A method for analyzing spatial energy evolution process of a multi-vortex system

Abstract

The application discloses a kind of multi-vortex system space energy evolution process analysis method, belong to the field of fluid mechanics and aerodynamics, it is in the purpose to solve the energy analysis error problem caused by integral radius dependence, single vortex hypothesis limitation in traditional method. The vortex core is located by Liutex method in the present application, and the vortex to be analyzed is translated to the origin, and the pseudo time sequence is generated in combination with the evolution direction slice, and the first modal energy distribution and evolution law are extracted using intrinsic orthogonal modal decomposition. The present application does not need to manually set the integral radius, and has strong adaptability, can comprehensively reflect the spatial variation process of the whole energy of the vortex, especially suitable for energy evolution analysis in the scene of complex vortex system such as bird flight, complex aircraft wake, etc. It provides a new perspective for vortex energy evolution, and has certain engineering practical value. The method of the present application can be used for quantitative analysis of complex vortex field scenes such as aircraft design, bionics, wind energy engineering, etc. It has important engineering application value.

Description

Technical Field

[0001] This application relates to the fields of fluid mechanics and aerodynamics, specifically entitled "A Method for Analyzing the Spatial Energy Evolution Process of a Multi-Vortex System." More specifically, this application provides an analytical method for the spatial energy evolution process of a multi-vortex system, which can be used for quantitative analysis of complex vortex field scenarios such as aircraft design, biomimetic studies (e.g., vortex system research on bird flight), and wind energy engineering. Background Technology

[0002] In aerodynamic studies, the multi-scale vortex structures formed by wingtip and feather movements during bird flight are complex, and their energy evolution directly affects flight efficiency and stability. Currently, traditional analysis methods are mainly based on single-vortex models, characterizing energy distribution through peak vortex intensity or circulation integral.

[0003] However, traditional analytical methods for analyzing the spatial energy evolution process of multi-vortex systems have the following drawbacks:

[0004] (1) Limitations of the single-vortex system assumption

[0005] The wake generated by bird flight typically contains multiple interacting vortex structures (such as initiation vortex, shedding vortex, etc.), and traditional single-vortex models cannot resolve the multi-vortex coupling effect.

[0006] (2) Sensitivity of circulation integral radius

[0007] The circulation calculation requires a pre-set integration radius. If the radius is too large, it is easy to include the interference of neighboring vortices, while if the radius is too small, the complete energy information of the target vortex will be missed, resulting in deviation of the results.

[0008] (3) Lack of energy distribution information

[0009] Peak vorticity only reflects the local maximum value and cannot characterize the spatial uniformity and diffusion law of vortex energy, making it difficult to support the dynamic evolution analysis of multi-vortex systems.

[0010] In existing technologies, energy analysis of multi-vortex systems largely relies on empirical parameters or simplified models, lacking adaptability. This makes it difficult to accurately quantify the energy evolution process of vortex systems in scenarios such as bird biomimicry and wind turbine wake optimization. Therefore, there is an urgent need for an analytical method that can adaptively separate multi-vortex interference and comprehensively reflect the spatial distribution characteristics of energy. Summary of the Invention

[0011] The purpose of this invention is to address the energy analysis errors caused by the dependence on the integral radius and the limitations of the single-vortex assumption in traditional methods, and to provide an adaptive method for energy evolution analysis of multi-vortex systems. This invention effectively solves the problems of integral radius dependence and the limitations of the single-vortex assumption in traditional energy analysis through adaptive coordinate transformation and mode decomposition.

[0012] To achieve the above objectives, this application adopts the following technical solution.

[0013] A method for analyzing the spatial energy evolution process of a multi-vortex system includes the following steps:

[0014] S1. Identify the vortex region

[0015] Calculate the velocity gradient tensor based on the existing velocity field data. With vorticity It is used to identify the vortex distribution in the flow field; based on the calculated velocity gradient tensor Solving for the Liutex vector yields the Liutex vector. ;

[0016] S2, Extract the vortex core line

[0017] Extracting the core lines of the vortex to be analyzed ;

[0018] S3, Pseudo-time series construction

[0019] The following steps are used to construct a pseudo-time series:

[0020] S3.1 Evolutionary Direction Slicing

[0021] For the vortex to be analyzed, the Liutex vector of its vortex core line is... The direction is denoted as Along the vortex core line, according to the vortex direction equidistant interception A slice in a flow field Each slice contains a vortex energy field ; i For natural numbers, i ≥1 and i Less than or equal to N ;

[0022] S3.2 Slice translation and rotation

[0023] Based on the vortex core line obtained in step S2 Determine the slice in the flow field vortex core center coordinates ; for slices Perform coordinate transformation to obtain the coordinates after slice transformation. The transformation formula is as follows:

[0024] Formula (8);

[0025] S3.3 One-dimensional data stretching

[0026] slice vortex energy field Expand into row vectors according to grid point order. The calculation formula is as follows:

[0027] Formula (9);

[0028] in, ; , , Slices at , , Number of grid nodes in the direction;

[0029] S3.4 Data Matrix Construction

[0030] Arrange all slice vectors sequentially along the vortex evolution direction to construct a pseudo-time data matrix. , ;in, The number of slices; the vortex evolution direction is the flow direction or the time axis direction;

[0031] S4. Intrinsic Orthogonal Mode Decomposition

[0032] S4.1 Evolutionary Direction Slicing

[0033] Calculate the data matrix covariance matrix , ;

[0034] For covariance matrix The eigenvalue decomposition is performed using the following formula:

[0035] Formula (12);

[0036] in, The eigenvalue matrix is ​​arranged in descending order; For each column of the eigenvector matrix, ... The time evolution coefficients characterizing a mode;

[0037] S4.2 Spatial Modal Reconstruction

[0038] Spatial modes are reconstructed through linear combinations of eigenvectors and data matrices. , ;

[0039] Each column Let represent the spatial energy distribution of the k-th mode, where k is a natural number, k≥1 and k≤N; The following orthogonality conditions must be met:

[0040] Formula (14);

[0041] S4.3 Dominant Energy Mode Extraction

[0042] By using intrinsic orthogonal mode decomposition, the energy contribution rate of each mode in different slices of the original vortex evolution direction is obtained; among which, the energy contribution rate of the k-th order mode is... The calculation formula is as follows:

[0043] Formula (16);

[0044] Among them, the first mode Dominant, its spatial distribution This reflects the energy density characteristics of the vortex to be analyzed.

[0045] In step S1, the velocity gradient tensor is calculated based on the existing velocity field data. With vorticity ;

[0046] Existing velocity field data includes ;in, These correspond to three coordinates in mutually orthogonal directions in space. Corresponding to Velocity in three directions;

[0047] velocity gradient tensor The calculation formula is as follows:

[0048] Formula (1);

[0049] vorticity The calculation formula is as follows:

[0050] Formula (2);

[0051] In the formula, It represents the partial differentials in the x, y, and z directions.

[0052] Based on the calculated velocity gradient tensor Solving for the Liutex vector yields the Liutex vector. The specific process is as follows: For the velocity gradient tensor Solve its characteristic equation; when the eigenvalue is a single real root and a pair of conjugate complex roots, a vortex is considered to exist at that location; when the eigenvalue has conjugate complex roots, solve for the eigenvalue and eigenvector, and record the real eigenvalue. The corresponding feature unit vector is The real and imaginary parts of the conjugate complex roots are denoted as , ... and The Liutex vector is obtained by solving using the following formula. :

[0053] Formula (3);

[0054] In the formula, for the Liutex vector The magnitude of the vector This represents the rotational intensity of a local fluid element; for the Liutex vector The direction of the vector The axis of rotation of the local fluid element is represented by the real eigenvalue corresponding to the real eigenvalue unit vector.

[0055] In step S2, based on the obtained Liutex vector Combined with adaptive integration factor Along the corrected vector field Extracting vortex core lines .

[0056] In step S2, the vortex core lines of the vortex to be analyzed are extracted. The specific operation is as follows: The vortex core line is obtained by integrating the vortex core line, using any point of the vortex to be analyzed as the seed point. The integration formula is as follows:

[0057] Formula (4);

[0058] In the formula, This is the integral field quantity of the vortex core line. The overline indicates that the vector is normalized, serving as an adaptive factor. The direction of the local rotation axis; Represents perpendicular to the local axis of rotation The rotational intensity gradient component;

[0059] The calculation formula is as follows:

[0060] Formula (5);

[0061] Adaptive factor By rotating along the direction of the projection gradient The step size is obtained by performing a one-dimensional line search. The following strong Wolfe condition must be met:

[0062] Formula (6);

[0063] In the formula, This represents the position of the trajectory line at the k-th integration step; This indicates the direction of gradient increase. Since the vortex core has its maximum value within a single vortex, it is necessary to ensure... ; , To set a constant;

[0064] The step size is obtained through a one-dimensional search. Then, the adaptive factor is obtained according to formula (7). :

[0065] Formula (7);

[0066] In the formula, It is a constant greater than 1; This indicates the gradient along the projection from the current integration node. The maximum distance from the direction to the next grid face.

[0067] To address the problems of traditional single-vortex analysis methods (such as peak vortex intensity and circulation integral) in multi-vortex systems, such as difficulty in defining the integral radius and inability to reflect the uniformity of energy distribution, the inventors have adopted a solution based on coordinate transformation and mode decomposition to provide a method for analyzing the spatial energy evolution process of multi-vortex systems.

[0068] This application uses the Liutex method to locate the vortex core, translates the vortex to be analyzed to the origin, generates a pseudo-time series by combining evolution direction slices, and extracts the energy distribution and evolution law of the first mode using intrinsic orthogonal mode decomposition (POD). Specifically, in this application, the velocity gradient tensor is first calculated based on the velocity field data. With vorticity First, it is used to identify the vortex distribution in the flow field; second, based on the obtained Liutex vector... Combined with adaptive integration factor Along the corrected vector field Extracting vortex core lines ; Again, along the vortex core line Equidistant intercept Two-dimensional slices were created, and the origin of each slice was shifted to the center of the vortex core to eliminate interference from other vortices; then, the energy field of the slices was... Stretching to a one-dimensional vector and arranging it in spatial order forms a matrix. Finally, the covariance matrix is ​​calculated. Reconstructing spatial modes after eigenvalue decomposition Extract the first mode This completes the spatial energy evolution of the multi-vortex system.

[0069] This invention eliminates the need for manual setting of the integration radius, exhibits strong adaptability, and can comprehensively reflect the energy spatial change process across the entire vortex domain. It is particularly suitable for energy evolution analysis in complex vortex scenarios such as bird flight and complex aircraft wakes, providing a new perspective on vortex energy evolution and possessing certain engineering practical value. Attached Figure Description

[0070] The present invention will be described by way of example and with reference to the accompanying drawings, wherein:

[0071] Figure 1 This is an example image of a typical flow field vortex structure in Example 1.

[0072] Figure 2 This is an example diagram showing a series of slices taken from a typical flow field in Example 1.

[0073] Figure 3 This is an example diagram of coordinate transformation of a series of slices according to the vortex to be analyzed in Example 1.

[0074] Figure 4 This is a comparison diagram of the original vortex energy distribution of a slice in Example 1 and the first mode energy distribution after intrinsic orthogonal decomposition and reconstruction. Figure 4 (a) represents the original flow field. Figure 4 (b) is the first mode after reconstruction.

[0075] Figure 5 This is a diagram showing the variation of the dominant mode energy along the flow direction in a typical vortex flow field in Example 1 (the flow direction here refers to the vortex evolution direction). Detailed Implementation

[0076] All features disclosed in this specification, or steps in all methods or processes disclosed herein, may be combined in any way, except for mutually exclusive features and / or steps.

[0077] Any feature disclosed in this specification, unless otherwise stated, may be replaced by other equivalent or similar features. That is, unless otherwise stated, each feature is merely one example of a series of equivalent or similar features.

[0078] Example 1

[0079] This invention provides a method for analyzing the spatial energy evolution process of a multi-vortex system, which includes the following steps.

[0080] S1. Identify the vortex region

[0081] For existing flow field data, the field quantities used for subsequent vortex core localization and vortex spatial energy analysis are calculated based on the velocity field. Specifically, this involves the following steps. The existing flow field data refers to experimental measurement data or data obtained through experimental calculations, and must include at least three directions (x, y, z) describing the spatial position, and the corresponding velocities in those three directions. .

[0082] S1.1 Calculate the velocity gradient tensor based on the velocity field data. (Formula 1) and vorticity (Formula 2) is used to identify the vortex distribution in the flow field.

[0083] Among them, the velocity gradient tensor The calculation formula is as follows:

[0084] Formula (1);

[0085] vorticity The calculation formula is as follows:

[0086] Formula (2);

[0087] In the formula, The three components representing the velocity field, It represents the partial differentials in the x, y, and z directions.

[0088] S1.2 Calculation of Vortex Energy Field

[0089] For the velocity gradient tensor Solve its characteristic equation; when the eigenvalue is a single real root and a pair of conjugate complex roots, according to the critical point theory of the flow field topology, it is assumed that a vortex exists at this location; when the eigenvalue has conjugate complex roots, solve for the eigenvalue and eigenvector, and record the real eigenvalue. The corresponding feature unit vector is The real and imaginary parts of the conjugate complex roots are denoted as , ... and Based on the equivalent formula of the third-generation eddy current identification method, the Liutex vector is obtained. The calculation formula is as follows:

[0090] Formula (3);

[0091] In the formula, for the Liutex vector The magnitude of the vector This represents the rotational intensity of a local fluid element; for the Liutex vector The direction of the vector The axis of rotation of a local fluid element is represented by the real eigenvalue corresponding to its real eigenvalue. A typical flow field vortex, through the isosurface generated by the Liutex intensity, is shown below. Figure 1 As shown.

[0092] S2, Extract the vortex core line

[0093] Based on a method for extracting vortex core lines within a Lagrange framework previously proposed by the inventors (Application No.: CN202311250711.X, Publication No.: CN117291117A, Publication Date: 2023-12-26), the vortex core lines to be analyzed are extracted. .

[0094] In a specific example, the vortex core line is extracted according to formulas (4)-(7). The specific steps are as follows.

[0095] The vortex core line is obtained by integrating from any point of the vortex to be analyzed, using the arbitrary point of the vortex as the seed point. The integration formula is as follows:

[0096] Formula (4);

[0097] In the formula, This is the integral field quantity of the vortex core line. The overline indicates that the vector is normalized, serving as an adaptive factor. The direction of the local rotation axis; Represents perpendicular to the local axis of rotation The rotational intensity gradient component, also known as the projected gradient, The calculation formula is as follows:

[0098] Formula (5);

[0099] Adaptive factor By rotating along the direction of the projection gradient The step size is obtained by performing a one-dimensional line search. The following strong Wolfe condition must be met:

[0100] Formula (6);

[0101] In the formula, This represents the position of the trajectory line at the k-th integration step; This indicates the direction of gradient increase. Since the vortex core has its maximum value within a single vortex, it is necessary to ensure... ;in, , To set constants, in this method, , .

[0102] The step size is obtained through a one-dimensional search. Then, the adaptive factor is obtained according to formula (7). :

[0103] Formula (7);

[0104] In the formula, It is a constant greater than 1, and is taken as 1.1 here; This indicates the gradient along the projection from the current integration node. The maximum distance from the direction to the next grid face.

[0105] S3, Pseudo-time series construction

[0106] S3.1 Evolutionary Direction Slicing

[0107] For the vortex to be analyzed, the Liutex vector of its vortex core line is... The direction is denoted as Along the vortex core line, according to the vortex direction equidistant interception A slice in a flow field Each slice contains a vortex energy field ,like Figure 2 As shown. i For natural numbers, i ≥1 and i Less than or equal to N .

[0108] S3.2 Slice translation and rotation

[0109] Based on the vortex core line obtained in step S2 Determine the slice in the flow field vortex core center coordinates , for slices Perform a coordinate transformation to align the slice center with the vortex core center, resulting in the transformed coordinates of the slice. The transformation formula is as follows:

[0110] Formula (8);

[0111] Figure 3 An example diagram of the vortex under analysis after coordinate transformation is provided. This transformation ensures that the energy field of the vortex under analysis remains at the center of the slice, thus maintaining the spatial structure of the vortex while diluting the spatial interference from other vortex structures.

[0112] S3.3 One-dimensional data stretching

[0113] slice vortex energy field Expand into row vectors according to grid point order. The calculation formula is as follows:

[0114] Formula (9);

[0115] in, ; , , Slices at , , The number of grid nodes in the direction.

[0116] S3.4 Data Matrix Construction

[0117] Arrange all slice vectors sequentially along the vortex evolution direction to construct a pseudo-time data matrix. , The vortex evolution direction can be either the flow direction or the time axis direction. The specific calculation formula is as follows:

[0118] Formula (10);

[0119] in, The matrix represents the number of slices; each column corresponds to the global energy distribution of a slice, and each row represents the dynamic change of the energy of a single grid point with the direction of evolution.

[0120] S4. Intrinsic Orthogonal Mode Decomposition (POD)

[0121] S4.1 Evolutionary Direction Slicing

[0122] Based on data matrix Calculate its covariance matrix , The covariance matrix characterizes the energy correlation between slices. The specific calculation formula is as follows:

[0123] Formula (11);

[0124] right The eigenvalue decomposition is performed using the following formula:

[0125] Formula (12);

[0126] in, The eigenvalue matrix is ​​arranged in descending order. ; For each column of the eigenvector matrix, ... The time evolution coefficients characterize a mode.

[0127] S4.2 Spatial Modal Reconstruction

[0128] Spatial modes are reconstructed through linear combinations of eigenvectors and data matrices. , Reconstructing spatial modes The specific calculation formula is as follows:

[0129] Formula (13);

[0130] Each column Let represent the spatial energy distribution of the k-th mode, where k is a natural number, k≥1 and k≤N; The following orthogonality conditions must be met:

[0131] Formula (14).

[0132] For example, the reconstruction space modal formula for the first mode is as follows:

[0133] Formula (15).

[0134] A schematic diagram of the reconstructed spatial modes is shown below. Figure 4 As shown; where, Figure 4 (a) represents the original flow field. Figure 4 (b) is the first mode after reconstruction.

[0135] S4.3 Dominant Energy Mode Extraction

[0136] By using intrinsic orthogonal mode decomposition, the energy contribution rate of each mode in different slices of the original vortex evolution direction can be obtained. Among them, the energy contribution rate of the k-th order mode... The calculation formula is as follows:

[0137] Formula (16).

[0138] First mode It usually dominates, and its spatial distribution This reflects the energy density characteristics of the vortex to be analyzed.

[0139] Based on the obtained first mode By observing the vortex energy density characteristics, one can understand the spatial energy evolution process of the vortex to be analyzed in a multi-vortex system. For example... Figure 5 As shown, it presents a diagram showing the variation of the dominant mode energy along the flow direction in a typical vortex flow field in this embodiment (the flow direction here is the direction of vortex evolution, and the dominant mode energy corresponds to the energy of the target vortex).

[0140] This invention is not limited to the specific embodiments described above. The invention extends to any new feature or combination disclosed in this specification, as well as any new method or process step or combination disclosed herein.

Claims

1. A method for analyzing the spatial energy evolution process of a multi-vortex system, characterized in that, Comprising the following steps: S1, vortex region recognition According to the existing velocity field data, the velocity gradient tensor is calculated and the vorticity is calculated for identifying the vortex distribution in the flow field; according to the calculated velocity gradient tensor , the Liutex vector is solved S2, vortex core line extraction Extracting vortex core lines for analysis of a vortex ; S3, pseudo-time sequence construction Pseudo-time sequence is constructed by the following steps: S3.1 Evolution direction slice interception For the vortex to be analyzed, the Liutex vector of its vortex core line is... The direction is denoted as ; along the vortex core line, according to the vortex direction , equidistantly taken slices in the flow field , each slice containing a vortex energy field ; i is a natural number, i ≥ 1 and i is less than or equal to N ; S3.2 Slice translation and rotation The vortex core line obtained according to step S2 , determining the vortex core center coordinates of the slice in the flow field ; performing coordinate transformation on the slice to obtain transformed coordinates of the slice , and the transformation formula is as follows: Equation (8); S3.3 One-dimensional data stretching Slices are cut vortex energy field unfolded as row vectors in the order of grid points The calculation formula is as follows: Formula (9): wherein ; , , are the number of grid nodes in the direction of the slice , , ; S3.4 Data matrix construction All the slice vectors are arranged in order along the vortex evolution direction to construct a pseudo-time data matrix , ; wherein, is the number of slices; the vortex evolution direction is the flow direction, or the time axis direction; S4, intrinsic orthogonal modal decomposition S4.1 Evolution direction slice interception Computing a covariance matrix of a data matrix , ; Eigenvalue decomposition of the covariance matrix is performed, and the following formula is used for the calculation Formula (12); wherein is a matrix of eigenvalues, arranged in descending order; is a matrix of eigenvectors, each column characterizes a time-evolution coefficient of a mode; S4.2 Spatial modal reconstruction reconstructing the spatial modes by a linear combination of the eigenvectors with the data matrix , ; each column represents a spatial energy distribution of the kth order mode, where k is a natural number, k≥1 and k≤N; satisfies the following orthogonality condition: Equation (14); S4.3 Dominant energy modal extraction The energy contribution rate of each mode in different slices of the original vortex evolution direction is obtained by intrinsic orthogonal modal decomposition; wherein the energy contribution rate of the kth mode is The calculation formula is as follows: Equation (16); wherein the first modality predominates, its spatial distribution reflects the energy density characteristics of the vortex to be analyzed.

2. The method of claim 1, wherein, In the step S1, the velocity gradient tensor is calculated according to the existing velocity field data and vorticity ; Existing velocity field data includes ;in, These correspond to three coordinates in mutually orthogonal directions in space. Corresponding to Velocity in three directions; Velocity gradient tensor The formula for calculating the velocity gradient tensor is as follows: Formula (1); vorticity The formula for calculating the vorticity is as follows: Formula (2); In the formula, represents the partial differential in the x, y, z directions.

3. The method of claim 1, wherein, According to the calculated velocity gradient tensor , the Liutex vector is solved , the specific process is as follows: for the velocity gradient tensor , the characteristic equation is solved; When the eigenvalue is a real root, a pair of conjugate complex roots, it is considered that the position exists vortex; when the eigenvalue exists conjugate complex roots, the eigenvalue and eigenvector are solved, and the real eigenvalue is recorded as The corresponding characteristic unit vector is The real part and the imaginary part of the conjugate complex root are recorded as and respectively; the Liutex vector is solved by the following formula : Equation (3); where, for the Liutex vector the modulus of the vector represents the rotational intensity of the local fluid parcel; For the Liutex vector , the direction of the vector represents the axis of rotation of the local fluid parcel, i.e., the real eigenvalue corresponds to the real eigenunit vector.

4. The method according to any one of claims 1 to 3, characterized in that, In the step S2, the Liutex vector is obtained according to the obtained , combined with the adaptive integral factor , along the modified vector field extract vortex core lines .

5. The method of claim 4, wherein, In the step S2, the vortex core line of the vortex to be analyzed is extracted The specific operation is as follows: the vortex core line is obtained by integration with an arbitrary point of the vortex to be analyzed as a seed point, and the integral formula is as follows: Equation (4); wherein is the vortex line integral field, is the adaptation factor, the overline indicates the vector is unitized, is the local rotation axis direction; represents the rotation intensity gradient component perpendicular to the local rotation axis ; The calculation formula is as follows: Equation (5); Adaptive factor By rotating the gradient direction along the projection A one-dimensional line search is performed to obtain a step size The following strong Wolfe conditions must be satisfied: Equation (6); wherein represents the position of the trajectory line when the kth step of integration is represented; represents the direction of the gradient ascent, and since the vortex core is the maximum value within a single vortex, it is necessary to ensure ; , is a constant set Obtaining step size by one-dimensional look-ahead After that, the adaptive factor is obtained according to formula (7) : Formula (7); wherein is a constant greater than 1 ; denotes the maximum distance from the current integration node along the projection gradient direction to the next grid face.

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