Multi-vortex system space energy evolution process analysis method

Abstract

The invention discloses a multi-vortex system space energy evolution process analysis method, belongs to the field of fluid mechanics and aerodynamics, and aims to solve the problem of energy analysis errors caused by integral radius dependence and single-vortex hypothesis limitation in a traditional method. According to the method, a vortex core is positioned through a Liutex method, a vortex to be analyzed is translated to an original point, a pseudo time sequence is generated in combination with evolution direction slices, and a first modal energy distribution and evolution law is extracted through eigen-orthogonal modal decomposition. According to the method, the integral radius does not need to be manually set, the adaptability is high, the vortex global energy space change process can be comprehensively reflected, the method is particularly suitable for energy evolution analysis in complex vortex system scenes such as bird flight and complex aircraft wake flow, a brand new view angle is provided for vortex energy evolution, and the method has certain engineering practical value. The method can be used for quantitative analysis of complex vortex field scenes such as aircraft design, biological bionics and wind energy engineering, and has important engineering application value.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of fluid mechanics and aerodynamics, in particular to a method for analyzing the spatial energy evolution process of a multi-vortex system. More specifically, the present application provides a method for analyzing 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, bionics (such as bird flight vortex system research), wind energy engineering, etc. BACKGROUND

[0002] In aerodynamic research, the multi-scale vortex system structure formed by the wing tip and feather movement during bird flight is complex, and its energy evolution directly affects the flight efficiency and stability. Currently, traditional analysis methods are mainly based on single vortex system model, and the energy distribution is characterized by peak vortex intensity or circulation integral.

[0003] However, when using traditional analysis methods to analyze the spatial energy evolution process of a multi-vortex system, the following defects exist: (1) Limitations of single vortex system hypothesis The wake generated by bird flight usually contains multiple interacting vortex structures (such as starting vortex, shedding vortex, etc.), and the traditional single vortex model cannot analyze the coupling effect of multiple vortices; (2) Sensitivity of circulation integral radius The circulation calculation requires the integral radius to be set in advance, and a large radius is easy to include the interference of adjacent vortices, and a small radius misses the complete energy information of the target vortex, resulting in result deviation; (3) Lack of energy distribution information The peak vortex intensity only reflects the local maximum value, and cannot characterize the uniformity and diffusion law of the vortex energy in space, making it difficult to support dynamic evolution analysis of multi-vortex systems.

[0004] In the prior art, the energy analysis of a multi-vortex system relies on empirical parameters or simplified models, lacks adaptability, and makes it difficult to accurately quantify the vortex energy evolution process in scenarios such as bird bionics and wind turbine wake optimization. Therefore, there is an urgent need for an analysis method that can adaptively separate multi-vortex interference and comprehensively reflect the energy spatial distribution characteristics. SUMMARY

[0005] The purpose of the present application is to solve the energy analysis error caused by the dependence of integral radius and the limitations of single vortex hypothesis in traditional methods, and to provide an adaptive multi-vortex system energy evolution analysis method. The present application effectively solves the problems of integral radius dependence and single vortex hypothesis limitation in traditional energy analysis through adaptive coordinate transformation and modal decomposition.

[0006] In order to achieve the above purpose, the present application adopts the following technical solutions.

[0007] A multi-vortex system space energy evolution process analysis method, comprising the following steps: S1, identifying vortex region According to the existing velocity field data, the velocity gradient tensor is calculated And the vorticity , used to identify the vortex distribution in the flow field; according to the calculated velocity gradient tensor , the Liutex vector is solved ; S2, extracting vortex core line Extract the vortex core line of the vortex to be analyzed ; S3, pseudo-time sequence construction The following steps are used to construct the pseudo-time sequence: S3.1 Evolution direction slice cutting For the vortex to be analyzed, the direction of the Liutex vector of the vortex core line is denoted as ; along the vortex core line, according to the vortex direction , cut slices in the flow field at equal intervals , each slice contains the 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 According to the vortex core line obtained in step S2, determine the vortex core center coordinates of the slice ; perform coordinate transformation on the slice to obtain the transformed coordinates of the slice , the transformation formula is as follows: Equation (8) S3.3 One-dimensional data stretching The vortex energy field of the slice is unfolded into a row vector according to the grid point order, and the calculation formula is as follows: Equation (9) Wherein, ; , , are the number of grid nodes of the slice in the , , direction respectively S3.4 Data matrix construction Along the vortex evolution direction, arrange all the slice vectors in order 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, Proper orthogonal modal decomposition S4.1 Evolution direction slice interception Calculate the covariance matrix of the data matrix ; , ; Perform eigenvalue decomposition on the covariance matrix , and the calculation formula is as follows: Equation (12); wherein, is the eigenvalue matrix arranged in descending order; is the eigenvector matrix, each column characterizes the time evolution coefficient of a mode; S4.2 Spatial mode reconstruction Reconstruct the spatial mode , ; Each column represents the spatial energy distribution of the kth mode, wherein k is a natural number, k≥1 and k≤N; satisfies the following orthogonality condition: Equation (14); S4.3 Dominant energy mode extraction Through the proper orthogonal modal decomposition, the energy contribution rate of each mode in the original vortex evolution direction of different slices is obtained; wherein, the energy contribution rate of the kth mode The calculation formula is as follows: Equation (16); wherein, the first mode is dominant, and its spatial distribution reflects the energy density characteristics of the vortex to be analyzed.

[0008] In the step S1, according to the existing velocity field data, the velocity gradient tensor and the vorticity are calculated; The existing velocity field data includes ; wherein, respectively correspond to three coordinates in mutually orthogonal directions in space, Corresponding to Velocity in three directions; velocity gradient tensor The calculation formula is as follows: Formula (1); vorticity The calculation formula is as follows: Formula (2); In the formula, It represents the partial differentials in the x, y, and z directions.

[0009] 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 using the following formula. : Formula (3); 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.

[0010] In step S2, based on the obtained Liutex vector Combined with adaptive integration factor Along the corrected vector field Extracting vortex core lines .

[0011] 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: Formula (4); In the formula, This is the integral field quantity of the vortex core line. is the adaptive factor, the underlined part is to unitize the vector, is the local rotation axis direction; represents the rotation intensity gradient component perpendicular to the local rotation axis ; The calculation formula is as follows: Formula (5); adaptive factor is obtained by performing one-dimensional line search along the projection rotation gradient direction , and the step size in the one-dimensional line search process must satisfy the following strong Wolfe condition: Formula (6); , wherein represents the position of the trajectory line when the k-th step integration is performed; represents the direction of gradient ascent, and since the vortex core is the maximum value in a single vortex, it is necessary to ensure ; , is a set constant; The step size is obtained by one-dimensional line search, and then the adaptive factor is obtained according to formula (7): Formula (7); , wherein is a constant greater than 1; represents the maximum distance from the current integration node to the next grid surface along the projection gradient direction.

[0012] In view of the problems in a multi-vortex system that the traditional single-vortex analysis method (such as peak vortex intensity and circulation integral) has difficulty in defining the integral radius and cannot reflect the energy distribution uniformity, the inventors adopt a solution based on coordinate transformation and modal decomposition to provide a multi-vortex system space energy evolution process analysis method.

[0013] The present application positions the vortex core by the Liutex method, translates the vortex to be analyzed to the origin, generates a pseudo-time sequence in combination with the evolution direction slice, and extracts the first modal energy distribution and evolution law by using proper orthogonal modal decomposition (POD). Specifically, in the present application, first, the velocity gradient tensor and the vorticity are calculated according to the velocity field data, which are used to identify the vortex distribution in the flow field; second, the Liutex vector is obtained, and the adaptive integral factor is combined to perform one-dimensional line search along the modified vector field extract vortex core line ; secondly, along the vortex core line equidistantly intercept two-dimensional slices, and translate the origin of each slice to the vortex core center to eliminate other vortex interference; then, stretch the slice energy field into a one-dimensional vector and arrange it in spatial order into a matrix ; finally, calculate the covariance matrix , and reconstruct the spatial mode after eigen decomposition , extract the first mode , and complete the spatial energy evolution of the multi-vortex system.

[0014] The present application does not need to manually set the integral radius, has strong adaptability, can comprehensively reflect the spatial energy change process of the vortex, is especially suitable for energy evolution analysis under complex vortex system scenes such as bird flight and complex aircraft wake, provides a new perspective for vortex energy evolution, and has certain engineering practical value. BRIEF DESCRIPTION OF DRAWINGS

[0015] The present application will be described by examples and with reference to the accompanying drawings, in which: Figure 1 Figure 1 is an example diagram of a typical vortex structure image of a flow field in Example 1.

[0016] Figure 2 Figure 2 is an example diagram of a series of slices intercepted from a typical flow field in Example 1.

[0017] Figure 3 Figure 3 is an example diagram of coordinate changes of a series of slices according to a vortex to be analyzed in Example 1.

[0018] Figure 4 Figure 4 is a comparison diagram of the original vortex energy distribution of a slice and the first mode energy distribution after eigen orthogonal decomposition reconstruction in Example 1. Figure 4 (a) is the original flow field, Figure 4 (b) is the first mode after reconstruction.

[0019] Figure 5 Figure 5 is a diagram of the energy change of the dominant mode of a typical vortex flow field along the flow direction (the flow direction here is the vortex evolution direction) in Example 1. DETAILED DESCRIPTION

[0020] All features disclosed in this specification, or the steps of any method or process specified in this specification, can be combined in any combination, except where features and / or steps are mutually exclusive.

[0021] Any feature disclosed in this specification, unless stated otherwise, can be replaced by alternative features serving the same, equivalent or a similar purpose. That is, unless stated otherwise, each feature disclosed is one example only of a generic series of equivalent or similar features.

[0022] Embodiment 1 The present application provides a multi-vortex system space energy evolution process analysis method, which comprises the following steps.

[0023] S1, identifying vortex region For existing flow field data, the field quantity used for subsequent vortex core positioning and vortex space energy analysis is calculated according to the velocity field. Specifically, it is divided into the following steps. The existing flow field data refers to the test measurement data or the data obtained by test calculation, which at least contains three directions x, y, z for describing the spatial position and the corresponding three direction velocities .

[0024] S1.1 Calculate the velocity gradient tensor according to the velocity field data (Formula 1) and the vorticity (Formula 2) for identifying the vortex distribution in the flow field.

[0025] Wherein, the calculation formula of the velocity gradient tensor is as follows: Formula (1); The calculation formula of the vorticity is as follows: Formula (2); In the formula, represents the three components of the velocity field, represents the partial derivative of x, y, z in three directions.

[0026] S1.2 Calculate the vortex energy field For the velocity gradient tensor , solve its characteristic equation; when the characteristic root is a real root and a pair of conjugate complex roots, according to the critical point theory of flow field topology, it is considered that there is a vortex at this position; when the characteristic root exists conjugate complex root, solve the characteristic root and characteristic vector, record the real characteristic root corresponding to the characteristic unit vector , the real part and the imaginary part of the conjugate complex root are recorded as and ; according to the equivalent formula of the third generation vortex identification method, the Liutex vector is solved, and the calculation formula is as follows: Formula (3); where, for the Liutex vector , the modulus of the vector represents the rotation intensity of the local fluid parcel; for the Liutex vector , the direction of the vector represents the rotation axis of the local fluid parcel, i.e., the real eigenvalue corresponding to the real eigenunit vector. A typical flow field vortex is shown in FIG. 1, where the vortex core line is generated by the Liutex intensity. Figure 1

[0027] S2, extracting the vortex core line According to the vortex core line extraction method in the Lagrangian framework proposed by the inventor before (application number: CN202311250711.X, publication number: CN117291117A, publication date: 2023-12-26), the vortex core line to be analyzed is extracted.

[0028] In one specific example, according to formulas (4)-(7), the vortex core line is extracted, and the specific steps are as follows.

[0029] The vortex core line is obtained by integrating the seed point of the vortex to be analyzed, where the integral formula is as follows: Formula (4); where, is the vortex core line integral field quantity, is an adaptive factor, and the overline indicates that the vector is unitized, is the local rotation axis direction; represents the rotation intensity gradient component perpendicular to the local rotation axis , which is simply referred to as the projected gradient, The calculation formula of Formula (5); The adaptive factor is obtained by performing one-dimensional line search along the projected rotation gradient direction , and the step size during the one-dimensional line search must satisfy the following strong Wolfe condition: Formula (6); where, represents the position of the trajectory line at the k-th step of integration; represents the direction of gradient ascent, and since the vortex core is the maximum value within a single vortex, it is necessary to ensure ; wherein, , are set constants, and in the present method, , .​

[0030] Step length is obtained by one-dimensional look-ahead Then, the adaptive factor is obtained according to formula (7) : Formula (7) is as follows: In the formula, is a constant greater than 1, and here 1.1 is taken; represents the maximum distance from the current integral node along the projection gradient direction to the next grid surface.

[0031] S3, pseudo-time sequence construction S3.1 Evolution direction slice interception For the vortex to be analyzed, the direction of the Liutex vector of the vortex core line is denoted as ; along the vortex core line, according to the vortex direction , an equal interval of slices in the flow field are intercepted, and each slice contains a vortex energy field , as shown in . Figure 2 is a natural number, i ≥1 and i is less than or equal to i . N .

[0032] S3.2 Slice translation and rotation According to the vortex core line obtained in step S2, the vortex core center coordinates of the slice in the flow field are determined, and the slice is subjected to coordinate transformation to align the slice center with the vortex core center, to obtain the transformed coordinates of the slice, and the transformation formula is as follows: Formula (8) is as follows: Figure 3 An example of coordinate transformation of the vortex to be analyzed is given. The transformation makes the vortex energy field to be analyzed always located at the center of the slice, ensuring that the vortex to be analyzed continuously maintains the spatial structure while diluting the spatial interference of other vortex structures.

[0033] S3.3 One-dimensional data stretching The vortex energy field of the slice is unfolded into a row vector according to the grid point order, and the calculation formula is as follows: Formula (9) is as follows: where, ; , , are the number of grid nodes in the direction of the slice in , , .

[0034] S3.4 Data matrix construction Arrange all slice vectors in the order of the vortex evolution direction to construct a pseudo-time data matrix , . The vortex evolution direction can be the flow direction or the time axis direction. The specific calculation formula is as follows: Equation (10); where, is the number of slices; each column of the matrix corresponds to the full-field energy distribution of a slice, and each row represents the dynamic change of the energy of a single grid point with the evolution direction.

[0035] S4, Proper orthogonal mode decomposition (POD) S4.1 Evolution direction slice interception Based on the data matrix , calculate its covariance matrix , , which represents the energy correlation between slices. The specific calculation formula of the covariance matrix is as follows: Equation (11); Perform eigenvalue decomposition on , and the calculation formula is as follows: Equation (12); where, is the eigenvalue matrix, arranged in descending order ; is the eigenvector matrix, and each column represents the time evolution coefficient of a mode.

[0036] S4.2 Spatial mode reconstruction Through linear combination of the eigenvector and the data matrix, the spatial mode , is reconstructed. The specific calculation formula of the reconstructed spatial mode is as follows: Equation (13); Each column represents the spatial energy distribution of the k-th mode, where k is a natural number, k≥1 and k≤N. satisfy the following orthogonality condition: Equation (14).

[0037] For example, the reconstruction space modal formula of the first modal is as follows: Equation (15).

[0038] The schematic diagram of the reconstruction space modal is shown in Figure 4 ; wherein, Figure 4 (a) is the original flow field, Figure 4 (b) is the first modal after reconstruction.

[0039] S4.3 Dominant energy modal extraction Through the intrinsic orthogonal modal decomposition, the energy contribution rate of each modal in different slices of the original vortex evolution direction can be obtained. The energy contribution rate of the kth order modal is The calculation formula is as follows: Equation (16).

[0040] The first modal usually occupies the dominant position, and the spatial distribution reflects the energy density characteristics of the vortex to be analyzed.

[0041] According to the vortex energy density characteristics of the first modal obtained, the spatial energy evolution process of the vortex to be analyzed in the multi-vortex system can be known. As shown in Figure 5 , it gives the dominant modal energy along the flow direction in the typical vortex flow field in this embodiment (the flow direction here is the vortex evolution direction, and the dominant modal energy corresponds to the energy of the target vortex).

[0042] The present application is not limited to the foregoing specific embodiments. The present application extends to any new feature or any new combination disclosed in this specification, and any new method or process steps disclosed or any new combination.

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 Unfolding into row vectors in grid point order 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 unitization of the vector, 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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