CN104239730A - Non-Gaussian turbulent flow analogy method based on Lagrange random particulate matter model - Google Patents

Abstract

The invention discloses a non-Gaussian turbulent flow analogy method based on a Lagrange random particulate matter model. The non-Gaussian turbulent flow analogy method particularly comprises the following steps of introducing a Pearson IV probability density function and making up the disadvantages of an original model simulation non-Gaussian turbulent flow condition by a random number generator applicable to an arbitrary probability density function based on a receiving-rejecting method, namely controlling the skewness and the kurtosis. The method is verified by measured data and the skewness and the kurtosis in a statistical magnitude can be effectively controlled on the basis of an original model so that the scheme can effectively simulate statistical characteristics to obey non-Gaussian distributed turbulent flows and the simulation degree of the model is improved.

Description

Based on the non-gaussian Turbulence Method of the random suspended particulates model of Lagrange

Technical field

The present invention relates to the non-gaussian turbulent flow modeling scheme based on the random suspended particulates model of Lagrange, specifically a kind of non-gaussian turbulent flow modeling scheme about near-earth air, the degree of bias in the simulation of Pearson's random number control turbulent flow and kurtosis can be utilized, improve the DIFFUSION PREDICTION ability of turbulent flow fidelity and suspended particulates, belong to micrometeorology field.

Background technology

Lagrange probabilistic model (Lagrangian Stochastic Model is called for short LSM), that a kind of suspended particulates are at three-dimensional tracks analogy model, LSM utilizes numerical simulation mode, substitute the Euler's spread of particles model using momentum mode to simulate, overcome accuracy and many restrictions in the application of Euler's model.LSM widely uses and popular a kind of spread of particles model up to now in the world most, is applied to air quality management more, natural resource management (as agricultural and dark woods manage and prevention and control), health management etc.

The suspended particulates (Aerosol is called for short particulate) that LSM pays close attention to have following character:

1) particulate of all simulations is single existence, and namely the motion process of each particulate is independently;

2) each particulate is enough little (being generally less than 10 μm), and its size is less than the minimum length scale (Kolmogorov scale) of turbulent flow, and therefore its expanding trajectory moves identical with DIFFUSION IN TURBULENCE;

3) interaction between particulate and particles agglomerate is not considered.

Spread of particles (Aerosol Dispersion): it is the core of LSM, its framework is expressed as on a movement dimension:

$\mathrm{dx}=(u+\stackrel{\‾}{u})\mathrm{dt}---\left(1\right)$

In formula, dx is the move distance under particle movement time dt; average velocity; U is velocity variations value; The average of usual u is 0, therefore determine particle speed mean of a probability distribution.

u＝qσ (2)

In formula, σ is particle speed variance of probability distribution.

q ^t+dt＝αq ^t+βr ^t+dt (3)

Formula (3) is a Markov chain (Markov Chain) about movement velocity change; Wherein α is acceleration factor, and it affected by historic state, and it is about position x, a funtcional relationship of u, t; And β is the coefficient of random force (Random Forcing is called for short RF) r; R is a random number, and model hypothesis RF Gaussian distributed.

Non-gaussian turbulent flow (Non-Gaussian Turbulence): LSM supposes turbulent flow be Gaussian distributed, and the average of actual measurement data distribution is really identical with Gaussian distribution with variance, but measured data distribution have the higher degree of bias and kurtosis.At present in actual applications, LSM still supposes turbulent flow statistical distribution Gaussian distributed, and in order to meet measured data foundation, simulate suspended particulates motion state more exactly, should revise its turbulent flow distributional assumption is non-gaussian distribution.

Good mixing condition (Well-Mixed Condition is called for short WMC): be that industry thinks most important, the standard of a rigorous evaluation model quality, has proved that other standard all can be summarized as WMC.WMC requires should ensure that statistical nature should be consistent with reality during Selection parameter in framework formula (1) to formula (3), and the statistical nature simulated should have stability and consistance.

LSM at present for the simulation of non-gaussian turbulent flow mainly adopts several measure of the following stated:

1) non-gaussian RF: allow the RF of script Gaussian distributed obey a kind of non-gaussian distribution, to control its degree of bias and kurtosis.Although the method can simulate the degree of bias and kurtosis comparatively accurately to a certain extent, its poor stability does not meet the requirement of WMC standard.

2) two Gauss's joint distribution (Two Gaussian Joint-PDFs): in conjunction with two Gaussian distribution, and control the attachment coefficient of two distributions, and then control the degree of bias and kurtosis.The method is feasible, but experimental result is effective not as original Gauss LSM, and its reason is that the destination probability distribution function obtained is inaccurate.

Summary of the invention

The improving eyesight of this is exactly the problems and shortcomings for above-mentioned background technology, a kind of non-gaussian turbulent flow modeling scheme based on the random suspended particulates model of Lagrange is provided, introduces Pearson's random number, can be more accurate, stably control the degree of bias and kurtosis, and then improve model emulation degree.

Non-gaussian Turbulence Method based on the random suspended particulates model of Lagrange involved in the present invention, implementation step is as follows:

1) anemoscope is used to obtain turbulent flow data on the spot: speed u with the wind, side direction wind speed v, vertical velocity w, real time temperature t;

2) staging treating analyze turbulent flow data, computation model inputs: mean wind speed and direction degree of bias S _w, kurtosis K _w, friction velocity u ^*, cloth Hough length L difficult to understand;

3) modeling non-gaussian turbulent motion state;

Further, described step 3) improve on master mould basis, be specially:

Formula (2) in model framework changes formula (4) into and adds formula (5) and formula (6)

u＝(ep+(1-e ²) ^1/2q)σ (4)

p＝p(M _I＝0,V _I＝1,S _I,K _I) (5)

In formula, p is Pearson's random number; E is attachment coefficient, average M after ensureing p and q combination _wwith variance V _wconstant, be respectively 0 and 1; S _w, K _wthe degree of bias and the kurtosis of measured data respectively; The input-mean M of p _iwith variance V _ifor default value 0 and 1;

{S _I,K _I,e}＝f(S _w,K _w) (6)

A. formula (6), the input degree of bias S of p _iwith kurtosis K _i, and e is about S _w, K _wfuntcional relationship, specific as follows:

A) according to K _wcalculate e, K _i:

e＝-25.9969K _w ²+161.4162K _w-249.9756,K _I＝4,K _w∈(3.0200,3.0968] (7)

e＝-0.6774K _w ²+5.1138K _w-8.7387,K _I＝4,K _w∈(3.0968,3.6041] (8)

e＝0.3137K _w-0.3275,K _I＝5,K _w∈(3.6041,3.9184] (9)

e＝0.2593K _w-0.1653,K _I＝6,K _w∈(3.9184，4.1089] (10)

e＝0.2335K _w-0.0989,K _I＝7,K _w∈(4.1089,4.2782] (11)

e＝0.9,K _I＝19.2881K _w-77.0168,K _w∈(4.2782,4.4155] (12)

B) according to S _wcalculate S _i:

S _I＝f _s(S _w)＝1.7172S _w+0.0568 (13)

B. in formula (4), p is made up of Pearson IV density function and random number generator, specific as follows:

A) Pearson IV density function is expressed as:

$f\left(x\right)=k{[1+{\left(\frac{x-\mathrm{\λ}}{\mathrm{\α}}\right)}^2]}^{-m}\exp [-{v\tan }^{-1}\left(\frac{x-\mathrm{\λ}}{\mathrm{\α}}\right)],(m>1/2)---\left(14\right)$

In formula (7), k is the regular coefficient of density function, is calculated obtain by formula (8-10):

$k=\frac{\mathrm{\Γ}\left(m\right)}{\sqrt{\mathrm{\π}}\mathrm{\Γ}(m-1/2)}{\left|\frac{\mathrm{\Γ}(m+\mathrm{iv}/2)}{\mathrm{\Γ}\left(m\right)}\right|}^2---\left(15\right)$

${\left|\frac{\mathrm{\Γ}(x+\mathrm{iy})}{\mathrm{\Γ}\left(x\right)}\right|}^2=\underset{n=0}{\overset{\∞}{\mathrm{\Π}}}{[1+{\left(\frac{y}{x+n}\right)}^2]}^{-1}---\left(16\right)$

$\mathrm{\Γ}\left(x\right)=\underset{0}{\overset{\∞}{\∫}}{t}^{t-1}{e}^{-x}\mathrm{dx}---\left(17\right)$

Input parameter λ in Pearson IV density function, α, m and ν and M _w, V _w, S _wand K _wrelation:

β ₁＝S _w ²,β ₂＝K _w-3 (18)

r＝6(β ₂-β ₁-1)/2β ₂-3β ₁-6 (19)

m＝0.5r+1(20)

$v=-r(r-2)\sqrt{{\mathrm{\β}}_1}/\sqrt{16(r-1)-{\mathrm{\β}}_1{(r-2)}^2}---\left(21\right)$

$\mathrm{\α}=0.25\sqrt{\mathrm{\σ}[16(r-1)-{\mathrm{\β}}_1{(r-2)}^2]}---\left(22\right)$

$\mathrm{\λ}=M_w-0.25(r-2)\sqrt{{\mathrm{\β}}_1}\sqrt{V_w}---\left(25\right)$

B) random number generator: adopt acceptance-refusal method (Acceptance-Rejection Method) close according to any probability

Degree function generates stable random number, and these algorithm concrete steps are as follows:

x＝r(min,max) (24)

s＝r(0,upper) (25)

y＝p(x) (26)

The first step, the minimum value min of input target random number, maximal value max and p most probable value upper; Second step, computing formula (24) and formula (25), wherein r is average random number generator, and p is target density function; 3rd step, as s>y, recurring formula (24) is to formula (26); Otherwise, return x as the random number generated; 4th step, repeats step until obtain the random number of predetermined quantity.

Present invention adds Pearson's random number and to model, according to measured data, its kurtosis and the degree of bias controlled, more realistic non-gaussian turbulent phenomenon, improve turbulent flow analog simulation degree; The incorporation way of Pearson's random number is identical with variance processing mode with the average of master mould, is all to be stripped out from die body (Markov chain), is more conducive to be separated physical significance and computing; Random number generator can generate stable, meets the random number of target density function, makes the program meet good mixing condition standard.

Accompanying drawing explanation

Fig. 1 to Figure 13 is measured data situation (Wind), master mould (Gaussian) and model (Pearson) analog case contrast after improving: comprising at three dimensions (i.e. downwind u, cross-wind direction v, vertical direction w), about statistic average (Mean), variance (Variance), the degree of bias (Skewness) and kurtosis (Kurtosis), and friction velocity (u ^*) contrast situation:

Fig. 1 to Fig. 3 is the contrast situation of the average of u, v, w three dimensionality respectively;

Fig. 4 to Fig. 6 is the contrast situation of the variance of u, v, w three dimensionality respectively;

Fig. 7 to Fig. 9 is the contrast situation of the degree of bias of u, v, w three dimensionality respectively;

Figure 10 to Figure 12 is the contrast situation of the kurtosis of u, v, w three dimensionality respectively;

Figure 13 is about friction velocity u ^*contrast situation.

Embodiment

Below in conjunction with drawings and Examples, the present invention is further described.Should understand this embodiment to be only not used in for illustration of the present invention and to limit the scope of the invention, the amendment of those skilled in the art to the various equivalent form of value of the present invention all falls within the application's claims limited range.

Embodiment:

The present embodiment is the example of the non-gaussian turbulent flow modeling scheme based on the random suspended particulates model of Lagrange, specific as follows:

1) the ultrasonic anemoscope of 10Hz 3D is utilized to detect the turbulent flow data of objective area: speed u with the wind, side direction wind speed v, vertical velocity w, real time temperature t;

2) carry out processing and analyzing based on measured data, calculating with 30 minutes is interval mode input, comprising:

A. each interval mean wind speed is calculated and direction

B. velocity of wave motion u' is calculated, v', w':

$u^{\′}=u-\stackrel{\‾}{u},v^{\′}=v-\stackrel{\‾}{v},w^{\′}=w-\stackrel{\‾}{w}---\left(27\right)$

C. each Interval Statistic average M is calculated _w, variance V _w, degree of bias S _w, kurtosis K _w;

D. friction velocity u is calculated ^*:

$u^{*}={({\left(\stackrel{\‾}{u^{\′}{w}^{\′}}\right)}^2+{\left(\stackrel{\‾}{v^{\′}{w}^{\′}}\right)}^2)}^{0.25}---\left(28\right)$

In formula, form refers to the related coefficient of x and y:

$\stackrel{\‾}{x^{\′}{y}^{\′}}=E\left((x-E\left(x\right))(y-E\left(y\right))\right)=E\left(\mathrm{xy}\right)-E\left(x\right)E\left(y\right)---\left(29\right)$

Wherein E (x) is the expectation of x;

E. cloth Hough length L difficult to understand is calculated:

$L=-{\left(u^{*}\right)}^3\stackrel{\‾}{\mathrm{\θ}}\left(\mathrm{Kg}\left(\stackrel{\‾}{w^{\′}{\mathrm{\θ}}^{\′}}\right)\right)---\left(30\right)$

In formula, K is Von Karman coefficient, is set to 0.4; G is acceleration of gravity, is set to 9.81; be replace θ by average potential temperature, its concrete account form is as follows:

$\mathrm{\θ}=T^{\′}{\left(P_{0}P\right)}^{(R/c_p)}---\left(31\right)$

In formula, P ₀be canonical reference virtual temperature, be set to 1000hPa; P measures ground atmospheric pressure, is set to 133.32hPa; T' is absolute temperature, equals t+273.15; R is gas law constant, is set to 8.314; c _pbe related coefficient, be set to 1.005;

F. according to u ^*with the average of L calculating simulation

In formula, coefficient in the same formula of K (30); z ₀be roughness, be set to 0.002;

G. according to u ^*with the variances sigma of L calculating simulation:

σ _u＝σ _v＝2.4u ^*,σ _w＝1.25u ^*,L≥0 (33)

σ _u＝σ _v＝u ^*(4+0.6(-z _i/L) ^2/3) ^1/2,σ _w＝1.25u ^*(1-3z/L) ^1/3,L＜0 (34)

In formula, z _ibe set to 1000;

3) simulation of non-gaussian turbulent flow is carried out according to Model Measured input:

A. model running is in a circular space, and particulate is all set out with central point to move according to downwind, and computer memory is converted to from (x, y, z) location status that (r, θ, z) preserves particulate; Wherein r is the radius of decentering starting point, and θ is and is 0 ° with the north and the angle counterclockwise increased, θ ₁be original angle, z is height:

$\left\{\begin{array}{c}x=r\cos (q-q_1)\\ y=r\sin (q-q_1)\\ z=z\end{array}\right.---\left(35\right)$

$\left\{\begin{array}{c}r=\sqrt{x^2+y^2}\\ q=\left\{\begin{array}{c}{q}_1+{\tan }^{-1}(y/x),(x>0)\\ q_1+{\tan }^{-1}(y/x)+p,(x<0)\end{array}\right.\\ z=z\end{array}\right.---\left(36\right)$

B. Markov chain is calculated:

$q_u^{t+\mathrm{dt}}={\mathrm{aq}}_u^t+b(c_u{r}_u^{t+\mathrm{dt}}+c_w{r}_w^{t+\mathrm{dt}})---\left(37\right)$

$q_v^{t+\mathrm{dt}}={\mathrm{aq}}_v^t+{\mathrm{br}}_v^{t+\mathrm{dt}}---\left(38\right)$

$q_w^{t+\mathrm{dt}}={\mathrm{aq}}_w^t+{\mathrm{br}}_w^{t+\mathrm{dt}}+{\mathrm{gt}}_L\frac{{\mathrm{ds}}_w}{\mathrm{dz}}---\left(39\right)$

In formula, α, beta, gamma, c _uand c _wparametric Representation is:

$\mathrm{\&alpha;}=1-\mathrm{dt}/{\mathrm{\&tau;}}_L,\mathrm{\&beta;}=\sqrt{1-{\mathrm{\&alpha;}}^2},\mathrm{\&gamma;}=1-\mathrm{\&alpha;}---\left(40\right)$

dt＝0.025τ _L,τ _L＝l/σ _w

$c_w=-u^{*}/{\mathrm{\&sigma;}}_w,c_u=\sqrt{1-{c_w}^2}---\left(42\right)$

Wherein, yardstick l is expressed as:

l＝0.5z(1+5z/L) ^-1,L≥0 (43)

l＝0.5z(1-6z/L) ^1/4,L＜0 (44)

C. velocity of wave motion is calculated:

$u=(e_u(c_u{R}_u+c_w{R}_w)+\sqrt{1-{e_u}^2}{q}_u^t){\mathrm{\&sigma;}}_u---\left(45\right)$

$v=(e_v{R}_v+\sqrt{1-{e_v}^2}){\mathrm{\&sigma;}}_v---\left(46\right)$

$w=(e_w{R}_w+\sqrt{1-{e_w}^2}){\mathrm{\&sigma;}}_w---\left(47\right)$

E in formula _u, e _v, e _wit is attachment coefficient; R _u, R _v, R _wbe Pearson's random number, elaborate in foregoing invention content, be made up of Pearson IV density function and random number generator:

$R_u=R_u(S_{I_u},K_{I_u}),R_v=R_v(S_{I_v},K_{I_v}),R_w=R_w(S_{I_w},K_{I_w})---\left(48\right)$

In formula, R _u, R _v, R _weach input be calculated as follows:

A) in formula, about K _wa function:

{e,K _I}＝f _k(K _w) (49)

Specifically calculate to formula (12) according to formula (7);

B) in formula, be calculated as follows:

S _I＝f _s(S _w) (50)

Specifically calculate according to formula (13);

D. particulate displacement calculates

$\mathrm{dx}=(\stackrel{\&OverBar;}{u}\left(z\right)+u)\mathrm{dt}---\left(51\right)$

dy＝vdt (52)

dz＝(w-v _s)dt (53)

V in formula _sbe average settlement speed, be set to-4.7303e-04;

E. the turbulent velocity gone out of each interval simulation is:

$U=\stackrel{\&OverBar;}{u}\left(z\right)+u,V=v,W=w-v_s---\left(54\right)$

F. the friction velocity u of each interval turbulent flow U, V, W is calculated ^*sum test statistics: average, variance, the degree of bias, kurtosis

G. Fig. 1 to Figure 13 compares and improves rear modeling situation (Pearson), with actual measurement data situation (Wind), and the u of master mould analog case (Gaussian) ^*and at three direction u, the statistic of v, w; After improvement, the degree of bias and kurtosis analog case are obviously improved, and very close with actual measurement situation, fidelity is high; Average, variance, friction velocity and master mould analog case similar.

Claims (1)

1. based on the non-gaussian turbulent flow modeling scheme of the random suspended particulates model of Lagrange, it is characterized in that the simulation of the program mainly for non-gaussian turbulent flow, the basis of former Gauss model is introduced Pearson's random number, to control its degree of bias and kurtosis, and ensure Stability and veracity, implementation step is as follows:

Spread of particles is the core of LSM, and its framework is expressed as on a movement dimension:

$\mathrm{dx}=(u+\stackrel{\&OverBar;}{u})\mathrm{dt}---\left(1\right)$

u＝qσ (2)

q ^t+dt＝αq ^t+βr ^t+dt (3)

Non-gaussian turbulent flow modeling scheme based on Lagrange random suspended particulates model is as follows:

1) anemoscope is used to obtain turbulent flow data on the spot: speed u with the wind, side direction wind speed v, vertical velocity w, real time temperature t;

2) staging treating analyze turbulent flow data, computation model inputs: mean wind speed and direction degree of bias S _w, kurtosis K _w, friction velocity u ^*, cloth Hough length L difficult to understand;

3) modeling non-gaussian turbulent motion state;

Further, described step 3) improve on master mould basis, be specially:

Formula (2) in model framework changes formula (4) into and adds formula (5) and formula (6)

u＝(ep+(1-e ²) ^1/2q)σ (4)

p＝p(M _I＝0,V _I＝1,S _I,K _I) (5)

In formula, p is Pearson's random number; E is attachment coefficient, average M after ensureing p and q combination _wwith variance V _wconstant, be respectively 0 and 1; S _w, K _wthe degree of bias and the kurtosis of measured data respectively; The input-mean M of p _iwith variance V _ifor default value 0 and 1;

{S _I,K _I,e}＝f(S _w,K _w) (6)

A. formula (6), the input degree of bias S of p _iwith kurtosis K _i, and e is about S _w, K _wfuntcional relationship, specific as follows:

A) according to K _wcalculate e, K _i:

e＝-25.9969K _w ²+161.4162K _w-249.9756,K _I＝4,K _w∈(3.0200,3.0968] (7)

e＝-0.6774K _w ²+5.1138K _w-8.7387,K _I＝4,K _w∈(3.0968,3.6041] (8)

e＝0.3137K _w-0.3275,K _I＝5,K _w∈(3.6041,3.9184] (9)

e＝0.2593K _w-0.1653,K _I＝6,K _w∈(3.9184，4.1089] (10)

e＝0.2335K _w-0.0989,K _I＝7,K _w∈(4.1089,4.2782] (11)

e＝0.9,K _I＝19.2881K _w-77.0168,K _w∈(4.2782,4.4155] (12)

B) according to S _wcalculate S _i:

S _I＝f _s(S _w)＝1.7172S _w+0.0568 (13)

B. in formula (4), p is made up of Pearson IV density function and random number generator, specific as follows:

A) Pearson IV density function is expressed as:

$f\left(x\right)=k{[1+{\left(\frac{x-\mathrm{\&lambda;}}{\mathrm{\&alpha;}}\right)}^2]}^{-m}\exp [-{v\tan }^{-1}\left(\frac{x-\mathrm{\&lambda;}}{\mathrm{\&alpha;}}\right)],(m>1/2)---\left(14\right)$

In formula (7), k is the regular coefficient of density function, is calculated obtain by formula (8-10):

$k=\frac{\mathrm{\&Gamma;}\left(m\right)}{\sqrt{\mathrm{\&pi;}}\mathrm{\&Gamma;}(m-1/2)}{\left|\frac{\mathrm{\&Gamma;}(m+\mathrm{iv}/2)}{\mathrm{\&Gamma;}\left(m\right)}\right|}^2---\left(15\right)$

${\left|\frac{\mathrm{\&Gamma;}(x+\mathrm{iy})}{\mathrm{\&Gamma;}\left(x\right)}\right|}^2=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Pi;}}}{[1+{\left(\frac{y}{x+n}\right)}^2]}^{-1}---\left(16\right)$

$\mathrm{\&Gamma;}\left(x\right)=\underset{0}{\overset{\&infin;}{\&Integral;}}{t}^{t-1}{e}^{-x}\mathrm{dx}---\left(17\right)$

Input parameter λ in Pearson IV density function, α, m and ν and M _w, V _w, S _wand K _wrelation:

β ₁＝S _w ²,β ₂＝K _w-3 (18)

r＝6(β ₂-β ₁-1)/2β ₂-3β ₁-6 (19)

m＝0.5r+1 (20)

$v=-r(r-2)\sqrt{{\mathrm{\&beta;}}_1}/\sqrt{16(r-1)-{\mathrm{\&beta;}}_1{(r-2)}^2}---\left(21\right)$

$\mathrm{\&alpha;}=0.25\sqrt{\mathrm{\&sigma;}[16(r-1)-{\mathrm{\&beta;}}_1{(r-2)}^2]}---\left(22\right)$

$\mathrm{\&lambda;}=M_w-0.25(r-2)\sqrt{{\mathrm{\&beta;}}_1}\sqrt{V_w}---\left(25\right)$

B) random number generator: adopt acceptance-refusal method (Acceptance-Rejection Method) basis

Independent Sources with Any Probability Density Function function generates stable random number, and these algorithm concrete steps are as follows:

x＝r(min,max) (24)

s＝r(0,upper) (25)

y＝p(x) (26)

The first step, the minimum value min of input target random number, maximal value max and p most probable value upper; Second step, computing formula (24) and formula (25), wherein r is average random number generator, and p is target density function; 3rd step, as s>y, recurring formula (24) is to formula (26); Otherwise, return x as the random number generated; 4th step, repeats step until obtain the random number of predetermined quantity.