A hybrid modal analysis method for unsteady flow feature extraction of turbomachinery

Abstract

The present invention discloses a hybrid modal analysis method for unsteady flow feature extraction of turbomachinery, which mainly includes the following steps: constructing a spatio-temporal matrix according to the unsteady three-dimensional velocity field; performing proper orthogonal decomposition on the spatio-temporal matrix, sorting the modes according to the modal energy magnitude and obtaining the modal dominant frequency; using the sparse enhanced dynamic modal decomposition method to obtain the sparse enhanced dynamic modal combination and calculating the modal frequency; obtaining the optimal dynamic modal combination sorting according to the proper orthogonal modal dominant frequency sorting. The present invention corrects the sorting of the sparse enhanced dynamic modes through the energy and dominant frequency information of the proper orthogonal modes of the unsteady flow of the turbomachinery, realizes the extraction of the main single-frequency dynamic modal information with good dynamic stability based on the energy scale, effectively overcomes the deficiencies of the traditional proper orthogonal decomposition method and dynamic modal decomposition method when applied to the unsteady flow of the turbomachinery, and provides a technical method for the accurate extraction of the complex unsteady flow features of the turbomachinery.

Description

Technical Field

[0001] The present invention relates to the field of aerodynamic thermodynamics design of aero-engine and gas turbine turbomachinery, and particularly to a hybrid modal analysis method for extracting unsteady flow characteristics of turbomachinery. Background Art

[0002] Aero-engines and gas turbines are major national needs and key development areas in high-end equipment manufacturing in China. The turbomachinery is one of the three core components of aero-engines and gas turbines. Advanced aerodynamic thermodynamics design of turbomachinery is one of the bottlenecks in the development of high-performance aero-engines and gas turbines. The internal flow of turbomachinery is very complex, and there are widely multi-scale secondary flow structures with strong unsteadiness and strong three-dimensionality, which have a great impact on the performance of turbomachinery and pose great challenges to the aerodynamic design of turbomachinery. In recent years, with the continuous improvement of the numerical simulation technology and experimental measurement fineness of the complex unsteady flow of turbomachinery, the obtained flow field data has increased explosively, bringing challenges to data analysis. Therefore, there is an urgent need to develop a turbomachinery unsteady flow field analysis method with strong applicability and high computational efficiency, so that it can quickly and accurately capture the main unsteady flow structures of turbomachinery to support the analysis of flow mechanisms and then guide the optimization of aerodynamic thermodynamics design of turbomachinery.

[0003] In recent years, the modal decomposition method based on data-driven has been widely used in analyzing complex unsteady flow mechanisms. By extracting the dominant flow characteristics contained in high-dimensional spatio-temporal data in a low-dimensional system, the analysis efficiency of the unsteady flow field is effectively improved. Typical modal decomposition methods include proper orthogonal decomposition and dynamic mode decomposition. The modes obtained by the proper orthogonal decomposition method can resolve the main flow structures and determine the main modes of the flow field through the energy scale information of the modes, but it is difficult to effectively distinguish multiple frequency components in a single mode. The dynamic mode decomposition method can obtain modes with a single frequency and their dynamic information, and has great advantages in analyzing the dynamic evolution characteristics of unsteady flows. However, when the dynamic mode decomposition is used to analyze the unsteady flow characteristics of turbomachinery, it cannot clearly reflect the modal energy scale characteristics, resulting in a fuzzy determination criterion for the main modes. In addition, due to its sensitivity to noise, the dynamic stability of the main modes extracted is weak.

[0004] To solve the above problems, the present invention proposes a hybrid modal analysis method for extracting unsteady flow characteristics of turbomachinery, which can effectively combine the advantages of the proper orthogonal decomposition method for extracting modal energy scale information and the dynamic mode decomposition method for extracting single-frequency modal dynamic information, realize the extraction of the main dynamic modes with clear energy scale characteristics and good dynamic stability in the unsteady flow of turbomachinery, analyze the unsteady flow structure faster and more accurately, and then guide the aerodynamic thermodynamics design of turbomachinery, providing technical support for the development of high-performance aero-engines and gas turbines. Summary of the Invention

[0005] (1) Technical Problem to be Solved

[0006] The object of the present invention is to propose a hybrid modal analysis method for extracting unsteady flow characteristics of a turbomachine, to overcome the problem that the traditional proper orthogonal decomposition method can only extract multi-frequency modal characteristics when applied to the complex unsteady flow of a turbomachine, and the problem that the main single-frequency modal determination criterion of the traditional dynamic modal decomposition method is fuzzy and the dynamic stability is poor when applied to the complex unsteady flow of a turbomachine.

[0007] (2) Technical Solution

[0008] To solve the above technical problem, the present invention provides a hybrid modal analysis method for extracting unsteady flow characteristics of a turbomachine, including the following steps:

[0009] Step 1: Construct a spatio-temporal matrix according to the unsteady three-dimensional velocity field;

[0010] Step 2: Perform proper orthogonal decomposition on the constructed spatio-temporal matrix;

[0011] Step 3: Sort the proper orthogonal modes according to the modal energy magnitude;

[0012] Step 4: Use the fast Fourier transform to obtain the dominant frequency of the proper orthogonal mode;

[0013] Step 5: Perform dynamic modal decomposition on the constructed spatio-temporal matrix;

[0014] Step 6: Use the sparse enhanced dynamic modal decomposition method to obtain a sparse enhanced dynamic modal combination;

[0015] Step 7: Calculate the frequency of the sparse enhanced dynamic modal combination;

[0016] Step 8: Obtain the optimal dynamic modal combination sorting according to the proper orthogonal mode dominant frequency sorting;

[0017] Steps 2 to 4 and steps 5 to 7 can be carried out simultaneously;

[0018] ① The construction of the spatio-temporal matrix according to the unsteady three-dimensional velocity field includes:

[0019] Based on the three-dimensional flow field data of the unsteady flow of the turbomachine analyzed, a spatio-temporal matrix for proper orthogonal decomposition and dynamic mode decomposition is constructed. Specifically, the velocity data in the x-direction, the velocity data in the y-direction, and the velocity data in the z-direction in the three-dimensional flow field data of the unsteady flow of the turbomachine are extracted. The velocity data in the x-direction, the velocity data in the y-direction, and the velocity data in the z-direction of each transient flow field are regarded as a flow field snapshot, and the data at each spatial grid point in the flow field is regarded as an element; for the unsteady flow field data with M spatial grid points and N snapshots, the three-dimensional velocity field U of the nth snapshot n can be expressed by the following formula:

[0020] where n = 1, 2, …, N;

[0021] where, u n is the column vector composed of the velocity data in the x-direction of all spatial grid points in the nth snapshot, v n is the column vector composed of the velocity data in the y-direction of all spatial grid points in the nth snapshot, w n is the column vector composed of the velocity data in the z-direction of all spatial grid points in the nth snapshot; the column vector u composed of the velocity data in the x-direction of all spatial grid points in the nth snapshot n is given by the following formula:

[0022] where n = 1, 2, …, N;

[0023] where, u mn is the velocity in the x-direction of the mth spatial grid point in the nth flow field snapshot; the representation methods of the column vector v n and the column vector w n are similar to the representation method of the column vector u n ;

[0024] Based on the three-dimensional flow field data of the unsteady flow of the turbomachine analyzed, a spatio-temporal matrix X based on the three-dimensional velocity pulsation field is constructed for proper orthogonal decomposition; the spatio-temporal matrix X based on the three-dimensional velocity pulsation field is given by the following formula:

[0025]

[0026] where, is the three-dimensional average velocity field of the unsteady flow of the turbomachine analyzed, is the column vector composed of the average velocity data in the x-direction of all spatial grid points, is a column vector composed of the average velocity data of all spatial grid points in the y direction, is a column vector composed of the average velocity data of all spatial grid points in the z direction;

[0027] According to the three-dimensional flow field data of the unsteady flow of the turbomachine to be analyzed, a first spatio-temporal matrix X1 based on the three-dimensional velocity field and a second spatio-temporal matrix X2 based on the three-dimensional velocity field are constructed for dynamic mode decomposition; the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field are given by the following formula:

[0028]

[0029] ② The performing of the proper orthogonal decomposition on the constructed spatio-temporal matrix includes:

[0030] Perform proper orthogonal decomposition on the spatio-temporal matrix X of the three-dimensional velocity pulsation field in step one to obtain the proper orthogonal mode space matrix Φ of the three-dimensional velocity pulsation field, the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field, and the proper orthogonal mode energy vector λ of the three-dimensional velocity pulsation field;

[0031] The proper orthogonal mode space matrix Φ of the three-dimensional velocity pulsation field is represented by the following formula:

[0032]

[0033] where, φ l is the k-th order proper orthogonal mode space vector of the three-dimensional velocity pulsation field, and the value range of l is 1 to N; the k-th order proper orthogonal mode space vector φ l of the three-dimensional velocity pulsation field is represented by the following formula:

[0034]

[0035] where, φ l,x is the l-th order proper orthogonal mode vector of the velocity pulsation in the x direction, φ l,y is the l-th order proper orthogonal mode vector of the velocity pulsation in the y direction, φ l,z is the l-th order proper orthogonal mode vector of the velocity pulsation in the z direction;

[0036] The proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field is represented by the following formula:

[0037]

[0038] where, the element in the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field Denote the modal coefficient of the \(l\)th proper orthogonal mode in the \(n\)th snapshot, where the subscript \(n\) ranges from 1 to \(N\), and the superscript \(l\) ranges from 1 to \(N\);

[0039] The spatio-temporal matrix \(X\) of the three-dimensional velocity fluctuation field, the proper orthogonal mode spatial matrix \(\varPhi\) of the three-dimensional velocity fluctuation field, and the proper orthogonal mode time coefficient matrix \(A\) of the three-dimensional velocity fluctuation field satisfy the following relationship:

[0040] \(X = \varPhi A\)

[0041] The proper orthogonal mode energy vector \(\lambda\) of the three-dimensional velocity fluctuation field is expressed by the following formula:

[0042]

[0043] where the \(l\)th element \(\lambda_l\) in the proper orthogonal mode energy vector \(\lambda\) of the three-dimensional velocity fluctuation field l is the energy of the \(l\)th proper orthogonal mode, and \(l\) ranges from 1 to \(N\); the energy \(\lambda_l\) of the \(l\)th proper orthogonal mode l is the \(l\)th eigenvalue of the autocovariance matrix \(C\) of the spatio-temporal matrix \(X\) of the three-dimensional velocity fluctuation field. The autocovariance matrix \(C\) of the spatio-temporal matrix \(X\) of the three-dimensional velocity fluctuation field is obtained from the following formula:

[0044]

[0045] ③ The sorting of the proper orthogonal modes according to the modal energy includes:

[0046] Arrange the elements in the proper orthogonal mode energy vector \(\lambda\) of the three-dimensional velocity fluctuation field in step 2 in descending order, that is, satisfy the following relationship:

[0047] \(\lambda_1>\lambda_2>\cdots>\lambda_N\) N

[0048] According to the order of the proper orthogonal mode energy, make corresponding adjustments to the column order of the proper orthogonal mode spatial matrix \(\varPhi\) of the three-dimensional velocity fluctuation field in step 2 and the row order of the proper orthogonal mode time coefficient matrix \(A\) of the three-dimensional velocity fluctuation field;

[0049] ④ The obtaining of the dominant frequency of the proper orthogonal mode by using the fast Fourier transform includes:

[0050] Extract the first \(r_1\) rows of data from the proper orthogonal mode time coefficient matrix \(A\) of the three-dimensional velocity fluctuation field in step 3 as the time coefficient discrete signal sequence of the first \(r_1\) proper orthogonal modes, where \(r_1 < N\); the time coefficient discrete signal sequence \(a_l\) of the \(l\)th proper orthogonal mode l is expressed by the following formula:

[0051]

[0052] Process the time coefficient signal sequence of the first r1 order proper orthogonal modes by the fast Fourier transform method, where r1 < N. For the processing result of the time coefficient signal sequence of each order of proper orthogonal modes, take the frequency with the highest amplitude as the main frequency of the corresponding order of proper orthogonal modes, calculate the main frequency of the first r1 order proper orthogonal modes and discard the frequencies with the same value, and form the main frequency vector f of the first r1 order proper orthogonal modes P , with the unit of Hz, expressed as:

[0053]

[0054] Among them, the main frequency vector f of the first r1 order proper orthogonal modes P The i-th element f Pi of is the i-th main frequency of the first r1 order proper orthogonal modes, and the value range of i is 1 to r2;

[0055] ⑤ The dynamic mode decomposition of the constructed spatio-temporal matrix includes:

[0056] Perform dynamic mode decomposition on the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field in step one to obtain the dynamic mode space matrix V of the three-dimensional velocity field, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field;

[0057] In the dynamic mode decomposition, the relationship between the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field can be represented by a linear dynamic system, as shown in the following formula:

[0058]

[0059] Among them, D is the linear coefficient matrix of the linear dynamic system; perform eigenvalue decomposition on the linear coefficient matrix D of the linear dynamic system, as shown in the following formula:

[0060] D = V·diag(μ1 μ2 … μ l … μ N-1 )·V H

[0061] Among them, μ l is the l-th order dynamic mode eigenvalue, and the value range of l is 1 to (N - 1); V is the dynamic mode space matrix of the three-dimensional velocity field, represented by the following formula:

[0062]

[0063] Among them, v lis the spatial vector of the k-th proper orthogonal mode, and the value range of l is 1 to (N - 1); the spatial vector of the k-th proper orthogonal mode is represented by the following formula:

[0064]

[0065] where, v l,x is the l-th order dynamic mode vector of the velocity in the x direction, v l,y is the l-th order dynamic mode vector of the velocity in the y direction, v l,z is the l-th order dynamic mode vector of the velocity in the z direction;

[0066] The dynamic mode eigenvalue vector μ of the three-dimensional velocity field is expressed as:

[0067]

[0068] where, μ l is the l-th order dynamic mode eigenvalue, and the value range of l is 1 to (N - 1); the dynamic mode amplitude vector α of the three-dimensional velocity field is obtained by the following formula:

[0069]

[0070] where, the element α in the dynamic mode amplitude vector α of the three-dimensional velocity field l is the l-th order dynamic mode amplitude, and the value range of l is 1 to (N - 1);

[0071] ⑥ The obtaining of the sparse enhanced dynamic mode combination by using the sparse enhanced dynamic mode decomposition method includes:

[0072] According to the first spatio-temporal matrix X1 based on the three-dimensional velocity field, the second spatio-temporal matrix X2 based on the three-dimensional velocity field, the dynamic mode spatial matrix V of the three-dimensional velocity field, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field in step five, the sparse enhanced dynamic mode decomposition method is used to screen out the sparse enhanced dynamic mode combination and obtain the sparse enhanced dynamic mode amplitude vector Specifically, the sparse enhanced dynamic mode decomposition method screens out r1 order dynamic modes through an optimization algorithm, where r1 < N, as the sparse enhanced dynamic mode combination, and the optimization problem involved is to minimize the value of the following formula:

[0073]

[0074] where, is the sparse enhanced dynamic mode amplitude diagonal matrix, is the i-th sparse enhanced dynamic mode amplitude, and the value range of i is 1 to r1; Vand is the dynamic modal evolution characteristic matrix, which is expressed as follows:

[0075]

[0076] ⑦ The frequencies for calculating the sparse-enhanced dynamic modal combination include:

[0077] According to the dynamic modal eigenvalue vector μ of the three-dimensional velocity field described in step five, calculate the frequency vector f of the sparse-enhanced dynamic modal combination described in step six D , with the unit of Hz, and is expressed as:

[0078]

[0079] Among them, the frequency vector f of the sparse-enhanced dynamic modal combination D The i-th element f Di is the i-th order dynamic modal frequency of the sparse-enhanced dynamic modal combination. The value range of i is 1 to r1. The frequency vector f of the sparse-enhanced dynamic modal combination D The i-th element f Di is obtained from the following formula:

[0080]

[0081] Among them, Δt is the time interval between adjacent snapshots;

[0082] ⑧ The obtaining of the optimal dynamic modal combination sorting according to the principal frequency sorting of the proper orthogonal modes includes:

[0083] According to the principal frequency vector f P of the first r1 order proper orthogonal modes, correct the sorting of the elements in the frequency vector f D of the sparse-enhanced dynamic modal combination in step seven to obtain the frequency vector of the optimal dynamic modal combination. According to the new sorting, obtain the eigenvalue vector of the optimal dynamic modal combination and calculate the dynamic modal growth rate vector Specifically, the frequency vector of the optimal dynamic modal combination is expressed as:

[0084]

[0085] Among them, the frequency vector of the optimal dynamic modal combination The i-th element is the frequency of the i-th order dynamic modal of the optimal dynamic modal combination. The value range of i is 1 to r2; The i-th element is defined as the frequency vector f of the sparse-enhanced dynamic mode combination D One of the elements closest to the i-th dominant frequency f of the first r1 proper orthogonal modes Pi is represented by the following formula:

[0086]

[0087] According to the sorting of the optimal dynamic mode combination, the eigenvalue vector of the optimal dynamic mode combination is obtained is represented by the following formula:

[0088]

[0089] wherein, the eigenvalue vector of the optimal dynamic mode combination The i-th element of is the eigenvalue of the i-th dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2;

[0090] The dynamic mode growth rate vector of the optimal dynamic mode combination is expressed as:

[0091]

[0092] wherein, the dynamic mode growth rate vector of the optimal dynamic mode combination The i-th element of is the growth rate of the i-th dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; the dynamic mode growth rate vector of the optimal dynamic mode combination The i-th element of is obtained by the following formula:

[0093] where i = 1, 2,..., r2.

[0094] (III) Beneficial Effects

[0095] A hybrid mode analysis method for unsteady flow characteristic extraction of a turbomachine provided by the present invention has the following beneficial effects:

[0096] By performing proper orthogonal decomposition and sparse-enhanced dynamic mode decomposition on the spatio-temporal matrix constructed based on the unsteady three-dimensional flow field of the turbomachine, the modal dominant frequency and energy information of the proper orthogonal modes, as well as the frequency and amplitude information of the sparse-enhanced dynamic modes, are obtained. According to the energy and dominant frequency information of the proper orthogonal modes, the sorting of the sparse-enhanced dynamic modes is corrected to obtain the optimal dynamic mode combination and its dynamic information, effectively overcoming the problem that the traditional proper orthogonal decomposition method can only extract the multi-frequency mode characteristics when applied to the complex unsteady flow of the turbomachine, and the problem that the main single-frequency mode determination criterion of the traditional dynamic mode decomposition method is fuzzy and the dynamic stability is poor when applied to the complex unsteady flow of the turbomachine, realizing the determination and extraction of the main single-frequency dynamic modes with good dynamic stability in the unsteady flow of the turbomachine based on the energy scale. The method of the present invention is concise in form and strong in portability, and can be well applied to the analysis of complex unsteady flow data with different flow characteristics in the turbomachine, realizing the extraction of flow field characteristics and further guiding the aerodynamic thermodynamics design of the turbomachine. The proposed method is convenient to be implanted into the existing flow field post-processing software for application and expansion, and has broad academic and engineering application prospects, providing a technical method for accurately extracting the characteristics of complex flows such as unsteady and multi-scale in the post-processing and analysis of the turbomachine flow field. Brief Description of the Drawings

[0097] Figure 1 is a flowchart of a hybrid mode analysis method for extracting the unsteady flow characteristics of a turbomachine according to the present invention;

[0098] Figure 2 is a distribution diagram of the relative energy proportion of different proper orthogonal modes sorted by energy provided by the embodiment of the present invention;

[0099] Figure 3 is a distribution diagram of the dominant frequency of the proper orthogonal mode and the frequency of the optimal dynamic mode provided by the embodiment of the present invention;

[0100] Figure 4 is a distribution diagram of the absolute value of the growth rate of the original dynamic mode and the optimal dynamic mode provided by the embodiment of the present invention. Detailed Description of the Invention

[0101] The following further details the specific implementation manners of the present invention by taking the unsteady flow analysis of the corner separation of a certain high-speed and high-load compressor cascade as an example in conjunction with the drawings. The following embodiments are used to illustrate the present invention but not to limit the scope of the present invention.

[0102] A hybrid mode analysis method for extracting the unsteady flow characteristics of a turbomachine provided by the present invention includes the following steps:

[0103] Step 1, constructing a spatio-temporal matrix according to the unsteady three-dimensional velocity field;

[0104] In this step, based on the three-dimensional flow field data of the unsteady flow in the turbomachine analyzed, a spatio-temporal matrix for proper orthogonal decomposition and dynamic mode decomposition is constructed. Here, the unsteady flow of corner separation in a high-speed and high-loaded compressor cascade is taken as an example. Specifically, the velocity data in the x-direction, the velocity data in the y-direction, and the velocity data in the z-direction in the three-dimensional flow field data of the unsteady flow in the turbomachine are extracted. The velocity data in the x-direction, the velocity data in the y-direction, and the velocity data in the z-direction of each transient flow field are regarded as a flow field snapshot, and the data at each spatial grid point in the flow field are regarded as an element; for the unsteady flow field data with M spatial grid points and N snapshots, the three-dimensional velocity field U of the nth snapshot n can be expressed by the following formula:

[0105] where n = 1, 2, …, N;

[0106] where, u n is the column vector composed of the velocity data in the x-direction of all spatial grid points in the nth snapshot, v n is the column vector composed of the velocity data in the y-direction of all spatial grid points in the nth snapshot, w n is the column vector composed of the velocity data in the z-direction of all spatial grid points in the nth snapshot; the column vector u composed of the velocity data in the x-direction of all spatial grid points in the nth snapshot n is given by the following formula:

[0107] where n = 1, 2, …, N;

[0108] where, u mn is the velocity in the x-direction of the mth spatial grid point in the nth flow field snapshot; the representation methods of the column vectors v n and the column vector w n are similar to the representation method of the column vector u n ;

[0109] According to the three-dimensional flow field data of the unsteady flow in the turbomachine analyzed, a spatio-temporal matrix X based on the three-dimensional velocity fluctuation field is constructed for proper orthogonal decomposition; the spatio-temporal matrix X based on the three-dimensional velocity fluctuation field is given by the following formula:

[0110]

[0111] where, is the three-dimensional mean velocity field of the unsteady flow in the turbomachine analyzed, is the column vector composed of the mean velocity data in the x-direction of all spatial grid points, is a column vector composed of the average velocity data of all spatial grid points in the y direction, is a column vector composed of the average velocity data of all spatial grid points in the z direction;

[0112] According to the three-dimensional flow field data of the unsteady flow of the turbomachine analyzed, a first spatio-temporal matrix X1 based on the three-dimensional velocity field and a second spatio-temporal matrix X2 based on the three-dimensional velocity field are constructed for dynamic mode decomposition; the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field are given by the following formula:

[0113]

[0114]

[0115] In the unsteady flow of corner separation of a certain high-speed and high-load compressor cascade in this embodiment, the number of spatial grid points M in the flow field is 6761060, the number of snapshots N is 500, and the data elements in the constructed spatio-temporal matrix X based on the three-dimensional velocity pulsation field, the first spatio-temporal matrix X1 based on the three-dimensional velocity field, and the second spatio-temporal matrix X2 based on the three-dimensional velocity field are arranged in sequence according to the spatial and snapshot time order;

[0116] Step two, perform proper orthogonal decomposition on the constructed spatio-temporal matrix;

[0117] In this step, perform proper orthogonal decomposition on the spatio-temporal matrix X based on the three-dimensional velocity pulsation field in step one to obtain the proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field, the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field, and the proper orthogonal mode energy vector λ of the three-dimensional velocity pulsation field;

[0118] The obtained proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field is represented by the following formula:

[0119]

[0120] Among them, φ l is the k-th order proper orthogonal mode spatial vector of the three-dimensional velocity pulsation field, and the value range of l is 1 to N; the k-th order proper orthogonal mode spatial vector φ l of the three-dimensional velocity pulsation field is represented by the following formula:

[0121]

[0122] Among them, φ l,x is the l-th order proper orthogonal mode vector of the velocity pulsation in the x direction, φ l,y is the l-th order proper orthogonal mode vector of the velocity pulsation in the y direction, φ l,zis the l-th proper orthogonal decomposition (POD) mode vector of the velocity fluctuation in the z direction;

[0123] The POD temporal coefficient matrix A of the three-dimensional velocity fluctuation field is expressed as follows:

[0124]

[0125] where the element in the POD temporal coefficient matrix A of the three-dimensional velocity fluctuation field represents the modal coefficient of the l-th POD mode in the n-th snapshot. The subscript n ranges from 1 to N, and the superscript l ranges from 1 to N;

[0126] Based on the spatio-temporal matrix X of the three-dimensional velocity fluctuation field, the POD spatial matrix Φ of the three-dimensional velocity fluctuation field, and the POD temporal coefficient matrix A of the three-dimensional velocity fluctuation field satisfy the following relationship:

[0127] X = ΦA

[0128] The POD energy vector λ of the three-dimensional velocity fluctuation field is expressed as follows:

[0129]

[0130] where the l-th element λ l in the POD energy vector λ of the three-dimensional velocity fluctuation field is the energy of the l-th POD mode, and l ranges from 1 to N; the energy λ l of the l-th POD mode is the l-th eigenvalue of the autocovariance matrix C of the spatio-temporal matrix X of the three-dimensional velocity fluctuation field. The autocovariance matrix C of the spatio-temporal matrix X of the three-dimensional velocity fluctuation field is obtained as follows:

[0131]

[0132] Step 3: Sort the POD modes according to the modal energy;

[0133] In this step, the elements in the POD energy vector λ of the three-dimensional velocity fluctuation field obtained in Step 2 are arranged in descending order, i.e., satisfying the following relationship:

[0134] λ1 > λ2 > … > λ N

[0135] According to the order of the POD energy, the column order of the POD spatial matrix Φ and the row order of the POD temporal coefficient matrix A of the three-dimensional velocity fluctuation field obtained in Step 2 are adjusted accordingly; Figure 2The relative energy proportion distribution diagram of different proper orthogonal modes sorted by energy provided for the unsteady flow in the corner separation of a certain high-speed and high-load compressor cascade in the embodiments of the present invention. It can be analyzed that as the mode order increases, the corresponding mode energy decreases.

[0136] Step four, use the fast Fourier transform to obtain the main frequency of the proper orthogonal mode;

[0137] In this step, extract the first r1 rows of data from the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field in step three as the time coefficient discrete signal sequence of the first r1 order proper orthogonal modes, where r1 < N; the time coefficient discrete signal sequence a of the l-th order proper orthogonal mode l is represented by the following formula:

[0138]

[0139] Process the time coefficient signal sequence of the first r1 order proper orthogonal modes by the fast Fourier transform method, where r1 < N. For the processing result of the time coefficient signal sequence of each order proper orthogonal mode, take the frequency with the highest amplitude as the main frequency of the corresponding order proper orthogonal mode, calculate the main frequency of the first r1 order proper orthogonal modes and discard the same frequencies, and form the main frequency vector f of the first r1 order proper orthogonal modes P , with the unit of Hz, is expressed as:

[0140]

[0141] where the main frequency vector f of the first r1 order proper orthogonal modes P the i-th element f Pi is the i-th main frequency of the first r1 order proper orthogonal modes, and the value range of i is 1 to r2; in the unsteady flow of the corner separation of a certain high-speed and high-load compressor cascade in this embodiment, r1 = 8, r2 = 6;

[0142] Step five, perform dynamic mode decomposition on the constructed spatio-temporal matrix;

[0143] In this step, perform dynamic mode decomposition on the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field in step one to obtain the dynamic mode space matrix V of the three-dimensional velocity field, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field;

[0144] In dynamic mode decomposition, the relationship between the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field can be represented by a linear dynamic system, as shown in the following formula:

[0145]

[0146] Among them, D is the linear coefficient matrix of the linear dynamic system; the eigenvalue decomposition of the linear coefficient matrix D of the linear dynamic system is carried out as shown in the following formula:

[0147] D = V·diag(μ1 μ2 … μ l … μ N-1 )·V H

[0148] Among them, μ l is the l-th order dynamic modal eigenvalue, and the value range of l is 1 to (N - 1); V is the dynamic modal space matrix of the three-dimensional velocity field, which is represented by the following formula:

[0149]

[0150] Among them, v l is the spatial vector of the k-th order proper orthogonal mode, and the value range of l is 1 to (N - 1); the spatial vector of the k-th order proper orthogonal mode is represented by the following formula:

[0151]

[0152] Among them, v l,x is the l-th order dynamic modal vector of the velocity in the x direction, v l,y is the l-th order dynamic modal vector of the velocity in the y direction, v l,z is the l-th order dynamic modal vector of the velocity in the z direction;

[0153] The dynamic modal eigenvalue vector μ of the three-dimensional velocity field is expressed as:

[0154]

[0155] Among them, μ l is the l-th order dynamic modal eigenvalue, and the value range of l is 1 to (N - 1); the dynamic modal amplitude vector α of the three-dimensional velocity field is obtained by the following formula:

[0156]

[0157] Among them, the element α l in the dynamic modal amplitude vector α of the three-dimensional velocity field is the l-th order dynamic modal amplitude, and the value range of l is 1 to (N - 1);

[0158] Step six, adopt the sparse enhanced dynamic modal decomposition method to obtain the sparse enhanced dynamic modal combination;

[0159] In this step, based on the first spatio-temporal matrix X1 of the three-dimensional velocity field, the second spatio-temporal matrix X2 of the three-dimensional velocity field, the dynamic mode space matrix V of the three-dimensional velocity field, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field in step five, the sparse enhanced dynamic mode decomposition method is used to screen out the sparse enhanced dynamic mode combination and obtain the sparse enhanced dynamic mode amplitude vector. Specifically, the sparse enhanced dynamic mode decomposition method screens out the r1-order dynamic modes through an optimization algorithm, where r1 < N, as the sparse enhanced dynamic mode combination. The optimization problem involved is to minimize the value of the following formula:

[0160]

[0161] where, is the sparse enhanced dynamic mode amplitude diagonal matrix, is the amplitude of the i-th sparse enhanced dynamic mode, and the value range of i is 1 to r1; V and is the dynamic mode evolution characteristic matrix, which is expressed as follows:

[0162]

[0163] Step seven, calculate the frequency of the sparse enhanced dynamic mode combination;

[0164] In this step, according to the dynamic mode eigenvalue vector μ of the three-dimensional velocity field in step five, the frequency vector f of the sparse enhanced dynamic mode combination in step six is calculated. D , with the unit of Hz, is expressed as:

[0165]

[0166] where, the frequency vector f of the sparse enhanced dynamic mode combination D The i-th element f Di of is the frequency of the i-th order dynamic mode of the sparse enhanced dynamic mode combination, and the value range of i is 1 to r1. The i-th element f D of the frequency vector f of the sparse enhanced dynamic mode combination Di is obtained from the following formula:

[0167]

[0168] where, Δt is the time interval between adjacent snapshots; in the unsteady flow of the corner separation of a certain high-speed and high-load compressor cascade in this embodiment, r1 = 8 and r2 = 6;

[0169] Step eight, obtain the optimal dynamic mode combination sorting according to the main frequency sorting of the proper orthogonal modes;

[0170] In this step, according to the main frequency vectors f of the first r1 order proper orthogonal modes P , the sorting of the elements in the frequency vector f of the sparse enhanced dynamic mode combination in step seven D is corrected to obtain the frequency vector of the optimal dynamic mode combination According to the new sorting, the eigenvalue vector of the optimal dynamic mode combination is obtained and the dynamic mode growth rate vector of the optimal dynamic mode combination is calculated Specifically, the frequency vector of the optimal dynamic mode combination is expressed as:

[0171]

[0172] where the i-th element of the frequency vector of the optimal dynamic mode combination is the frequency of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; the i-th element of the frequency vector of the optimal dynamic mode combination is defined as the one in the elements of the frequency vector f of the sparse enhanced dynamic mode combination that is closest to the i-th main frequency f D of the first r1 order proper orthogonal modes, and is expressed by the following formula: Pi

[0173]

[0174] According to the sorting of the optimal dynamic mode combination, the eigenvalue vector of the optimal dynamic mode combination is obtained and is expressed by the following formula:

[0175]

[0176] where the i-th element of the eigenvalue vector of the optimal dynamic mode combination is the eigenvalue of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2;

[0177] The dynamic mode growth rate vector of the optimal dynamic mode combination is expressed as:

[0178]

[0179] where the i-th element of the dynamic mode growth rate vector of the optimal dynamic mode combination is the growth rate of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; the dynamic mode growth rate vector of the optimal dynamic mode combination the i-th element of is obtained from:

[0180] where i = 1, 2, …, r2.

[0181] In this embodiment, data on the unsteady flow of corner separation in a certain high-speed and high-load compressor cascade is analyzed. After introducing the analysis method proposed in this patent, an optimal dynamic mode combination with both energy scale characteristics and good dynamic stability is obtained.

[0182] Figure 3 It is the distribution diagram of the dominant frequency of the proper orthogonal mode and the optimal dynamic mode frequency. It can be analyzed that the order of the proper orthogonal modes is arranged in descending order of the proper orthogonal mode energy. The optimal dynamic mode combination is obtained by correcting the sorting of the sparse-enhanced dynamic mode combination according to the dominant frequency and energy scale of the proper orthogonal mode. At the same mode order, the optimal dynamic mode frequency is close to the dominant frequency of the proper orthogonal mode, indicating that the optimal dynamic mode combination not only reflects the size of the energy scale but also the single-frequency characteristics of the dynamic mode.

[0183] Figure 4 It is the distribution diagram of the absolute value of the growth rate of the original dynamic mode and the optimal dynamic mode. It can be analyzed that the growth rates of the first 6 original dynamic modes obtained based on the traditional dynamic mode decomposition method are generally high, indicating poor dynamic characteristics; while after introducing the method proposed in this patent, the 6 dynamic modes in the obtained optimal dynamic mode combination all have low growth rates, indicating good dynamic stability, which is conducive to the extraction of the main characteristics of unsteady flow and dynamic analysis.

[0184] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

[0185] In summary, the present invention performs proper orthogonal decomposition and sparse-enhanced dynamic mode decomposition on the spatio-temporal matrix constructed based on the unsteady three-dimensional flow field of the turbomachine, obtaining the modal dominant frequency and energy information of the proper orthogonal modes, as well as the frequency and amplitude information of the sparse-enhanced dynamic modes. According to the energy and dominant frequency information of the proper orthogonal modes, the sorting of the sparse-enhanced dynamic modes is corrected, obtaining the optimal dynamic mode combination and its dynamic information, effectively overcoming the problem that the traditional proper orthogonal decomposition method can only extract multi-frequency modal characteristics when applied to the complex unsteady flow of the turbomachine, and the problem that the main single-frequency modal determination criterion of the traditional dynamic mode decomposition method is fuzzy and the dynamic stability is poor when applied to the complex unsteady flow of the turbomachine, realizing the determination and extraction of the main single-frequency dynamic modes with good dynamic stability in the unsteady flow of the turbomachine based on the energy scale. It has the characteristics of simple form and strong portability, can be well applied to the analysis of complex unsteady flow data with different flow characteristics in the turbomachine, realize the extraction of flow field characteristics, and further guide the aerodynamic thermodynamics design of the turbomachine, providing a technical method for the accurate extraction of complex flow characteristics such as unsteady and multi-scale in the post-processing and analysis of the turbomachine flow field.

Claims

1. A hybrid modal analysis method for unsteady flow feature extraction of turbomachines, characterized in that It includes the following steps: Step 1, construct a spatio-temporal matrix according to the unsteady three-dimensional velocity field; Step 2, perform proper orthogonal decomposition on the constructed spatio-temporal matrix; Step 3, sort the proper orthogonal modes according to the modal energy magnitude; Step 4, use the fast Fourier transform to obtain the main frequency of the proper orthogonal modes; Step 5, perform dynamic mode decomposition on the constructed spatio-temporal matrix; Step 6, use the sparse enhanced dynamic mode decomposition method to obtain a sparse enhanced dynamic mode combination; Step 7, calculate the frequency of the sparse enhanced dynamic mode combination; Step 8, obtain the optimal dynamic mode combination sorting according to the sorting of the main frequencies of the proper orthogonal modes; Steps 2 to 4 and steps 5 to 7 can be carried out simultaneously; ① The construction of the spatio-temporal matrix according to the unsteady three-dimensional velocity field includes: According to the three-dimensional flow field data of the unsteady flow of the turbomachine analyzed, a spatio-temporal matrix for proper orthogonal decomposition and dynamic mode decomposition is constructed. Specifically, the velocity data in the x direction, the velocity data in the y direction, and the velocity data in the z direction in the three-dimensional flow field data of the unsteady flow of the turbomachine are extracted. The velocity data in the x direction, the velocity data in the y direction, and the velocity data in the z direction of each transient flow field are regarded as a flow field snapshot, and the data at each spatial grid point in the flow field are regarded as an element; for the unsteady flow field data with M spatial grid points and N snapshots, the three-dimensional velocity field U of the nth snapshot n can be expressed by the following formula: where n = 1, 2, …, N; where u n is a column vector formed by the velocity data of all spatial grid points in the x - direction in the nth snapshot, v n is a column vector formed by the velocity data of all spatial grid points in the y - direction in the nth snapshot, w n is a column vector formed by the velocity data of all spatial grid points in the z - direction in the nth snapshot; the column vector u n formed by the velocity data of all spatial grid points in the x - direction in the nth snapshot is given by the following formula: where n = 1, 2, …, N; where, u mn is the velocity of the m-th spatial grid point in the n-th flow field snapshot in the x direction; the column vector v n , the column vector w n are represented in a similar way to the column vector u n ; Construct a spatio-temporal matrix X based on the three-dimensional velocity pulsation field according to the three-dimensional flow field data of the unsteady flow of the turbomachine for proper orthogonal decomposition; the spatio-temporal matrix X based on the three-dimensional velocity pulsation field is given by the following formula: Among them, is the three-dimensional average velocity field of the unsteady flow of the turbomachinery to be analyzed, is a column vector composed of the average velocity data of all spatial grid points in the x direction, is a column vector composed of the average velocity data of all spatial grid points in the y direction, is a column vector composed of the average velocity data of all spatial grid points in the z direction; Construct a first spatio-temporal matrix X1 based on the three-dimensional velocity field and a second spatio-temporal matrix X2 based on the three-dimensional velocity field according to the three-dimensional flow field data of the unsteady flow of the turbomachine for dynamic mode decomposition; the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field are given by the following formula: ② The proper orthogonal decomposition of the constructed spatio-temporal matrix includes: Perform proper orthogonal decomposition on the spatio-temporal matrix X based on the three-dimensional velocity pulsation field in Step 1 to obtain a proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field, a proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field, and a proper orthogonal mode energy vector λ of the three-dimensional velocity pulsation field; The proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field is represented by the following formula: where φ l is the k-th order proper orthogonal decomposition (POD) spatial vector of the three-dimensional velocity fluctuation field, and l ranges from 1 to N; the k-th order POD spatial vector φ l of the three-dimensional velocity fluctuation field is expressed as follows: where, φ l,x is the l-th order proper orthogonal mode vector of the velocity pulsation in the x direction, and φ l,y is the l-th order proper orthogonal mode vector of the velocity pulsation in the y direction, and φ l,z is the l-th order proper orthogonal mode vector of the velocity pulsation in the z direction; The proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field is represented by the following formula: Among them, the elements in the proper orthogonal decomposition (POD) temporal coefficient matrix A of the three-dimensional velocity fluctuation field represent the modal coefficients of the l-th POD mode in the n-th snapshot, where the subscript n ranges from 1 to N, and the superscript l ranges from 1 to N; The spatio-temporal matrix X based on the three-dimensional velocity pulsation field, the proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field, and the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field satisfy the following relational expression: X = ΦA The proper orthogonal mode energy vector λ of the three-dimensional velocity pulsation field is represented by the following formula: Among them, the l-th element λ in the proper orthogonal decomposition (POD) energy vector λ of the three-dimensional velocity fluctuation field l is the l-th order POD energy, and the value range of l is from 1 to N; the l-th order POD energy λ l is the l-th eigenvalue of the autocovariance matrix C of the spatio-temporal matrix X based on the three-dimensional velocity fluctuation field, and the autocovariance matrix C of the spatio-temporal matrix X based on the three-dimensional velocity fluctuation field is obtained by the following formula: ③ The sorting of the proper orthogonal modes according to the modal energy magnitude includes: Arrange the elements in the proper orthogonal mode energy vector λ of the three-dimensional velocity pulsation field in Step 2 in descending order, that is, satisfy the following relationship: λ1 > λ2 > … > λ N Make corresponding adjustments to the column order of the proper orthogonal mode spatial matrix Φ of the three-dimensional velocity pulsation field and the row order of the proper orthogonal mode time coefficient matrix A of the three-dimensional velocity pulsation field in Step 2 according to the order of the proper orthogonal mode energy; ④ The use of the fast Fourier transform to obtain the main frequency of the proper orthogonal modes includes: Extract the data of the first r1 rows of the proper orthogonal decomposition (POD) time coefficient matrix A of the three-dimensional velocity fluctuation field described in Step 3 as the discrete signal sequence of the time coefficients of the first r1 POD modes, where r1 < N; the discrete signal sequence al of the time coefficients of the l-th POD mode l is represented by the following formula: Processing the time coefficient signal sequence of the first r1 proper orthogonal modes by the fast Fourier transform method, where r1 < N. For the processing result of the time coefficient signal sequence of each order of proper orthogonal modes, the frequency with the highest amplitude is taken as the main frequency of the corresponding order of proper orthogonal modes. Calculate the main frequencies of the first r1 order proper orthogonal modes and discard the frequencies with the same values to form the main frequency vector f of the first r1 order proper orthogonal modes P , with the unit of Hz, expressed as: Among them, the main frequency vector f of the first r1 order proper orthogonal modes P The i-th element f Pi is the i-th main frequency of the first r1 order proper orthogonal modes, and the value range of i is 1 to r2; ⑤ The dynamic mode decomposition of the constructed spatio-temporal matrix includes: Perform dynamic mode decomposition on the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field to obtain the dynamic mode space matrix V of the three-dimensional velocity field, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field; In the dynamic mode decomposition, the relationship between the first spatio-temporal matrix X1 based on the three-dimensional velocity field and the second spatio-temporal matrix X2 based on the three-dimensional velocity field can be represented by a linear dynamical system, as shown in the following formula: where D is the linear coefficient matrix of the linear dynamical system; perform eigenvalue decomposition on the linear coefficient matrix D of the linear dynamical system, as shown in the following formula: D = V·diag(μ1 μ2 … μ l … μ N-1 )·V H where μ l is the l-th order dynamic modal eigenvalue, and the value range of l is 1 to (N - 1); V is the dynamic modal space matrix of the three-dimensional velocity field, which is expressed by the following formula: where, v l is the spatial vector of the k-th order proper orthogonal mode, and the value range of l is from 1 to (N - 1); the spatial vector of the k-th order proper orthogonal mode is expressed by the following formula: where, v l,x is the l-th order dynamic mode vector of the velocity in the x direction, v l,y is the l-th order dynamic mode vector of the velocity in the y direction, v l,z is the l-th order dynamic mode vector of the velocity in the z direction; The dynamic mode eigenvalue vector μ of the three-dimensional velocity field is expressed as: where, μ l is the l-th order dynamic modal eigenvalue, and the value range of l is 1 to (N - 1); the dynamic modal amplitude vector α of the three-dimensional velocity field is obtained by the following formula: Among them, the element α in the dynamic mode amplitude vector α of the three-dimensional velocity field l is the l-th order dynamic mode amplitude, and the value range of l is 1 to (N - 1); ⑥ The obtaining of the sparse enhanced dynamic mode combination by using the sparse enhanced dynamic mode decomposition method includes: Based on the first spatio-temporal matrix X1 of the three-dimensional velocity field described in step 1, the second spatio-temporal matrix X2 of the three-dimensional velocity field, the dynamic mode space matrix V of the three-dimensional velocity field described in step 5, the dynamic mode eigenvalue vector μ of the three-dimensional velocity field, and the dynamic mode amplitude vector α of the three-dimensional velocity field, the sparse enhanced dynamic mode decomposition method is used to screen out the sparse enhanced dynamic mode combination and obtain the sparse enhanced dynamic mode amplitude vector Specifically, the sparse enhanced dynamic mode decomposition method screens out the r1-order dynamic modes through an optimization algorithm, where r1 < N, as the sparse enhanced dynamic mode combination. The optimization problem involved is to minimize the value of the following formula: Among them, is the diagonal matrix of the dynamic modal amplitude with sparse enhancement, is the dynamic modal amplitude with sparse enhancement of the i-th, and the value range of i is 1 to r1; V and is the dynamic modal evolution characteristic matrix, which is expressed by the following formula: ⑦ The calculating of the frequency of the sparse enhanced dynamic mode combination includes: Calculate the frequency vector \(f\) of the sparse-enhanced dynamic mode combination described in step six based on the dynamic mode eigenvalue vector \(\mu\) of the three-dimensional velocity field described in step five D , with the unit of Hz, expressed as: Among them, the i-th element \(f_{ D}\) of the frequency vector \(\mathbf{f}\) of the sparse-enhanced dynamic mode combination D is the i-th order dynamic mode frequency of the sparse-enhanced dynamic mode combination. The value range of i is 1 to r1. The i-th element \(f_{ D}\) of the frequency vector \(\mathbf{f}\) of the sparse-enhanced dynamic mode combination Di is obtained by the following formula: D is the i-th element \(f_{ Di}\) of the frequency vector \(\mathbf{f}\) of the sparse-enhanced dynamic mode combination Di is obtained by the following formula: where Δt is the time interval between adjacent snapshots; ⑧ The obtaining of the optimal dynamic mode combination sorting according to the natural orthogonal mode main frequency sorting includes: According to the main frequency vector f of the first r1 order proper orthogonal modes P , correct the sorting of the elements in the frequency vector f of the sparsity-enhanced dynamic mode combination in step seven D to obtain the frequency vector of the optimal dynamic mode combination Obtain the eigenvalue vector of the optimal dynamic mode combination according to the new sorting and calculate the dynamic mode growth rate vector of the optimal dynamic mode combination Specifically, the frequency vector of the optimal dynamic mode combination is expressed as: Among them, the frequency vector of the optimal dynamic mode combination The i-th element is the frequency of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; the frequency vector of the optimal dynamic mode combination The i-th element is defined as the frequency vector f of the sparse enhanced dynamic mode combination D One of the elements closest to the i-th main frequency f of the first r1 eigenorthogonal modes Pi is represented by the following formula: According to the sorting of the optimal dynamic mode combination, the eigenvalue vector of the optimal dynamic mode combination is obtained It is represented by the following formula: Among them, the eigenvalue vector of the optimal dynamic mode combination The i-th element is the eigenvalue of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; The dynamic mode growth rate vector of the optimal dynamic mode combination is expressed as: Among them, the dynamic mode growth rate vector of the optimal dynamic mode combination The i-th element is the growth rate of the i-th order dynamic mode of the optimal dynamic mode combination, and the value range of i is 1 to r2; the dynamic mode growth rate vector of the optimal dynamic mode combination The i-th element is obtained by the following formula: where i = 1, 2, …, r2.

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