CN114896912B - Three-dimensional zero-divergence pulsating wind field generation method based on composite vortex - Google Patents
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Abstract
The invention discloses a three-dimensional zero-divergence pulsating wind field generation method based on a synthetic vortex, which comprises the steps of establishing a three-dimensional expression of a vortex-shaped function, establishing the relation between the strength of a target pulsating wind field and the integral scale of turbulence and the strength and scale of the vortex, realizing anisotropic pulsating wind simulation based on the synthetic vortex, and establishing a pulsating wind generation method which jointly considers the strength and the scale anisotropy of the vortex and the variability of a vortex group by optimizing probability distribution of the strength and the scale parameter of the vortex on the basis. The method can generate scattered wind speed fields meeting the target three-dimensional pulsating wind energy spectrum and the spatial correlation thereof, realizes high-precision pulsating wind field synthesis, and has remarkable effectiveness and applicability.
Description
Technical Field
The invention relates to the technical field of wind engineering, in particular to a three-dimensional zero-divergence pulsating wind field generation method based on a synthetic vortex.
Background
The incoming flow pulsating wind field is a precondition for researching fluid movement and fluid-solid coupling problems in a designated area. An efficient pulsating wind field generation method can provide reliable incoming flow boundary conditions for a designated area. The boundary condition is one of key factors influencing the numerical simulation of fluid movement and the correctness of experimental results. The existing pulsating wind field generation method can be classified into three main types of a random Fourier method, a digital filtering method and a synthetic vortex method. The random fourier method is an approximate synthesis method based on fourier decomposition techniques (e.g., harmonic synthesis is a representative method of the random fourier method, which treats a pulsating wind field as a superposition of a large number of harmonics). The random fourier method can better realize the spatial correlation of the wind field, but has insufficient performance on the time correlation. In this way, gaussian filtering is introduced, and digital filtering is implemented on the wind field time sequence through convolution processing, so that the synthesized wind field has good time correlation. This method is called digital filtering. Unlike the random fourier method and the digital filtering method, the synthetic swirl method is a result of superimposing a limited number of swirls from the physical structure of the wind field. The synthetic swirling method has great development potential as a novel pulsating wind field generation method. The basic idea of the method is that a certain number of virtual vortexes are randomly generated in a certain space range, each vortexes carries a local speed field determined by a specific function (called a vortex shape function), and the speed fields carried by all vortexes are superimposed together to obtain the target pulsating wind field. The speed, intensity, scale and shape functions of the movement of the vortices directly influence the spatial and temporal correlation of the pulsating wind field.
However, the existing synthetic swirl method does not reasonably define the physical relationship between the swirl scale and the wind field turbulence integration scale, and the consideration of the swirl anisotropy and the swirl group variability is insufficient, so that a pulsating wind field which simultaneously satisfies a plurality of space correlation conditions cannot be effectively generated.
Disclosure of Invention
The invention provides a three-dimensional zero-divergence pulsating wind field generation method based on a synthetic vortex, which solves the problems and the defects existing in the existing pulsating wind field simulation technology.
In order to achieve the above purpose, the present invention provides the following technical solutions: a three-dimensional zero-divergence pulsation wind field generation method based on a composite vortex, which takes air vortex as a basic object and wind field energy spectrum and spatial correlation thereof as an implementation target, comprises the following steps:
s1, constructing a vector field expression of a scattered wind field;
s2, building and solving relationship between the Reynolds stress of the pulsating wind field and the integral scale and the vortex strength and scale;
s3, introducing random vectors on the basis of the S1 and S2 models, and establishing a vortex group variation model by taking the variability of the intensity and the scale of the vortex group into consideration;
s4, generating a space vortex group changing along with time, and further obtaining a pulsating wind speed field through vortex superposition.
Preferably, the step S1 specifically includes the following steps:
1) Expressing the fluctuating wind speed u (x) as a coordinate transformation matrix [ A ]]＝[a 1 a 2 a 3 ]And velocity field u * (x) Where a is the product of 1 ，a 2 ，a 3 For column vectors, i.e. u (x) =au * (x) And introducing each vortex into a three-dimensional Cartesian space x= [ x ] 1 x 2 x 3 ]Intensity vector epsilon= [ epsilon ] on the upper 1 ε 2 ε 3 ]And a scale tensor [ sigma ]]＝[σ 1 σ 2 σ 3 ] T Further provides a new vortex scale shape function expression f σi The formula is as follows:
wherein V is the target space volume; i, j=1, 2,3; k=1, 2, …, N is the number of vortices in the target space;
2) Construct to satisfy the condition of scattered degreeIs = [ ψ = [ psi ] 1 ψ 2 ψ 3 ]The expression is as follows:
wherein + -1 represents the direction of movement of the vortex, +1 is positive along the axis, -1 is negative along the axis.
Preferably, the step S2 specifically includes the following steps:
1) Generating a spatial correlation function of the vector field psi, obtaining a spectrum expression of the vector field psi by developing Fourier transform, and establishing a velocity field u * (x) The relation between the frequency spectrum and the frequency spectrum of the vector field psi is used for generating a speed field u by performing Fourier inversion * (x) Is a spatial correlation function of (1); introducing a coordinate transformation matrix [ A ]]Finally obtaining the frequency spectrum of the fluctuating wind speed u (x)Expression, and thus obtaining the pulsating wind field integral scale tensor [ L ]]And velocity field u * (x) Integral scale tensor [ L ] * ]Is the relation of:
wherein ,Lij and Lij * An integral scale representing i-direction velocity with respect to j-direction; τ ij and τij * Velocity fields u (x) and u, respectively * (x) Reynolds stress tensor of (2);
2) The velocity field u * (x) Is a function of the intensity and scale of the vortex by introducing a fourier transform f of the vortex size function F (n), where n= [ n ] 1 n 2 n 3 ]Representing wavenumbers in three orthogonal directions, thereby establishing u * (x) Reynolds stress tensor [ tau ] * ]Expression for vortex intensity and scale parameters:
u * (x) Is expressed by the energy spectrum expression:
wherein ,γi (i=1, 2, 3) is the swirl strength ε i k Is a variance of (2); sigma (sigma) ij Is a vortex scale tensor; c (C) 0 Is Fourier transform type f with vortex scale shape function F (n) a coefficient of correlation, the formula of which is:the shape function and its Fourier transform have spherical symmetry, then C 0 F in the expression F (n)＝f F (n), and->E ij * (i, j=1, 2, 3) is the velocity fieldu * (x) An energy spectrum of the i-direction component with respect to the j-direction; />Is a Fourier transform f of a form function F (n) a related function having the formula: />And i noteq noteqp;
according to the essential relation between the energy spectrum and the integral scale, u can be further obtained * (x) Integral scale of [ L ] * ]The expression for the vortex intensity and scale parameter is as follows:
wherein ,C1 Is Fourier transform type f with vortex scale shape function F (n) a coefficient of correlation, the formula of which is:the shape function and its Fourier transform have spherical symmetry, then C 1 F in the expression F (n)＝f F (n), and->
3) And (3) giving the Reynolds stress and the energy spectrum of the target pulsating wind field and the spatial correlation thereof through wind field actual measurement or engineering assignment, selecting a vortex scale function form, and adopting a Newton iteration method to jointly solve the nonlinear equation set in 1) and 2) to obtain the values of the vortex intensity parameter gamma and the scale parameter [ sigma ] in the target space.
Preferably, the step S3 specifically includes the following steps:
1) Introducing a random vector λ= [ λ ] 1 λ 2 λ 3 ]To take into account the variability of the intensity and scale of the vortex population, each vortex intensity parameter and scale parameter is expressed asWhereby the object isThe intensity and scale of each vortex in the space have randomness, and the probability distribution characteristics of the vortex are similar to the random variable lambda i Probability density p of (2) i (lambda) correlation; wherein i=1, 2,3; k represents a swirl number;
2) Bringing the vortex intensity parameter and scale parameter expressions described in 1) into u in S2 * (x) The energy spectrum expression of (2) to obtain u * (x) Is represented by the new energy spectrum:
and will beU brought into S2 * (x) Integral scale of [ L ] * ]Expression for vortex intensity and scale parameters, yielding u * (x) Integral scale of [ L ] * ]New relations regarding swirl strength and scale parameters:
wherein ,is related to a random variable lambda i Is given by ∈K>Velocity field u * (x) Is = [ ψ = [ psi ] 1 ψ 2 ψ 3 ]The formula can be:
further expressed as the formula:
wherein ,
3) Taking the variability of vortex groups into consideration, constructing probability density p i Functional formula of (λ):
wherein ,Cp,i ，α p,i ，λ i,min and λi,max Is a probability distribution parameter to be optimized, and i=1, 2,3;
and (3) constructing an optimization objective function formula by considering energy spectrum error minimization:
wherein ,Eij The energy spectrum simulation value of the wind speed in the direction j in the direction i is given by the following formula: the measured value or the target value of the energy spectrum of the wind speed in the i direction in the j direction is obtained; n is n ub Is the upper wavenumber limit;
combining probability density p i Fitting probability distribution parameters C to be optimized by adopting an interior point method by adopting a (lambda) functional formula and an optimization objective function p,i ，α p,i ，λ i,min and λi,max Is a value of (2).
Preferably, the step S4 specifically includes the following steps:
1) Generating a time-varying spatial vortex group, the generation of the vortex group comprising vortex initialization and vortex regeneration, the vortex initialization being as follows:
the probability density function shown is premised on generating a random vector lambda k ＝[λ 1 k λ 2 k λ 3 k ]According to the formula:
imparting k strength and scale to the vortex, where k is the vortex number, k=1, 2, …, N; for the vortex k, a corresponding space B is generated k Such as the formula:
wherein ,
and χ is a limiting factor of the swirl travel distance; based on the space uniform distribution, the vortex initial coordinate x is randomly generated k And a direction of motion, wherein +1 is positive along the coordinate axis, -1 is negative along the coordinate axis;
2) Knowing the target pulsating wind field reynolds stress tensor [ tau ]]Calculate the coordinate transformation matrix [ A ]]Its column vector a i Is [ tau ]]According to the formulas in the steps S1, S2 and S3, combining the random vector lambda k And the vortex intensity parameter gamma and the scale parameter [ sigma ]]Generating a pulsating wind speed field u (x, t) at an initial time t;
3) Performing vortex regeneration, and updating the vortex position x by taking the time increment delta t into consideration k (t+Δt)＝x k (t) +u·Δt, wherein U is the average wind speed of the target pulsating wind field; identify that space B has been left k Is a vortex of (i), i.eAccording to the probability distribution characteristics of the separated vortex, the formula is as follows: /> wherein spani I-side representing a spatial function B (lambda)Directional span, regeneration of random vector lambda k And corresponding space B Δt k Sampling to generate new vortex coordinates and motion directions, and giving new strength and scale to the vortex to realize the supplementation of the vortex; and further obtaining a fluctuating wind speed field u (x, t+Δt) at the time t+Δt through vortex superposition.
Compared with the prior art, the invention has the beneficial effects that: by constructing a three-dimensional expression of the vortex function, establishing the relation between the target wind field intensity and the turbulence integral scale and the vortex intensity and scale, the anisotropic pulsating wind simulation based on the synthesized vortex is realized, and on the basis, the pulsating wind generation method which jointly considers the vortex intensity and the scale anisotropy and the vortex group variability is established by optimizing the probability distribution of the vortex intensity and the scale parameter. The method can generate scattered wind speed fields meeting the target three-dimensional pulsating wind energy spectrum and the spatial correlation thereof, realizes high-precision pulsating wind field synthesis, and has remarkable effectiveness and applicability.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate the invention and together with the description serve to explain the invention.
In the drawings:
FIG. 1 is a flow chart of a method of generating a three-dimensional zero divergence pulsating wind field based on synthetic vortexes of the present invention;
FIG. 2 is a cloud of wind speed distribution according to a preferred embodiment of the invention;
FIG. 3 is a plot of regional center point wind speed in a preferred embodiment of the present invention;
FIG. 4 is a graph of the energy spectrum of the pulsatile wind velocity at 1/2 height from the bottom of the pipeline in the preferred embodiment of the present invention;
FIG. 5 is a graph of the energy spectrum of the pulsatile wind velocity at 1/5 height from the bottom of the pipeline in the preferred embodiment of the present invention;
FIG. 6 is a graph of the energy spectrum of the pulsatile wind velocity at 1/20 height from the bottom of the pipeline in the preferred embodiment of the present invention;
Detailed Description
The preferred embodiments of the present invention will be described below with reference to the accompanying drawings, it being understood that the preferred embodiments described herein are for illustration and explanation of the present invention only, and are not intended to limit the present invention.
Examples: a three-dimensional zero-divergence pulsating wind field generation method based on a synthetic vortex with air vortex as a basic object and wind field energy spectrum and spatial correlation thereof as an implementation target comprises the following steps:
a first part: complete steps of this embodiment
(one) explicitly input parameters and parameter models
1) Wind field parameters and model thereof
Establishing a target wind farm parameter or parameter model, comprising: (1) an average wind speed U; (2) reynolds stress tensor [ tau ]]The method comprises the steps of carrying out a first treatment on the surface of the (3) Three-dimensional spatial correlation function R ij Or a correlation function characteristic and a corresponding turbulence integral scale L ij Wherein i is a correlation function of wind velocity in the j direction, i, j=1, 2,3; (4) wind spectrum or energy spectrum E of three-dimensional pulsating wind along three orthogonal directions in sequence ij . The above target wind farm parameters or parameter models may be established based on wind farm measurements or specified directly based on engineering application context.
2) Target space parameters
Specifying a target space S, and taking a limited number of space points N according to the variation range of the space S in the three orthogonal directions S ＝N 1 ×N 2 ×N 3, wherein N1 ,N 2 ,N 3 And taking the discrete points in the three orthogonal directions in the target space as vortex synthesis points, and recording coordinates of all points.
3) Shape function model
And selecting a shape function expression. Table 1 lists several general classes of form functions for reference.
Table 1 several form function expressions and their corresponding fourier transforms
(II) solving the vortex intensity parameter gamma and the scale parameter [ sigma ]
The solving steps of the vortex intensity parameter gamma and the scale parameter [ sigma ] are summarized as follows:
1) Knowing the target wind field Reynolds stress tension [ tau ]]Solving for its eigenvector a i (i=1, 2, 3) to obtain a coordinate transformation matrix [ a ]]；
2) The swirl function form (see table 1) was chosen according to the formula:
solving speed u * (x) Obtaining a velocity field u * (x) Reynolds stress tensor [ tau ] * ]；
3) Knowing the target wind field turbulence integral scale tensor [ L ]]And Reynolds stress tension [ tau ]]Wind speed field u * (x) Reynolds stress tensor [ tau ] * ]And a coordinate transformation matrix [ A ]]According to the formula:
solving wind velocity field u * (x) Integral scale tensor [ L ] * ]；
4) Known velocity field u * (x) Reynolds stress tensor [ tau ] * ]And integral scale tensor [ L ] * ]Solving the formula by combining the selected vortex-shaped function expression:
and obtaining the values of the vortex intensity parameter gamma and the scale parameter [ sigma ] in the target space by the nonlinear equation system.
(III) optimizing the vortex random variable lambda i Probability distribution parameters of (2)
At p i (lambda) probability density function:
in a basic form, consider energy spectrum error minimization according to the formula:
the optimization objective function is shown, and the parameter C is fitted by adopting an interior point method p,i ，α p,i ，λ i,min and λi,max (i=1, 2, 3), the specific steps are as follows:
1) Assigned to C p,i ，α p,i ，λ i,min and λi,max Initial value, sequentially taking: 0.5,1.0,0.001,1.0;
2) According to the formula:
substituted into C p,i ，α p,i ，λ i,min and λi,max Value, initial step substituting initial value, other steps substituting updated value, determining random variable lambda i Probability density function p of (2) i (lambda) based on which N lambda are extracted i A sample, wherein N is the number of vortexes, and the value is not less than 10000;
3) According to lambda i Sampling results, combining the vortex intensity parameter gamma and the scale parameter [ sigma ] obtained in (II)]According to the formula:
obtaining an intensity vector and a scale tensor of each vortex;
4) Combining the selected vortex-shaped function and Fourier transform thereof, and according to the formula:
generating a velocity field u * (x) Three-dimensional energy spectrum E of (2) ij * Further introducing a coordinate transformation matrix [ A ]]According to the formula
Synthesis of three-dimensional wind field energy Spectrum E ij ；
5) Knowing the target wind field energy spectrumAccording to the formula:
calculating an energy spectrum error coefficient of the wind field: if the error requirement is not satisfied, updating C p,i ，α p,i ，λ i,min and λi,max Value and return to step 2); if the error requirement is met, C p,i ，α p,i ，λ i,min and λi,max The value is established as an optimized value and the optimization process is ended.
(IV) vortex synthesized pulsating wind field
Considering the spatiotemporal histories of the vortex motion, the vortex synthesis process comprises two aspects of vortex initialization and vortex regeneration. The method comprises the following specific steps:
1) Introduction of C p,i ，α p,i ，λ i,min and λi,max Optimizing the value, in the formula:
the probability distribution shown is premised on the generation of a random vector lambda k ＝[λ 1 k λ 2 k λ 3 k ]；
2) According to the formula:
imparting intensity and scale to the vortex k, where k is the vortex number, k=1, 2, …, N, and for the vortex k, generating the formula:
space B shown k The limit factor χ of the vortex moving distance is determined according to specific engineering requirements;
3) Based on space B k Uniformly distributed, randomly generated vortex initial coordinate x k And the movement direction, +1 is positive along the coordinate axis, and-1 is negative along the coordinate axis;
4) According to the formula:
u(x)＝Au * (x)
u(x)＝Au * (x)
and the formula:
and combine the random vector lambda k And (2) the vortex intensity parameter gamma and the scale parameter [ sigma ] in (two)]Generating a pulsating wind speed field u (x, t) at the moment t;
5) Knowing the average wind speed U of the target wind field, and considering the time increment delta t, updating the vortex position x k (t+Δt)＝ x k (t)+U·Δt；
6) Identify that space B has been left k Is a vortex of (i), i.eAccording to the formula:
the probability distribution of the eddies that have left off is shown, regenerating the random vector lambda k And corresponding space B Δt k Sampling to generate new vortex coordinates and motion directions, and giving new strength and scale to the new vortex coordinates and motion directions to realize vortex regeneration;
7) Obtaining a pulsating wind speed field u (x, t+Δt) at the time t+Δt by vortex synthesis;
8) Let t=t+Δt, return to step 5).
By combining the above, the three-dimensional zero-divergence pulsating wind field generation method based on the synthetic vortex comprises four steps: parameters and parameter models are definitely input, vortex strength parameters and scale parameters are solved, and vortex random variable lambda is optimized i The implementation of the probability distribution parameters and vortex synthesis pulsating wind field can be summarized as fig. 1.
A second part: method utility display
The three-dimensional zero-divergence pulsating wind field generation method based on the synthetic vortex provided by the invention is adopted to simulate and generate a pulsating wind field formed by air flow in a pipeline by taking a typical square pipeline flow, such as a ventilating pipeline, as an application scene. The air flowing longitudinally along the duct, wherein the duct is longitudinally set to x 1 An axis focusing on the flow field at 1/2 height, 1/5 height and 1/20 height from the bottom of the pipe, wherein the height direction is set as x 2 The flow field characteristics at the axis, the target height of the pipe, may be x 1 -x 2 The plane of construction is described, the duct being set transversely to x 3 A shaft. The target flow field parameters and other input parameters and their models are shown in table 2.
TABLE 2 input parameters or parameter models
* And (3) injection: the values of the average flow velocity, the Reynolds stress tensor and the integral scale tensor in the table are normalized dimensionless values.
Taking the object of interest at a height of 1/2 from the bottom of the pipeline, FIG. 2 (a) gives x 1 -x 3 Wind speed distribution cloud pictures on a plane; FIG. 3 (a) shows x at this height 1 -x 3 A pulsatile wind speed schedule for a center point in a plane; FIGS. 4 (a) - (f) show comparison of the results of the pulse wind energy spectrum simulation with the target spectrum at 1/2 height from the bottom of the pipeline, where (a) E 11 ；(b)E 21 ；(c)E 31 ；(d)E 13 ；(e)E 23 ；(f)E 33 。
Taking the object of interest at a height of 1/5 from the bottom of the pipeline, FIG. 2 (b) gives x 1 -x 3 Wind speed distribution cloud pictures on a plane; FIG. 3 (b) shows x at this height 1 -x 3 A pulsatile wind speed schedule for a center point in a plane; FIGS. 5 (a) - (f) show 1/5 height from the bottom of the pipeComparing the pulse wind energy spectrum simulation result with the target spectrum, wherein (a) E 11 ；(b)E 21 ；(c)E 31 ；(d)E 13 ；(e)E 23 ；(f)E 33 。
Taking the object of interest at a height of 1/20 from the bottom of the pipeline, FIG. 2 (c) gives x 1 -x 3 Wind speed distribution cloud graph on plane; FIG. 3 (c) shows x at this height 1 -x 3 A pulsatile wind speed schedule for a center point in a plane; FIGS. 6 (a) - (f) show comparison of the results of the pulse wind energy spectrum simulation with the target spectrum at a distance of 1/20 of the height from the bottom of the pipeline, where (a) E 11 ；(b)E 21 ；(c)E 31 ；(d)E 13 ；(e)E 23 ；(f)E 33 。
The wind field simulation results and comparison results shown in fig. 2 to 6 show that: the method can realize high-precision pulsating wind field synthesis and has remarkable effectiveness and applicability.
Finally, it should be noted that: the foregoing is merely a preferred example of the present invention, and the present invention is not limited thereto, but it is to be understood that modifications and equivalents of some of the technical features described in the foregoing embodiments may be made by those skilled in the art, although the present invention has been described in detail with reference to the foregoing embodiments. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (3)
1. The three-dimensional zero-divergence pulsating wind field generation method based on the composite vortex is characterized by comprising the following steps of:
s1, constructing a vector field expression of a scattered wind field;
s2, building and solving relationship between the Reynolds stress of the pulsating wind field and the integral scale and the vortex strength and scale;
s3, introducing random vectors on the basis of the S1 and S2 models, and establishing a vortex group variation model by taking the variability of the intensity and the scale of the vortex group into consideration;
s4, generating a space vortex group which changes along with time, and further obtaining a pulsating wind speed field through vortex superposition;
the step S1 specifically comprises the following steps:
1) Expressing the fluctuating wind speed u (x) as a coordinate transformation matrix [ A ]]＝[a 1 a 2 a 3 ]And velocity field u * (x) Where a is the product of 1 ，a 2 ，a 3 For column vectors, i.e. u (x) =au * (x) And introducing each vortex into a three-dimensional Cartesian space x= [ x ] 1 x 2 x 3 ]Intensity vector epsilon= [ epsilon ] on the upper 1 ε 2 ε 3 ]And a scale tensor [ sigma ]]＝[σ 1 σ 2 σ 3 ] T Further provides a new vortex scale shape function expression f σi The formula is as follows:
wherein V is the target space volume; i, j=1, 2,3; k=1, 2, …, N is the number of vortices in the target space;
2) Construct to satisfy the condition of scattered degreeIs = [ ψ = [ psi ] 1 ψ 2 ψ 3 ]The expression is as follows:
wherein + -1 represents the direction of movement of the vortex, +1 is positive along the axis, -1 is negative along the axis;
the step S2 specifically comprises the following steps:
1) Generating a spatial correlation function of the vector field psi, obtaining a frequency spectrum expression of the vector field psi by developing Fourier transformation, and establishing a speed field u * (x) The relation between the spectrum and the spectrum of the vector field ψ is used for generating a velocity field u by performing an inverse fourier transform * (x) A kind of electronic deviceA spatial correlation function; introducing a coordinate transformation matrix [ A ]]Finally obtaining a frequency spectrum expression of the fluctuating wind speed u (x), and obtaining a fluctuation wind field integral scale tensor [ L ]]And velocity field u * (x) Integral scale tensor [ L ] * ]Is the relation of:
wherein ,Lij and Lij * An integral scale representing i-direction velocity with respect to j-direction; τ ij and τij * Velocity fields u (x) and u, respectively * (x) Reynolds stress tensor of (2);
2) The velocity field u * (x) Is a function of the intensity and scale of the vortex by introducing a fourier transform f of the vortex scale shape function F (n), where n= [ n ] 1 n 2 n 3 ]Representing wavenumbers in three orthogonal directions, thereby establishing u * (x) Reynolds stress tensor [ tau ] * ]Expression for vortex intensity and scale parameters:
u * (x) Is expressed by the energy spectrum expression:
wherein ,γi (i=1, 2, 3) is the swirl strength ε i k Is a variance of (2); sigma (sigma) ij Is a vortex scale tensor; c (C) 0 Is Fourier transform type f with vortex scale shape function F (n) a coefficient of correlation, the formula of which is:the shape function and its Fourier transform have spherical symmetry, then C 0 F in the expression F (n)＝f F (n), and->E ij * (i, j=1, 2, 3) is the velocity field u * (x) An energy spectrum of the i-direction component with respect to the j-direction; />Is a Fourier transform f of a form function F (n) a related function having the formula:
i, j, p=1, 2,3, and i+.j+.p;
according to the essential relation between the energy spectrum and the integral scale, u can be further obtained * (x) Integral scale of [ L ] * ]The expression for the vortex intensity and scale parameter is as follows:
wherein ,C1 Is Fourier transform type f with vortex scale shape function F (n) a coefficient of correlation, the formula of which is:
the shape function and its Fourier transform have spherical symmetry, then C 1 F in the expression F (n)＝f F (n), and->
3) And (3) giving the Reynolds stress and the energy spectrum of the target pulsating wind field and the spatial correlation thereof through wind field actual measurement or engineering assignment, selecting a vortex scale function form, and adopting a Newton iteration method to jointly solve the nonlinear equation set in 1) and 2) to obtain the values of the vortex intensity parameter gamma and the scale parameter [ sigma ] in the target space.
2. The three-dimensional zero divergence pulsating wind field generation method based on the composite vortex according to claim 1, wherein the step S3 specifically comprises the following steps:
1) Introducing a random vector λ= [ λ ] 1 λ 2 λ 3 ]To take into account the variability of the intensity and scale of the vortex population, each vortex intensity parameter and scale parameter is expressed asWhereby the intensity and scale of each vortex in the target space has randomness, the probability distribution characteristics of which are related to the random variable lambda i Probability density p of (2) i (lambda) correlation; wherein i=1, 2,3; k represents a swirl number;
2) Bringing the vortex intensity parameter and scale parameter expressions described in 1) into u in S2 * (x) The energy spectrum expression of (2) to obtain u * (x) Is represented by the new energy spectrum:
and will beU brought into S2 * (x) Integral scale of [ L ] * ]Expression for vortex intensity and scale parameters, yielding u * (x) Integral scale of [ L ] * ]New relations regarding swirl strength and scale parameters:
wherein ,is related to followingVariable lambda i Is given by ∈K>Velocity field u * (x) Is = [ ψ = [ psi ] 1 ψ 2 ψ 3 ]The formula can be:
further expressed as the formula:
wherein ,
3) Taking the variability of vortex groups into consideration, constructing probability density p i Functional formula of (λ):
wherein ,Cp,i ，α p,i ，λ i,min and λi,max Is a probability distribution parameter to be optimized, and i=1, 2,3;
and (3) constructing an optimization objective function formula by considering energy spectrum error minimization:
wherein ,Eij The energy spectrum simulation value of the wind speed in the direction j in the direction i is given by the following formula: the measured value or the target value of the energy spectrum of the wind speed in the i direction in the j direction is obtained; n is n ub Is the upper wavenumber limit;
combining probability density p i Fitting probability distribution parameters C to be optimized by adopting an interior point method and adopting a lambda functional formula and an optimization objective function p,i ，α p,i ，λ i,min and λi,max Is a value of (2).
3. The three-dimensional zero divergence pulsating wind field generation method based on the composite vortex according to claim 2, wherein the step S4 specifically comprises the following steps:
1) Generating a time-varying spatial vortex group, the generation of the vortex group comprising vortex initialization and vortex regeneration, the vortex initialization being as follows:
the probability density function shown is premised on generating a random vector lambda k ＝[λ 1 k λ 2 k λ 3 k ]According to the formula:
imparting k strength and scale to the vortex, where k is the vortex number, k=1, 2, …, N; for the vortex k, a corresponding space B is generated k Such as the formula:
wherein ,
and χ is a limiting factor of the swirl travel distance; based on the space uniform distribution, the vortex initial coordinate x is randomly generated k And a direction of motion, wherein +1 is positive along the coordinate axis, -1 is negative along the coordinate axis;
2) Knowing the target pulsating wind field reynolds stress tensor [ tau ]]Calculate the coordinate transformation matrix [ A ]]Its column vector a i Is [ tau ]]According to the formulas in the steps S1, S2 and S3, combining the random vector lambda k And the vortex intensity parameter gamma and the scale parameter [ sigma ]]Generating a pulsating wind speed field u (x, t) at an initial time t;
3) Performing vortex regeneration, and updating the vortex position x by taking the time increment delta t into consideration k (t+Δt)＝x k (t) +U.DELTA.t, wherein U is the average wind speed of the target pulsating wind field; identify that space B has been left k Is a vortex of (i), i.eAccording to the probability distribution characteristics of the separated vortex, the formula is as follows: /> wherein spani Representing the span in the i-direction of the spatial function B (lambda), the random vector lambda is regenerated k And corresponding space B Δt k Sampling to generate new vortex coordinates and motion directions, and giving new strength and scale to the vortex to realize the supplementation of the vortex; and further obtaining a fluctuating wind speed field u (x, t+Δt) at the time t+Δt through vortex superposition.
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