of the runner.Due to the leakage flow of the blade tip, the eccentricity of the runner causes uneven distribution of the blade tip along the circumferential direction, resulting in uneven load and circumferential pressure distribution on each blade.This generates a fluid excitation force perpendicular to the eccentric direction of the impeller, which is called the Alford effect [1], as shown in Figure 1.This poor self-excited vibration force can seriously affect the safe and stable operation of hydraulic machinery, When the Alford force exceeds a certain threshold, the balance between it and the bearing force, as well as the additional centrifugal force caused by the deviation of the center of mass, is disrupted.The Alford force and the uneven clearance are mutually excited and enhanced, causing strong self-excited vibration.In severe cases, it can cause collisions between the runner and the runner chamber, endanger the safe operation of the unit, and even cause production accidents [2][3][4][5].Due to the complexity of the generation and mechanism of fluid radial excitation force, it is difficult to measure this radial excitation force through experiments.Therefore, numerical simulation has become the main research method for studying radial excitation force [6,7].At present, numerical simulation methods can effectively represent and demonstrate the radial excitation force on the runner, but how to analyze the signal of this temporal change has become the key to revealing the eccentric Alford effect.
In the process of numerical simulation development, FFT method has become the most widely used method for time-frequency decoupling of flow field information [8][9][10].Due to its simple, convenient application, and intuitive results, it has been widely used in various one-dimensional signals with time series changes.However, due to the high-dimensional complexity and nonlinearity of the flow field, the fast Fourier transform (FFT) method has poor spectral analysis performance for local features and nonperiodic signals, and has not shown significant advantages in the analysis of eccentric radial excitation forces.In recent years, in the analysis of flow fields, research methods such as linearizing high-order complex flow fields and using reduced-order modelling methods to capture the main characteristics of flow fields have gradually been widely used.Among them, the dynamic mode decomposition (DMD) method is simple to implement, has clear frequency, and has obvious advantages [11][12][13].
However, this reduced-order modelling method is based on matrix decomposition theory and cannot be directly applied to the decomposition of one-dimensional time series signals.To solve this problem in the future, this study uses numerical simulation methods to obtain different radial excitation forces of a certain tubular water turbine with six eccentricities.According to the Takens embedding theorem, a delayed embedding method is used to construct the Hankel matrix of one-dimensional radial excitation force signals, and a Hankel-based DMD method is proposed to analyze and short-term predict six different radial excitation forces, provide a new method for analyzing radial excitation force caused by eccentric runner [14], and provide guarantees for the safe and stable operation of tubular water turbines.
The unit speed n 11 and unit flowrate q 11 of the turbine are 2.733 and 170.33, respectively, with a specific speed n s of 854.63.The gap of the model turbine is 1mm.The computational domain of numerical simulation includes four parts: inlet channel, guide vane, runner, and outlet channel.To ensure the accuracy of numerical simulation results, the fluid domain extends the main axis of the runner from the bulb body.The specific situation of the computational domain is shown in Figure 2. Figure 2 also shows the grid division of key parts in the fluid domain.In this study, hexahedral grids were used for components with simple structures, while tetrahedral grids were used for runner domains with complex structure.The grid convergence index (GCI) criterion based on Richardson extrapolation method was used to check the convergence of the grid, as detailed in reference [15].While ensuring the independence of the grid, the final number of grids was determined to be 3.47 million.

Calculation model and reliability verification
Perform transient calculations on the internal flow of the flow field, using the SST k- model for turbulence calculation.The inlet and outlet boundaries of the fluid domain are set as pressure boundaries, and all solid walls are treated as non-slip walls.The time step for numerical simulation calculation is 1.598×10 -4 s, which means 360 time steps are calculated for each rotation of the runner, and the maximum iterations per step is 20 to meet the convergence residual standard of 1×10 -5 .
This study mainly focuses on rated operating condition.The blade angle of the turbine under rated operating conditions is 11°, and the guide vane angle is 65°.The external characteristics of the turbine at the rated point were obtained through experiments and numerical simulations, as shown in Table 2.  2, it can be seen that the relative error between the numerical simulation and experimental results of the external characteristics of the rated point does not exceed 3.0%, proving that a c the calculation method used in this study can accurately simulate the internal flow and effectively ensure the reliability of the analysis.Due to the Alford effect caused by the eccentricity of the runner, it can cause significant changes in radial force.Therefore, for tubular turbines, the influence of eccentricity on radial excitation force is very important.The force characteristics of rotating machinery are closely related to the rotational speed of the runner, so the time-frequency analysis method of radial excitation force is mainly carried out through the fast Fourier transform (FFT) method commonly used in flow field analysis.In recent years, capturing and analyzing one-dimensional signal features through reduced-order modelling methods has become an important analysis method for unstable signals.Below, the dynamic mode decomposition (DMD) method in mode decomposition methods will be used as the basis for analyzing radial excitation forces under different eccentricities.

Dynamic mode decomposition
According to the flow field snapshot sequence, obtain the snapshot sequence X of the real flow field information, and then form two data matrices from the snapshot sequence: By assuming that the snapshot is a discrete linear system, it can be considered that: 3 Therefore, we can link the above two data matrices.

* X
A X = 4 Matrix A contains the specific flow field information of the whole flow field snapshot.According to the theory of singular value decomposition, the flow field information matrix A can be obtained by the following formula: Eigenvalue  j and eigenvector  j can be obtained by eigenvalue decomposition of matrix A. The mode of flow field information matrix A can be obtained according to the following formula: In order to find out the main mode information affecting the flow field structure, the mode amplitude  is defined as the contribution of the modal to the initial snapshot, namely: N represents the temporal variation of the snapshot.As the DMD method is a linear method, it has short-term prediction ability.By increasing the number of N, which means increasing the width of the Vandermonde matrix, linear prediction can be made for future snapshot situations after snapshot restoration.

Hankel-based DMD
In fact, DMD can only extract the main modes when the cardinality of the linearly independent signal set is greater than or equal to the number of basic modes, which means that the state variables in the data must cross the main modes of the time series This is why DMD is commonly used for highdimensional datasets, i.e ≫.However, the number of dimension of the radial excitation force signal is relatively low, which is one-dimensional data, i.e. <.The DMD method failed in low dimensional situations and the results were inaccurate because the SVD processing of the data matrix X −1 would generate a non-zero singular value r, where  is much smaller than  and −1 (≪, −1).This feature makes it impossible to fully capture dynamic features on a time snapshot with a small number of DMD eigenvalues and modes.
Inspired by the construction of Hankel matrices, a time-delay embedding method is used to construct Hankel matrices to approximate linear dynamics.According to the Takens embedding theorem, under certain assumptions, we can recover all system information by observing a system state variable and adding the time-delay embedding method.In other nonlinear systems, this method is also applicable.The Hankel matrix is as follows: The specific steps to execute the Hankel based DMD algorithm are as follows: 1. Normalization and generation of one-dimensional time series data using Hankel matrix； 2. Feature decomposition and dynamic mode decomposition; 3. Initial prediction and de-normalization; 4. Data rearrangement for final load forecast.

Analysis of eccentric self-excited vibration based on Hankel-based DMD
The Hankel-based DMD method was used to perform mode decomposition on the radial force situation of five rotation cycles under six types of eccentricity, and the first five order modes with the greatest impact were selected for analysis.The specific analysis results are shown in Figure 3.The left side of the figure shows the time variation of each mode, namely the time coefficient (C t ), while the right side shows the FFT time-frequency domain decomposition of the time coefficient.
From the figure, it can be seen that the Hankel-based DMD method can effectively capture the characteristics of one-dimensional radial excitation force signals.Due to the basic theory of the DMD method, the first-order mode of the DMD method is represented as the average of the signal, while the other-order modes are extremely low-order decomposition of pulsating signals.Therefore, the first-order modal frequencies of the six different eccentricities are displayed as 0Hz, so they are not shown in the figure .From the time coefficient (C t ) and frequency domain decomposition of the second to fifth order modes, it can be seen that the radial excitation force varies with the eccentricity, and the main modes' frequencies captured by the DMD method constantly change.The second-order modal frequencies all exhibit low-order frequencies.The third-order mode gradually shifts from 48 shaft frequency (f s ) to 16f s and then rapidly increases to 56f s at an eccentricity of 0.9mm.Similar to the third-order mode, the frequencies of the fourth and fifth order modes show significant shifts with increasing eccentricity.In the basic theory of the DMD method, it can be known that the DMD method is a linear modal reduction method based on frequency, so each mode obtained by its reduction contains a fixed frequency.As shown in Figure 3, according to the time coefficients of each mode, the time coefficients of each level of mode contain a fixed pulsating frequency.FFT decomposition of the time coefficients can obtain the specific situation of this frequency.Due to the basic theory of DMD, the first-order mode is generally the average of the signal, and the frequency situation exhibited by this mode is generally 0Hz.Therefore, it is not particularly considered, and only the frequency situation of the remaining 2-4th order modes is analyzed.
From Table 3, it can be seen that the error between the frequency of each order of DMD mode with six eccentricities and the frequency obtained from FFT time-frequency analysis of the time coefficient is very small when the blade passes through the frequency (69.52Hz) or above, and can even be considered almost zero error.However, in the low-frequency range of shaft frequency (17.38Hz) and below, due to the fact that the lowest low-frequency range of the FFT transformation signal can only exhibit shaft frequency, a large number of low-frequency signals lower than the shaft frequency (9.30Hz, 11.22Hz, 6.64Hz, etc.) will be summarized as shaft frequency.From the table, it can be seen that a large number of low-frequency signals are not effectively distinguished under FFT transformation, while the DMD method can clearly distinguish and capture them, From this, it can be seen that the DMD method has obvious advantages in signal feature extraction.

Time series prediction of eccentric self-excited vibration force based on Hankel-based DMD
In order to better evaluate the convergence of the Hankel-based DMD method with the increase of the number of low order modes, the loss function is used as the evaluation index.The specific formula is as follows [16]: where,  CFD represents the snapshot of the real flow field obtained from numerical simulation, and  recon represents the flow field obtained from low order mode reconstruction.Figure 4 reflects the accuracy of modal inversion with different numbers of modes.It can be seen from the figure that as the number of modes increases, the inversion accuracy at each eccentricity decreases significantly, and the final error can decrease to about 15%.The size of eccentricity has a significant impact on the accuracy of modal inversion, and as the eccentricity increases, the accuracy of modal inversion continues to increase.
Taking into account the inversion accuracy of six different eccentricities radial forces and the impact of the number of modes on the inversion time, the first ten order modes were ultimately selected as the final number of modes for radial excitation force inversion.According to the principle of DMD method, due to the linearization feature of DMD, the time series prediction of radial excitation force can be achieved by increasing the dimension of the Vandermonde matrix, as shown in Figure 5.    From the figure, it can be seen that the inversion of low order modes can accurately capture the key information of radial force changes.The radial excitation force obtained from the inversion of a few low order modes can reflect the temporal changes of radial force, but the amplitude of the pulsation of the inverted radial excitation force is much smaller than the numerical simulation results.As the low order modes mainly reflect the key information of radial force, So the ignored high-order mode information (information after the tenth-order mode) contains radial excitation force information, which is a pulsating signal with high amplitude.Due to the extremely large pulsating amplitude of the radial excitation force obtained from numerical simulation, the pulsating amplitude of the radial excitation force obtained from the inversion of the first ten low order modes is only one-third of the numerical simulation results.Therefore, the error of inversion of different numbers of low order modes has always been higher than 20%, The energy contained in these amplitudes is mostly characterized by higher-order modes.From the figure, it can be seen that when the eccentricity is greater than 0.3mm, the periodicity of the radial excitation force is significantly enhanced due to the Alford effect.At this time, the radial excitation force obtained from the lower order mode inversion is more consistent with the actual situation.However, when there is no eccentricity (e=0mm) and the eccentricity is small (e=0.1mm), the periodicity of the radial excitation force is significantly smaller.The DMD method is difficult to accurately grasp the key information and basic situation of the radial excitation force signal, Due to the inaccuracy of the selected low order modes, the signal error of their inverse performance is much higher than that of the case with large eccentricity.
Analyzing the short-term time series prediction of radial excitation force under various eccentricities, it can be seen that the low order modal inversion obtained from the first five cycles can effectively reflect the changes in radial excitation force.However, based on the characteristics of the Vandermonde matrix, predicting the radial force signal for three cycles also achieved significant results.Compared with the real situation, it can be seen that the Hankel-based DMD method can effectively predict signals in the short term.However, the accuracy of prediction is closely related to the inversion of low order modes.Due to the main characteristics of radial excitation force changes captured by the first ten modes, it is not possible to accurately predict the rapidly changing amplitude of the real excitation force, but it has an accurate grasp of the overall trend of changes in the last three cycles, It can be imagined that if the number of predicted modes is increased, the radial excitation force of the last three cycles predicted by the DMD method will be more accurate.Consistent with the previous inversion accuracy analysis, the periodic variation of radial excitation force is significantly smaller when there is no eccentricity (e=0mm) and the eccentricity is small (e=0.1mm).At this time, it is difficult for the DMD method to accurately grasp the key information and basic situation of the radial excitation force signal.Due to the inaccurate selection of low order modes, this also affects the predictive ability of the DMD method.As can be seen from Figures 5(a) and (b), The prediction accuracy has been greatly affected, and there has been significant deformation in the last cycle of prediction.Therefore, it can be seen that accurately selecting the low order modes of the signal is crucial for the applicability of the DMD method.

Conclusion
This study investigated the phenomenon of eccentric self-excited vibration of a tubular turbine model runner through numerical simulation.The Hankel-based DMD method was used to extract the main low-order features and invert the radial excitation forces of six different runner eccentricities.At the same time, the characteristics of the DMD method were used to predict and analyze the short-term radial excitation forces.The main research conclusions are as follows: (1) The Hankel-based DMD method can effectively solve the problem of signal dimensionality, accurately capture the basic feature changes of signals at different frequencies, and provide a new approach and method for self-excited vibration analysis of the Alford effect of runner eccentricity.The DMD method for analyzing flow field characteristics in flow fields has been widely studied, but due to the influence of snapshot dimensions, the DMD method has always been a blank in the analysis of one-dimensional signals.This study effectively solves the problems of conventional DMD methods using the Hankel matrix method, providing a powerful means for the analysis and prediction of onedimensional signals in rotating machinery.

Figure 1 .
Figure 1.Eccentricity of runner.Due to the complexity of the generation and mechanism of fluid radial excitation force, it is difficult to measure this radial excitation force through experiments.Therefore, numerical simulation has become the main research method for studying radial excitation force[6,7].At present, numerical simulation methods can effectively represent and demonstrate the radial excitation force on the runner, but how to analyze the signal of this temporal change has become the key to revealing the eccentric Alford effect.In the process of numerical simulation development, FFT method has become the most widely used method for time-frequency decoupling of flow field information[8][9][10].Due to its simple, convenient application, and intuitive results, it has been widely used in various one-dimensional signals with time series changes.However, due to the high-dimensional complexity and nonlinearity of the flow field, the fast Fourier transform (FFT) method has poor spectral analysis performance for local features and nonperiodic signals, and has not shown significant advantages in the analysis of eccentric radial excitation forces.In recent years, in the analysis of flow fields, research methods such as linearizing high-order complex flow fields and using reduced-order modelling methods to capture the main characteristics of flow fields have gradually been widely used.Among them, the dynamic mode decomposition (DMD) method is simple to implement, has clear frequency, and has obvious advantages[11][12][13].However, this reduced-order modelling method is based on matrix decomposition theory and cannot be directly applied to the decomposition of one-dimensional time series signals.To solve this problem in the future, this study uses numerical simulation methods to obtain different radial excitation forces of a certain tubular water turbine with six eccentricities.According to the Takens embedding theorem, a delayed embedding method is used to construct the Hankel matrix of one-dimensional radial excitation force signals, and a Hankel-based DMD method is proposed to analyze and short-term predict six different radial excitation forces, provide a new method for analyzing radial excitation force caused by eccentric runner[14], and provide guarantees for the safe and stable operation of tubular water turbines.

Figure 2 .
Figure 2. fluid domain and grid division of tubular turbine.((a) represents the fluid domain of tubular turbine, (b) represents the grid division of tubular turbine, (c) represents the runner clearance).The computational domain of numerical simulation includes four parts: inlet channel, guide vane, runner, and outlet channel.To ensure the accuracy of numerical simulation results, the fluid domain extends the main axis of the runner from the bulb body.The specific situation of the computational domain is shown in Figure2.Figure2also shows the grid division of key parts in the fluid domain.In this study, hexahedral grids were used for components with simple structures, while tetrahedral grids were used for runner domains with complex structure.The grid convergence index (GCI) criterion based on Richardson extrapolation method was used to check the convergence of the grid, as detailed in reference[15].While ensuring the independence of the grid, the final number of grids was determined to be 3.47 million.
method, the real situation of the flow field can be predicted by ordering the modes with this value.of the system can be obtained from the real part of the eigenvalues  j , and the change frequency of the system can be obtained from the imaginary part  j , which is the change frequency of each mode order.Formula 5 represents the decomposition of the snapshot, and V in the formula is represented by the Vandermonde matrix, which is: The 17th Asian International Conference on Fluid Machinery (AICFM 17 2023) Journal of Physics: Conference Series 2707 (2024) 012068 IOP Publishing doi:10.1088/1742-6596/2707/1/012068

Figure 3 .
Figure 3.Time coefficients and frequency domain analysis of radial excitation forces.Table3.Comparison of modal frequency and FFT frequency.

Figure 5 .
Figure 5.Comparison of radial excitation force inversion and prediction.Figure 5 reflects the comparison between the inversion results (red line, five cycles) and numerical simulation results (black line, eight cycles) of the first ten order modes of the six eccentricity cases of the turbine runner from non-eccentricity (e=0mm) to large eccentricity (e=0.9mm), and also tests the predictive ability of the DMD method.The last three cycles (blue line) of the radial force in the figure are directly predicted based on the results of the first ten order modes of the DMD.From the figure, it can be seen that the inversion of low order modes can accurately capture the key information of radial force changes.The radial excitation force obtained from the inversion of a few low

Figure 5
Figure 5.Comparison of radial excitation force inversion and prediction.Figure 5 reflects the comparison between the inversion results (red line, five cycles) and numerical simulation results (black line, eight cycles) of the first ten order modes of the six eccentricity cases of the turbine runner from non-eccentricity (e=0mm) to large eccentricity (e=0.9mm), and also tests the predictive ability of the DMD method.The last three cycles (blue line) of the radial force in the figure are directly predicted based on the results of the first ten order modes of the DMD.From the figure, it can be seen that the inversion of low order modes can accurately capture the key information of radial force changes.The radial excitation force obtained from the inversion of a few low

( 2 )
Analyzed the advantages of Hankel-based DMD method over FFT method.The Hankel-based DMD method can effectively capture a large number of low-frequency signals that have not been effectively distinguished under FFT transformation.(3)Analyzed the low order inversion and prediction ability of the Hankel-based DMD method for radial excitation force signals.The radial excitation force obtained from the inversion of a few low order modes can reflect the temporal variation of the radial force.Compared with the real situation, it can be seen that the Hankel based DMD method can effectively predict the short-term signal.

Table 1 .
Main parameters and values of tubular turbine.

Table 2 .
Comparison of performance under rated condition.

Table 3 .
Comparison of modal frequency and FFT frequency.