Method for extracting unsteady flow field characteristics of cavitation water jet

Abstract

The invention relates to the technical field of fluid mechanics, in particular to a method for extracting cavitation water jet unsteady flow field characteristics, which comprises the following steps: S1, performing three-dimensional modeling on a nozzle for cavitation water jet calculation and a fluid domain thereof to generate a three-dimensional model; s2, performing grid division on the three-dimensional model, and importing Fluent for numerical simulation to obtain flow field simulation data; s3, importing the flow field time sequence data into Tecplot, and exporting the grid node data into a flow field snapshot sequence in batches; s4, through a Python program, extracting flow field data to form a matrix; s5, POD decomposition is carried out in the compiled Python program; s6, carrying out DMD (Digital Micromirror Device) decomposition in the compiled Python program; s7, the modal data obtained after POD and DMD decomposition are subjected to topological distribution to the grids to generate the spatial modal diagrams. According to the method, the analysis flexibility is improved, and the method is suitable for various scenes.

Description

Technical Field

[0001] The invention relates to the technical field of fluid mechanics, and in particular to a method for extracting unsteady flow field characteristics of cavitation water jet. Background Art

[0002] Cavitation refers to the formation of bubbles or vapor bubbles in a fluid due to a decrease in local pressure. This phenomenon can be applied in the field of cleaning.

[0003] In the study of cavitation jets, the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) methods have played an important role. The POD method analyzes experimental or simulation data, distinguishes the spatiotemporal coherent structure of the flow field, decomposes the time evolution sequence of the flow data into a set of basic spatial modes and time coefficients, and obtains the energy contribution of different modes to the entire flow field, thereby revealing the key factors affecting the cleaning effect and flow behavior. This process helps researchers understand the dominant mechanisms in complex flows and provides guidance for equipment design optimization. On the other hand, the DMD method focuses on analyzing the dynamic behavior of cavitation jets. DMD is more effective than POD in decomposing complex flow fields into uncoupled coherent structures and extracting the dynamic modes and time evolution characteristics of the system. This dynamic analysis capability enables researchers to gain a deep understanding of the instability and transient behavior of the flow, thereby optimizing the application effect of cavitation jets.

[0004] The Chinese invention patent application with application publication number CN118821661A discloses a modal reduction method and system for cavitation flow field of a centrifugal pump based on dynamic modal decomposition, including: S1, three-dimensional modeling of the centrifugal pump, and the flow calculation domain is composed of the impeller, volute, and inlet and outlet sections of the centrifugal pump; S2, according to the design parameters of the centrifugal pump, geometric modeling is performed in Solidworks according to the size and the flow calculation domain modeling is geometrically processed, the geometric model constructed by Solidworks is imported into the ICEM software to mesh the flow calculation domain, and the meshes of the various components of the centrifugal pump are generated in sequence; S3, the divided meshes are imported into the software fluent for numerical simulation to obtain simulation data; S4, the inlet pressure is changed in the software fluent to obtain different cavitation states, and the flow field snapshots at different times are obtained under transient simulation, and exported as Tecplot files. After data processing, DMD modal decomposition is performed in the compiled MATLAB program. The above-mentioned public scheme is of certain significance for quickly identifying cavitation in centrifugal pumps and preventing cavitation hazards caused by vaporization, thereby preventing centrifugal pumps from malfunctioning. However, data processing requires data interpolation in Tecplot, which has high requirements on computing resources and can only process two-dimensional data. The calculation method has limitations and cannot meet diverse computing needs. Therefore, it is very necessary to design an intrinsic orthogonal decomposition and dynamic mode decomposition method that can simultaneously meet two-dimensional and three-dimensional computing needs. Summary of the Invention

[0005] Technical Objective: In order to overcome the deficiencies existing in the prior art, the present invention provides a method for extracting the unsteady flow field characteristics of cavitating water jets.

[0006] Technical Solution: To achieve the above objective, a method for extracting the unsteady flow field characteristics of cavitating water jets disclosed by the present invention includes the following steps:

[0007] S1: Three-dimensionally model the nozzle and its fluid domain for cavitating water jet calculation to generate a three-dimensional model;

[0008] S2: After meshing the three-dimensional model, import it into Fluent for numerical simulation to obtain flow field simulation data;

[0009] S3: Import the flow field time series data in the flow field simulation data into Tecplot, and batch export the grid node data as a sequence of flow field snapshots;

[0010] S4: Identify the coordinate information and flow field data information of the grid node data in the sequence of flow field snapshots through a Python program, and extract the flow field data to form a matrix;

[0011] S5: Perform POD decomposition in the compiled Python program;

[0012] S6: Perform DMD decomposition in the compiled Python program;

[0013] S7: Topologically assign the modal data after POD and DMD decompositions to the grid to generate a spatial modal diagram.

[0014] Further, the POD decomposition includes the following steps:

[0015] S101: Prepare a sequence of flow field snapshots U(x, t) composed of N snapshots. The snapshot sequence U(x, t) includes a mean part and a pulsating part U′(x, t);

[0016] S102: After performing eigenvalue decomposition on the covariance matrix R, solve for the spatial mode φ j , and represent U′(x, t) as a linear combination of the spatial mode φ j . Sort the eigenvalues λ j of the covariance matrix R, select the top k modes with the largest energy contribution for superposition, and obtain a linear reconstruction expression for the flow field after POD decomposition;

[0017] S103: In POD decomposition, POD energy is used to measure the contribution of different modes to the total energy of the system. The POD energy includes the energy E of a single modej and the cumulative energy Ec j , solve for the energy E of a single mode j and the cumulative energy Ec j .

[0018] Furthermore, the expression of the snapshot sequence U(x,t) in step S101 is:

[0019] U(x,t) = [u(x,t 1 ), u(x,t 2 ),..., u(x,t i )..., u(x,t N )],

[0020]

[0021] where u(x,t i ) is the i-th snapshot in the snapshot sequence of the flow field;

[0022] The formula for the mean part is:

[0023]

[0024] The formula for the pulsating part U′(x,t) is:

[0025] U′(x,t) = [u′(x,t 1 ), u′(x,t 2 ),..., u′(x,t i )..., u′(x,t N )].

[0026] Furthermore, the formula for the eigenvalue decomposition of the covariance matrix R in step S102 is:

[0027]

[0028] R = U′(x,t) T U′(x,t),

[0029] where λ j and are the eigenvalues and eigenvectors of the covariance matrix R respectively, and U′(x,t) T is the transpose of U′(x,t);

[0030] The formula for the spatial mode φ j is:

[0031]

[0032] The pulsating part U′(x,t) is expressed as a linear combination of spatial modes φ j , and its expression is:

[0033]

[0034] where a j (t) is the corresponding time coefficient, and its calculation formula is:

[0035]

[0036] The linear reconstruction expression is:

[0037]

[0038] Furthermore, in step S103, the energy E j of the single mode is the contribution percentage of mode j to the total energy, and its calculation formula is:

[0039]

[0040] where λ j is the energy corresponding to mode j, is the sum of all eigenvalues, representing the total energy of the system;

[0041] The cumulative energy Ec j is the percentage of the energy of the first j modes in the total energy of the system, and its calculation formula is:

[0042]

[0043] where, is the sum of the eigenvalues of the first j modes.

[0044] Furthermore, the DMD decomposition includes the following steps:

[0045] S201: Arrange the N snapshot data in step S101 into two matrices, which are respectively defined as X and Y, determine the linear mapping relationship between X and Y, and obtain the matrix

[0046] S202: Perform eigenvalue decomposition on the matrix to obtain the calculation expression of the DMD spatial mode and the growth and decay rates g j and frequencies ω j of each order mode, as well as the time coefficients corresponding to the DMD spatial mode;

[0047] S203: In the DMD decomposition, the modal amplitude b, the norm of the eigenmode, and the time coefficient are combined as criteria to represent the energy contribution of the DMD mode.

[0048] Furthermore, the expressions of matrix X and matrix Y in step S201 are respectively:

[0049] X = [u(x, t 1 ), u(x, t 2 ),..., u(x, t i )..., u(x, t N-1 )],

[0050] Y = [u(x, t 2 ), u(x, t 3 ),..., u(x, t i )..., u(x, t N )],

[0051] where u(x, t i ) is the i-th snapshot in the sequence of flow field snapshots;

[0052] According to the Koopman analysis method, the linear mapping between X and Y is defined as:

[0053] Y = AX,

[0054] A = YX',

[0055] where A is the system matrix, which contains the dynamic evolution information of the flow field, and X' is the pseudo-inverse matrix of X;

[0056] By combining the Koopman theory and the singular value decomposition SVD of the data matrix, we get: X = U∑V * , where V * is the conjugate transpose of V;

[0057] The expression of matrix is:

[0058]

[0059] where U T is the transpose of matrix U.

[0060] Furthermore, the formula for eigenvalue decomposition of matrix in step S202 is:

[0061]

[0062] where γ = [γ 1 , γ 2 ,..., γ j ,..., γ N-1 is the eigenvalue matrix, W is the eigenvector, and γ jis the eigenvalue of the j-th order mode;

[0063] The calculation expression of the DMD spatial mode is:

[0064] ψ = YV∑ -1 W;

[0065] The growth and decay rate g j , the frequency ω j are obtained from the corresponding eigenvalue γ j , and the calculation formula is:

[0066]

[0067] where Δt represents the time interval between snapshots;

[0068] The time coefficients corresponding to the DMD spatial mode are included in the Vandermonde matrix V and , and the expression of the matrix V and is:

[0069]

[0070] Furthermore, the expression of the modal amplitude b in the step S203 is:

[0071] b = ψ -1 u 1 , b = [b 1 ,..., b N T ,

[0072] The expression taking the comprehensive modal amplitude b, the norm of the eigenmode, and the time coefficient as criteria is:

[0073]

[0074] where || || 2 is the L 2 norm of the vector, b j is the j-th modal amplitude b j , which is used to represent the contribution of the initial snapshot.

[0075] The beneficial effects of the present invention are:

[0076] ​The present invention uses the proper orthogonal decomposition and dynamic mode decomposition methods to decompose and reconstruct the grid data snapshots, extract the flow field mode information, extract the main features from the snapshot data, reduce the data dimension, and perform static and dynamic analysis of the system, providing an effective tool for pattern recognition and prediction and strong support for data analysis and understanding. The grid data can be freely exported as two-dimensional or three-dimensional snapshots. The original grid flow field data can be recognized and extracted through a Python program and exported as a snapshot matrix, improving the flexibility of analysis and being applicable to various scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] Figure 1 is the overall method flow chart of the present invention;

[0078] Figure 2 is a schematic diagram of the nozzle jet fluid domain modeling and boundary conditions in the present invention;

[0079] Figure 3 is the POD modal energy and energy accumulation distribution diagram based on steam in the present invention;

[0080] Figure 4 is the schematic diagram of the POD spatial mode based on steam in the present invention;

[0081] Figure 5 is the DMD modal eigenvalue distribution diagram based on steam in the present invention;

[0082] Figure 6 is the DMD modal growth / decay rate frequency distribution diagram based on steam in the present invention;

[0083] Figure 7 is the schematic diagram of the DMD spatial mode based on steam in the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0084] The principles and features of the present invention will be described below in conjunction with the attached Figure 1 to the attached Figure 7 The examples given are only used to explain the present invention and are not intended to limit the scope of the present invention.

[0085] A method for extracting the unsteady flow field characteristics of a cavitating water jet, as Figure 1 - Figure 2 shown, a method for extracting the unsteady flow field characteristics of a cavitating water jet based on proper orthogonal decomposition and dynamic mode decomposition, comprising the following steps:

[0086] S1: Perform three-dimensional modeling on the nozzle and its fluid domain used for cavitating water jet calculation to generate a three-dimensional model;

[0087] Specifically, before performing three-dimensional modeling, the size data of the nozzle and the size of the fluid domain need to be selected, and three-dimensional modeling is performed through the CATIA modeling software.

[0088] S2: After meshing the three-dimensional model, import it into Fluent for numerical simulation to obtain flow field simulation data;

[0089] Specifically, mesh the three-dimensional model for subsequent calculations. In this embodiment, the meshing software is selected as ICEM, and hexahedral structured meshes are used for simulation, which can effectively capture the cavitation vortex structure. Import the meshed grids into the CFD calculation software Fluent for numerical simulation, set appropriate boundary conditions, physical models, physical time, etc. The calculation settings in this embodiment are as Figure 2 shown, and a calculation result with good convergence can be obtained. Export the required flow field data to be decomposed, such as steam volume fraction, velocity, etc. In this embodiment, the flow field steam volume fraction is selected as the exported data.

[0090] S3: Import the flow field time series data in the flow field simulation data into Tecplot, and batch export the grid node data as a sequence of flow field snapshots;

[0091] Specifically, in this embodiment, import the flow field steam volume fraction data into Tecplot for data processing, and export the original grid data as two-dimensional or three-dimensional data snapshots according to the time series.

[0092] S4: Identify the coordinate information of the grid node data and the flow field data information in the flow field snapshot sequence through a Python program, and extract the flow field data to form a matrix. This step is a well-known operation for those skilled in the art and will not be elaborated here.

[0093] S5: Perform POD decomposition in the compiled Python program;

[0094] Furthermore, use the flow field snapshot sequence in step S3 as the snapshot source, and perform proper orthogonal decomposition (POD decomposition) in the program compiled in Python. In this embodiment, take the pulsating part of the steam field as an example for POD decomposition. The POD decomposition includes the following steps:

[0095] S101: Prepare a steam field snapshot sequence U(x,t) consisting of N snapshots. The snapshot sequence U(x,t) includes the mean part and the pulsating part U′(x,t); Considering both calculation accuracy and calculation cost, 650 snapshots are used as the snapshot source in this embodiment.

[0096] Specifically, the expression of the snapshot sequence U(x,t) is:

[0097] U(x,t) = [u(x,t 1 ),u(x,t 2 ),...,u(x,t i )...,u(x,tN )],

[0098]

[0099] where \(u(x,t\) i ) is the \(i\)-th snapshot in the sequence of flow field snapshots;

[0100] The mean part is calculated as follows:

[0101]

[0102] The formula for calculating the fluctuating part \(U′(x,t)\) is:

[0103] U′(x,t) = [u′(x,t 1 ), u′(x,t 2 ),..., u′(x,t i ),..., u′(x,t N )].

[0104] S102: Solve for the spatial mode \(\varphi\) after performing eigenvalue decomposition on the covariance matrix \(R\) j , and represent \(U′(x,t)\) as a linear combination of the spatial mode \(\varphi\) j . Sort the eigenvalues \(\lambda\) j of the covariance matrix \(R\), select the top \(k\) modes with the largest energy contribution for superposition, and obtain the linear reconstruction expression of the flow field after POD decomposition;

[0105] Specifically, represent the fluctuating part \(U′(x,t)\) as a linear combination of the spatial mode \(\varphi\) j , and its expression is:

[0106]

[0107] where \(a\) j \((t)\) is the corresponding time coefficient, and its calculation formula is:

[0108] a j (t i ) = \(\langle u′(x,t i ), \varphi j \rangle\);

[0109] In the POD decomposition method, to solve for the spatial mode \(\varphi\) j , first perform eigenvalue decomposition on the covariance matrix \(R\). The formula for performing eigenvalue decomposition on the covariance matrix \(R\) is:

[0110]

[0111] R = U′(x,t) TU′(x,t),

[0112] where λ j and are the eigenvalue and eigenvector of the covariance matrix R respectively, and U′(x,t) T is the transpose of U′(x,t);

[0113] The spatial mode φ j is calculated as follows:

[0114]

[0115] In addition, the eigenvalue λ j reflects the energy contribution of the corresponding mode to the original flow field. By sorting λ j and selecting the top k modes with the largest energy contributions for superposition, the linear reconstruction expression for the flow field after POD decomposition is obtained as:

[0116]

[0117] S103: In POD decomposition, the POD energy is a key indicator for measuring the contribution of different modes to the total energy of the system. The POD energy includes the energy E j of a single mode and the cumulative energy Ec j . Solve the energy E j of a single mode and the cumulative energy Ec j .

[0118] The energy E j of a single mode is the percentage of the contribution of mode j to the total energy, and its calculation formula is:

[0119]

[0120] where λ j is the energy corresponding to mode j, is the sum of all eigenvalues, representing the total energy of the system;

[0121] The cumulative energy Ec j is the percentage of the energy of the first j modes in the total energy of the system, and its calculation formula is:

[0122]

[0123] where is the sum of the eigenvalues of the first j modes;

[0124] S6: Perform DMD decomposition in the compiled Python program;

[0125] The DMD decomposition includes the following steps:

[0126] S201: Arrange the N snapshot data in step S101 into two matrices, which are respectively defined as X and Y, determine the linear mapping relationship between X and Y, and obtain the matrix

[0127] Specifically, similar to POD, the process of DMD decomposition requires snapshot data as input. Using the same snapshot source as POD decomposition, the dynamic mode decomposition, i.e., DMD decomposition, is performed by a program compiled in Python, and the snapshot data is arranged into the following two matrices X and Y.

[0128] The expressions of matrix X and matrix Y are respectively:

[0129] X = [u(x, t 1 ), u(x, t 2 ),..., u(x, t i )..., u(x, t N-1 )],

[0130] Y = [u(x, t 2 ), u(x, t 3 ),..., u(x, t i )..., u(x, t N )],

[0131] where u(x, t i ) is the i-th snapshot in the flow field snapshot sequence;

[0132] According to the Koopman analysis method, the linear mapping between X and Y is defined as:

[0133] Y = AX,

[0134] A = YX′,

[0135] where A is the system matrix, which contains the dynamic evolution information of the flow field, and X′ is the pseudo-inverse matrix of X; given the matrix dimensions, directly calculating matrix A is inefficient. Therefore, DMD decomposition attempts to use a specific numerical method to approximate the Koopman decomposition method, which is achieved by combining the Koopman theory and the singular value decomposition SVD of the data matrix:

[0136] X = U∑V * ,

[0137] where V * is the conjugate transpose of V;

[0138] Through the above matrices, the expression of matrix can be obtained, and the expression of matrix is:

[0139]

[0140] Among them, U T is the transpose of matrix U.

[0141] S202: Perform eigenvalue decomposition on the matrix to obtain the calculation expression of the DMD spatial mode and the growth and decay rates g j , frequencies ω j and the time coefficients corresponding to the DMD spatial modes;

[0142] The formula for eigenvalue decomposition of the matrix is:

[0143]

[0144] Among them, γ = [γ 1 , γ 2 ,..., γ j ,..., γ N-1 is the eigenvalue matrix, W is the eigenvector, and γ j is the eigenvalue of the j-th mode;

[0145] The calculation expression of the DMD spatial mode is:

[0146] ψ = YV∑ -1 W;

[0147] The growth and decay rates g j , frequencies ω j of each stage are obtained from the corresponding eigenvalue γ j , and the calculation formula is:

[0148]

[0149] Among them, Δt represents the time interval between snapshots;

[0150] The time coefficients corresponding to the DMD spatial modes are included in the Vandermonde matrix V and , and the expression of the matrix V and is:

[0151]

[0152] S203: In the DMD decomposition, the combined modal amplitude b, the norm of the eigenmode, and the time coefficient are used as criteria to represent the energy contribution of the DMD mode;

[0153] Different from POD decomposition, there is no clear criterion for selecting dominant modes in DMD decomposition. In the standard DMD method, the modal amplitude b can be calculated to represent the energy contribution of DMD modes. However, the evaluation criterion of the modal amplitude b cannot cover the entire evolution process of the snapshot sequence. The norm of the eigenmode can also be used to determine the reference of energy contribution.

[0154] The expression of the modal amplitude b is:

[0155] b = ψ -1 u 1 , b = [b 1 ,..., b N T ,

[0156] The expression combining the modal amplitude b, the norm of the eigenmode, and the temporal coefficient as a criterion is:

[0157]

[0158] where, || || 2 is the L 2 norm of the vector, b j is the j-th modal amplitude b j , which is used to represent the contribution of the initial snapshot.

[0159] S7: Topologically assign the modal data after POD and DMD decompositions to the grid to generate a spatial modal map.

[0160] Figure 3 is the distribution of the energy contribution E j and the cumulative energy Ec j for the steam-based POD modes, where E j is represented by black dots, and is represented by red dots. As the number of modes increases, the contribution of a single mode gradually decreases and finally approaches zero. The energy contribution of mode 1 is the highest, at 13.3%. From mode 1 to mode 6, the decline trend of E j is almost in a linear pattern. After mode 6, the decline rate of E j slows down. The first 8 modes contribute a total of 48.2% of the total energy and can better reflect the fluctuation characteristics of the flow field.

[0161] Figure 4 are the turbulent coherent structures obtained by extracting spatial modal data through POD decomposition based on the steam field. The high-energy coherent structures in red and blue reflect the pulsation of the flow field, including the growth, contraction, shedding, and expansion of the cavity structure.

[0162] Figure 5 ​It is the eigenvalue distribution of the steam-based DMD in the complex plane. The black circles represent the unit circle. Most eigenvalues are located around the unit circle. The points inside the unit circle correspond to the convergent modes with decay rates, the points outside the unit circle correspond to the divergent modes with growth rates, and the points on the unit circle correspond to the stable periodic modes. The distribution of eigenvalues indicates the overall stability of the flow field.

[0163] Figure 6 Shows the growth / decay rate g j and frequency, which are calculated by g j =(1 / Δt)Re(log(γ j )) and Im(log(γ j )) / (2πΔt) respectively. The positive value of g j indicates growth, and the negative value indicates decay. The eigenvalues appear in the form of conjugate complexes, showing a symmetric spectrum in frequency. Note that each DMD mode corresponds to a single frequency. It is found that the unstable modes mainly appear in the high-frequency band, indicating the rapid growth and disappearance of small-scale pulsations. Most modes have a decay trend, ensuring the convergence of the flow system.

[0164] Figure 7 Are the coherent structures and their corresponding frequencies obtained by the DMD modal decomposition of the first 8 steam fields, where the spatial structures are visualized through the real part of the modes.

[0165] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for extracting unsteady flow field characteristics of cavitation water jet, characterized by: The following steps are involved: S1: Generate a 3D model by performing 3D modeling on the nozzle and its fluid domain used for cavitation water jet calculation; S2: meshing the three-dimensional model and importing it into Fluent for numerical simulation to obtain flow field simulation data; S3: importing the flow field time series data in the flow field simulation data into Tecplot, and exporting the grid node data in batches as a flow field snapshot sequence; S4: identifying the coordinate information of the grid node data and the flow field data information in the flow field snapshot sequence through a Python program, and extracting the flow field data composition matrix; S5: POD decomposition in compiled Python programs; S6: DMD decomposition in compiled Python program; S7: The modal data after POD and DMD decomposition are topologically assigned to the grid to generate the spatial modal diagram.

2. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 1, characterized in that: The POD decomposition comprises the following steps: S101: Prepare a flow field snapshot sequence U(x, t) consisting of N snapshots. The snapshot sequence U(x, t) includes a mean value part. and the pulsating part U′(x,t); S102: Perform eigenvalue decomposition on the covariance matrix R and solve the spatial mode φ j , and represent U′(x,t) as the spatial mode φ j The linear combination of the covariance matrix R has the eigenvalue λ j Sorting is performed, and the first k modes with the largest energy contribution are selected for superposition to obtain the linear reconstruction expression of the flow field after POD decomposition; S103: In POD decomposition, POD energy is used to measure the contribution of different modes to the overall energy of the system. The POD energy includes the energy E of a single mode. j and the accumulated energy Ec j , solve for the energy E of a single mode j and the accumulated energy Ec j .

3. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 2, characterized in that: The expression of the snapshot sequence U(x, t) in step S101 is: U(x,t)=[u(x,t1),u(x,t2),...,u(x,t i )...,u(x,t N )], Among them, u(x,t i ) is the i-th snapshot in the flow field snapshot sequence; The mean portion The calculation formula is: The calculation formula of the pulsating part U′(x, t) is: U′(x,t)=[u′(x,t1),u′(x,t2),...,u′(x,t i )...,u′(x,t N )]。 4. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 3, characterized in that: The formula for performing eigenvalue decomposition on the covariance matrix R in step S102 is: R=U′(x,t) T U′(x,t), Among them, λ j and are the eigenvalues ​​and eigenvectors of the covariance matrix R, U′(x,t) T is the transpose of U′(x,t); The spatial mode φ j The calculation formula is: The pulsating part U′(x, t) is expressed as the spatial mode φ j The linear combination of is expressed as: Among them, a j (t) is the corresponding time coefficient, and its calculation formula is: a j (t i )= <u′(x,t i ),f j >; The linear reconstruction expression is:

5. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 4, characterized in that: The energy E of the single mode in step S103 j is the contribution percentage of mode j to the total energy, and its calculation formula is: Among them, λ j is the energy corresponding to mode j, is the sum of all eigenvalues, representing the total energy of the system; The accumulated energy Ec j is the percentage of the energy of the first j modes to the total energy of the system, and its calculation formula is: in, is the sum of the eigenvalues ​​of the j previous modes.

6. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 2, characterized in that: The DMD decomposition comprises the following steps: S201: Arrange the N snapshot data in step S101 into two matrices, the two matrices are defined as X and Y respectively, determine the linear mapping relationship between X and Y, and obtain the matrix S202: the matrix Perform eigendecomposition to obtain the calculation expression of DMD spatial mode and the growth and decay rate g of each order mode j , frequency ω j and the time coefficient corresponding to the DMD spatial mode; S203: In DMD decomposition, the comprehensive modal amplitude b, the norm of the eigenmode and the time coefficient are used as criteria to represent the energy contribution of the DMD mode.

7. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 6, characterized in that: The expressions of the matrix X and the matrix Y in step S201 are respectively: X=[u(x,t1),u(x,t2),...,u(x,t i )...,u(x,t N-1 )], Y=[u(x,t2),u(x,t3),...,u(x,t i )...,u(x,t N )], Among them, u(x,t i ) is the i-th snapshot in the flow field snapshot sequence; According to the Koopman analysis method, the linear mapping between X and Y is defined as: Y=AX, A=YX′, Among them, A is the system matrix, which contains the dynamic evolution information of the flow field, and X′ is the pseudo-inverse matrix of X; By combining Koopman theory and the singular value decomposition SVD of the data matrix: X = U∑V * , where V * is the conjugate transpose of V; matrix The expression is: Among them, U T is the transpose of the matrix U.

8. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 7, characterized in that: In step S202, the matrix The formula for eigenvalue decomposition is: Where γ=[γ1,γ2,...,γ j ,...,γ N-1 ] is the eigenvalue matrix, W is the eigenvector, γ j is the eigenvalue of the j-th mode; The calculation expression of the DMD spatial mode is: ψ=YV∑ -1 W; The growth and decay rates g j , frequency ω j By the corresponding eigenvalue γ j The calculation formula is: Where Δt represents the time interval between snapshots; The time coefficients corresponding to the DMD spatial modes are contained in the Vandermonde matrix V and In the matrix V and The expression is:

9. The method for extracting unsteady flow field characteristics of cavitation water jet according to claim 8, characterized in that: The expression of the modal amplitude b in step S203 is: b=ψ -1 u1,b=[b1,...,b N ] T , The expression of the comprehensive modal amplitude b, the norm of the eigenmode and the time coefficient as the criterion is: Among them, || ||2 is the L of the vector 2 norm, b j is the jth mode amplitude b j , which is used to represent the contribution of the initial snapshot.

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