CN104077478A - Numerical simulation method for downburst non-stationary fluctuating wind speed - Google Patents

Abstract

The invention provides a numerical simulation method for a downburst non-stationary fluctuating wind speed. The method comprises the steps of modulating a power spectrum through a time varying average wind speed model of the downburst and an inhomogeneous modulation function related to time frequency so as to obtain the time varying power spectrum of the downburst non-stationary fluctuating wind speed; building a TAR (Terrain Avoidance Radar) time varying model of a stationary stochastic process by further considering the time varying characteristic of an AR (Access Rate) model of the stationary stochastic process; and effectively stimulating the downburst non-stationary fluctuating wind speed by the time varying power spectrum and the TAR time varying model. The relevance of the downburst non-stationary fluctuating wind speed simulated by the numerical simulation method provided by the invention can be weakened along with the increasing of the distances among different positions, the amplitude can be improved to be consistent with the characteristics of a practical wind field as the time varying average wind speed is improved; the power spectrum of the simulated downburst non-stationary fluctuating wind speed has an obvious time varying characteristic, and is consistent with the time varying characteristic of a target time varying spectrum, and the average power spectrum of sample statistics and relevant functions are consistent with the targets.

Description

A kind of method for numerical simulation of downburst non-stationary pulsation wind speed

Technical field

The invention belongs to storm analogue technique field, relate in particular to a kind of method for numerical simulation of downburst non-stationary pulsation wind speed.

Background technology

In Thunderstorm Weather, tend to form local strong down draft and fiercely impact ground, produce that a kind of to have sudden and destructive high wind be downburst.People recognize that the disastrous of downburst is in the aviation accident occurring when the 727 flight landing of Boeing of the U.S. in 1975 first.After this accident investigation, Fujita has proposed downburst first, can produce on ground a kind of strong down draft of the above divergence wind of 17.9m/s ^[1].Chay and Letchford ^[2]from detailed in essence elaboration the generation type of downburst wind speed.In thunderstorm environment, the frequency that the micro-downburst that yardstick is less occurs is very high, at the maximum wind velocity of generation near the ground, can reach 75m/s ^[3].Obviously its destructiveness is large especially, belongs to disastrous high wind near the ground.Downburst all has generation in the world wides such as the U.S., Japan, China, Australia, and buildings main body and building enclosure thereof can produce serious damage, destruction under its effect, even collapse ^[4,5].

People recognize importance and the necessity of downburst research gradually, and researchist has launched research widely by methods such as field observation, experimental simulation, theoretical analysis and numerical simulations to downburst both at home and abroad.By the several to downburst, survey, it is near the ground that foreign scholar finds in downburst wind speed field that maximum wind velocity appears at, and relatively little apart from the wind speed of ground higher position, this obvious difference and traditional atmospheric boundary layer wind speed field distribute ^[3].Holmes and Oliver ^[6]by impacting jet theoretical research the mean wind speed that produces of a certain fixed position in downburst traveling process, and the model of temporal evolution mean wind speed has been proposed.Wood etc. ^[7]by the Wind Tunnel Data of down draft, the wind speed field mean wind speed model matching with Computational Fluid Dynamic Analysis model has been proposed.Savory etc. ^[8]use model investigation that Holmes sets up the electric transmission pole tower that hits under sudden and violent stream effect destroy situation.Because this model is not considered the random fluctuation composition of wind speed field, this may underestimate the dynamic response of structure under Downburst Wind Loads effect.Downburst wind speed has very strong non-stationary.China Wind Engineering domain expert Qu Weilian professor ^[9]downburst is decomposed into a pulsation wind speed that becomes mean wind speed and a modulation non-stationary when definite, has carried out numerical simulation study.Li Chunxiang etc. ^[10]the even modulation Nonstationary random field analogy method that adopts Deodatis to propose has been simulated downburst non-stationary pulsation wind speed.Zhang Wenfu etc. ^[11]steady pulsation wind speed by the even modulation of time-varying function based on AR model has become to have the downburst non-stationary pulsation wind speed of spatial coherence next life.The Evolutionary Spectral proposing according to Priestley is theoretical ^[12], the simulation of non-stationary pulsation wind speed, the time-varying power spectrum that should modulate acquisition non-stationary wind speed to power spectrum by a non-uniformly modulated function relevant with temporal frequency is Evolutionary Spectral, then by the analogy method of nonstationary random process, simulates.Yet, in order to simplify, conventionally non-uniformly modulated function is assumed to only relevant with the time even modulating function, the simulation of nonstationary random process is converted into the modulation of even modulating function to stationary stochastic process, avoided according to the difficult point of time-varying power spectrum simulation nonstationary random process.And, at present for the simulation of non-stationary pulsation wind speed, conventionally by time the even modulating function that equal function of wind speed is assumed to steady pulsation wind speed that flattens do not have correlation theory foundation.For this reason, Li Jinhua etc. ^[13]the non-stationary wind speed of pulsing is separated into some sections is can be approximated to be the time series in short-term of steady pulsation wind speed in enough short time Δ t, thereby strictly derive the Evolutionary Spectral of the non-uniformly modulated function relevant with time and frequency and corresponding non-stationary pulsation wind speed.

Summary of the invention

The object of the present invention is to provide a kind of method for numerical simulation of downburst non-stationary pulsation wind speed, be intended to solve in prior art and avoided according to the problem of time-varying power spectrum simulation nonstationary random process difficult point.

The present invention is achieved in that a kind of method for numerical simulation of downburst non-stationary pulsation wind speed, comprises the following steps:

S1, theoretical according to Evolutionary Spectral, by downburst time become the mean wind speed model non-uniformly modulated function relevant with temporal frequency with power spectrum modulated to obtain the downburst non-stationary wind speed time-varying power spectrum of pulsing;

S2, the AR model of stationary stochastic process is further considered to time varying characteristic sets up the TAR time-varying model of nonstationary random process;

S3, by described time-varying power spectrum, TAR time-varying model, downburst non-stationary pulsation wind speed is effectively simulated.

Preferably, in step S1, described non-uniformly modulated function is Kaimal modulating function:

$A(\mathrm{\ω},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{U}_j}}{1+50\frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}}$

In formula: ω is circular frequency; z _jvertical ground height for space point; for some place, space time become mean wind speed; statistical average wind speed for some place, space non-stationary wind speed.

Preferably, described downburst non-stationary pulsation wind speed time-varying power spectrum comprises the non-uniformly modulated function after Davenport spectrum, Harris spectrum, the modulation of Simiu spectrum.

Preferably, in step S1, the non-uniformly modulated function after the modulation of described Davenport spectrum is:

$A(\mathrm{\ω},t)={\left(\frac{1+{\left(\frac{1200\mathrm{\ω}}{2\mathrm{\π}{U}_{10}}\right)}^2}{1+{\left(\frac{1200\mathrm{\ω}}{2\mathrm{\π}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{2/3};$

Non-uniformly modulated function after the modulation of described Harris spectrum is:

$A(\mathrm{\ω},t)=\sqrt{\frac{{\tilde{U}}_{10}\left(t\right)}{U_{10}}{\left(\frac{2+{\left(\frac{1800\mathrm{\ω}}{2\mathrm{\π}{U}_{10}}\right)}^2}{2+{\left(\frac{1800\mathrm{\ω}}{2\mathrm{\π}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{5/6}}$

Non-uniformly modulated function after the modulation of described Simiu spectrum is:

$\left\{\begin{array}{cc}A(\mathrm{\ω},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{U}_j}}{1+50\frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}},& \frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{\tilde{U}}_j\left(t\right)}\≤0.2\left(1\right)\\ A\left(t\right)={\left(\frac{{\tilde{U}}_j\left(t\right)}{U_j}\right)}^{4/3},& \frac{\mathrm{\ω}{z}_j}{2\mathrm{\π}{\tilde{U}}_j\left(t\right)}>0.2\left(2\right)\end{array}\right.;$

In various above, ω is circular frequency; z _jvertical ground height for space point; for some place, space time become mean wind speed; statistical average wind speed for some place, space non-stationary wind speed; be 10 meters of eminences time become mean wind speed; it is the statistical average wind speed of 10 meters of eminence non-stationary wind speed.

Preferably, in step S2, described AR model is autoregressive model:

$f\left(t\right)=\underset{i=1}{\overset{p}{\mathrm{\Σ}}}{A}_{i}f(t-\mathrm{i\Δt})+\mathrm{Lw}\left(t\right);$

In formula, A _irepresent autoregressive model coefficient, L represents model coefficient undetermined.

Preferably, in step S2, described TAR time-varying model function definition is:

$f\left(t\right)=\underset{i=1}{\overset{p}{\mathrm{\Σ}}}{A}_i\left(t\right)f(t-\mathrm{i\Δt})+L\left(t\right)w\left(t\right);$

In formula, time become autoregressive model coefficient A _i(t) available functions is defined as:

$R_{\mathrm{ff}}(t,\mathrm{j\Δt})=\underset{i=1}{\overset{p}{\mathrm{\Σ}}}{A}_i\left(t\right)R_{\mathrm{ff}}(t-\mathrm{i\Δt},\mathrm{j\Δt}-\mathrm{i\Δt}),$

Time-varying model coefficient L (t) by function definition is:

$L\left(t\right)=\sqrt{\underset{i=0}{\overset{p}{\mathrm{\Σ}}}{A}_i\left(t\right)R_{\mathrm{ff}}(t-\mathrm{i\Δt},-\mathrm{i\Δt})}$

In above formula, t=1,2,3, F (t) is zero-mean nonstationary random process; R _ff(t, j Δ t) be f (t) at t related function constantly; A _i(t) for time become autoregressive model coefficient; L (t) is time-varying model coefficient undetermined; The white noise sequence that w (t) is 1 for variance, and meet related function $R_{\mathrm{ww}}(t,\mathrm{j\Δt})=E\left[w\left(t\right)w(t-\mathrm{j\Δt})\right]=\left\{\begin{array}{cc}1,& j=0\\ 0,& j\≠0\end{array}\right.$ , p is model order.

Downburst wind speed has very strong non-stationary, and not only mean wind speed has time varying characteristic, and the power spectrum of pulsation wind speed also has time variation, shows obvious non-stationary characteristic.The Evolutionary Spectral proposing according to Priestley is theoretical, the simulation of non-stationary pulsation wind speed, the time-varying power spectrum that should modulate acquisition non-stationary wind speed to power spectrum by a non-uniformly modulated function relevant with temporal frequency is Evolutionary Spectral, then based on this Evolutionary Spectral, nonstationary random process is simulated.Yet, for the simplification of calculating, for the simulation of non-stationary pulsation wind speed, conventionally modulating function is assumed to only relevant with time function at present, make the simulation of non-stationary pulsation wind speed be converted into the modulation of the function of time to steady pulsation wind speed, this does not have correlation theory foundation.For this reason, the present invention will be theoretical according to Evolutionary Spectral, first by downburst time become the mean wind speed model non-uniformly modulated function relevant with temporal frequency with power spectrum modulated to obtain downburst non-stationary pulsation wind speed time-varying power spectrum, then the AR model of stationary stochastic process is further considered to time varying characteristic sets up the TAR time-varying model of nonstationary random process, finally pass through time-varying power spectrum, TAR time-varying model is realized the simulation of downburst non-stationary pulsation wind speed, the correlativity of the downburst non-stationary pulsation wind speed by simulation, power spectrumanalysis has been verified the validity of simulation.

Accompanying drawing explanation

Fig. 1 is the flow chart of steps of the method for numerical simulation of downburst non-stationary pulsation wind speed of the present invention;

Fig. 2 is the downburst vector composite diagram of wind speed and point-to-point speed radially in the embodiment of the present invention;

Fig. 3 is the equal anemobiagraph that flattens during downburst in the embodiment of the present invention;

Fig. 4 is the Kaimal non-uniformly modulated functional arrangement of downburst in the embodiment of the present invention;

Fig. 5 is downburst Wind Velocity History and non-stationary ripple component figure in the embodiment of the present invention;

Fig. 6 is the time-varying power spectrum figure of downburst non-stationary pulsation wind speed in the embodiment of the present invention;

Fig. 7 is that in the embodiment of the present invention, the comparison diagram that spectrum and target are composed is estimated in the statistical average of single sample downburst non-stationary pulsation wind speed;

Fig. 8 is the comparison diagram of single sample downburst non-stationary pulsation wind speed statistical average autocorrelation function and target autocorrelation function in the embodiment of the present invention;

Fig. 9 is single sample downburst non-stationary pulsation wind speed statistical average autocorrelation function in the embodiment of the present invention.

Embodiment

In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with drawings and Examples, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.

The method for numerical simulation that the invention provides a kind of downburst non-stationary pulsation wind speed, as shown in Figure 1, comprises the following steps:

S1, theoretical according to Evolutionary Spectral, by downburst time become the mean wind speed model non-uniformly modulated function relevant with temporal frequency with power spectrum modulated to obtain the downburst non-stationary wind speed time-varying power spectrum of pulsing

In step S1, during downburst, become mean wind speed specifically describe into:

The describing method of wind speed in reference atmosphere boundary layer is considered the non-stationary of downburst simultaneously, at a certain position z height t downburst wind speed constantly, can be expressed as time dependent mean wind speed with pulsation wind speed u (z, t) sum.Different from atmospheric boundary layer near-earth wind, downburst Wind Velocity History has stronger non-stationary property, and downburst mean wind speed is a kind of instantaneous average time-varying process.The downburst mean wind speed with time varying characteristic, can be expressed as the product of a vertical distribution function and a function of time, that is:

$\tilde{U}(z,t)=V\left(z\right)\×f\left(t\right)$

(1)

In formula: f (t) is the function of time, its maximal value is 1; V (z) is the vertical distribution function of maximum mean wind speed, and Vicroy, Wood, OsegueraandBowles have proposed respectively 3 kinds of maximum mean wind speed vertical wind sections ^[14].

The vertical distributed model of Vicroy

$V\left(z\right)=1.22\×[e^{-0.15z/z_{\max }}-e^{-3.2175z/z_{\max }}]\×V_{\max }$

(2)

The vertical distributed model of Wood

$V\left(z\right)=1.55{\left(\frac{z}{\mathrm{\δ}}\right)}^{1/6}[1-\mathrm{erf}\left(0.7\frac{z}{\mathrm{\δ}}\right)]\×V_{\max }$

(3)

The vertical distributed model of OsegueraandBowles

$V\left(z\right)=\left(\frac{\mathrm{\λ}{R}^2}{2r}\right)[1-e^{-{(r/R)}^2}](e^{-z/z^{*}}-e^{-z/\mathrm{\ϵ}})$

(4)

In above formula, V _maxfor the maximum wind velocity in vertical distribution wind speed; z _maxfor the residing height and position of maximum wind velocity; Erf is error function; δ is half residing height and position of maximum wind velocity; R is the radial position apart from downburst wind field center; R is the radiation radius of downburst wind field; z ^*for the feature height beyond atmospheric boundary layer; ε is the feature height in atmospheric boundary layer; λ is scale factor.

In impacting jet theory, radially wind speed is axisymmetric, a certain At The Height radially wind speed can be expressed as ^[6]:

$V_r\left(r\right)=\left\{\begin{array}{cc}\mathrm{\&Pi;}\&times;V_{r,\max }\&times;(r/r_{\max })& (0\&le;r<r_{\max })\\ \mathrm{\&Pi;}\&times;V_{r,\max }\&times;\exp (-{[(r-r_{\max })/R_r]}^2)& (r\&GreaterEqual;r_{\max })\end{array}\right.$

(5)

In formula: V _{r, max}it is the maximum wind velocity of a certain At The Height in wind speed field; r _maxit is the horizontal range between maximum wind velocity point and downburst center; R is the distance between observation station and downburst center; R _rit is a radical length scale-up factor; Π is strength factor, shows thunderstorm intensity over time.Consider that downburst is certainly in movement, so the distance r of observation station and thunderstorm center _tbe time dependent, the mean wind speed of observation station can be expressed as radially wind speed V _rwith downburst point-to-point speed V _ovector synthetic,

V _p(t)＝V _r(r _t)+V _ο

(6)

Function of time f (t) may be defined as the mean wind speed of spatial point any time and the ratio of maximum mean wind speed, is expressed as the time dependent function of vertical wind section, that is:

f(t)＝V _p(t)/max|V _p(t)|

(7)

In step S1, downburst non-stationary pulsation wind speed is specially:

Different from atmospheric boundary layer wind speed, the mean wind speed of downburst has obvious time varying characteristic, and its pulsation wind speed has stronger non-stationary characteristic.Evolutionary Spectral based on nonstationary random process is theoretical, and first the simulation of the non-stationary pulsation wind speed of downburst, can be modulated and obtain time-varying power spectrum a certain power spectrum by a modulating function, then by nonstationary random process modeling algorithm, simulates.This simulation process not only needs to obtain effective modulating function, but also need set up the modeling algorithm of nonstationary random process.

In step S1, described non-uniformly modulated function is specially:

According to the simulation theory of nonstationary random process, the numerical simulation key of non-stationary pulsation wind speed is to obtain time-varying power spectrum.Obtaining of time-varying power spectrum can be by field measurement non-stationary wind speed is carried out to time varying spectrum estimation, or according to Evolutionary Spectral theory, by modulating function, power spectrum to be modulated to obtain time-varying power spectrum be Evolutionary Spectral.Field measurement is also infeasible for most numerical simulation study person, and convenient feasible method is to adopt power spectrum modulation.In order to obtain effective modulating function, Li Jinhua etc. ^[13]the non-stationary wind speed of pulsing is separated into the some sections of time serieses in short-term that can be approximated to be steady pulsation wind speed in enough short time Δ t, based on Kaimal spectrum, derive the Kaimal spectrum non-uniformly modulated function relevant with time and frequency, be called for short Kaimal modulating function:

$A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{U}_j}}{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}}$

(8)

In formula: ω is circular frequency; z _jvertical ground height for space point; for some place, space time become mean wind speed; statistical average wind speed for some place, space non-stationary wind speed.By the method, can further derive acquisition Davenport, Harris, Simiu compose corresponding non-uniformly modulated function.Wherein, Davenport modulating function:

$A(\mathrm{\&omega;},t)={\left(\frac{1+{\left(\frac{1200\mathrm{\&omega;}}{2\mathrm{\&pi;}{U}_{10}}\right)}^2}{1+{\left(\frac{1200\mathrm{\&omega;}}{2\mathrm{\&pi;}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{2/3}$

(9)

Harris modulating function:

$A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_{10}\left(t\right)}{U_{10}}{\left(\frac{2+{\left(\frac{1800\mathrm{\&omega;}}{2\mathrm{\&pi;}{U}_{10}}\right)}^2}{2+{\left(\frac{1800\mathrm{\&omega;}}{2\mathrm{\&pi;}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{5/6}}$

(10)

Simiu modulating function:

$\left\{\begin{array}{cc}A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{U}_j}}{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}},& \frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}\&le;0.2\left(1\right)\\ A\left(t\right)={\left(\frac{{\tilde{U}}_j\left(t\right)}{U_j}\right)}^{4/3},& \frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}>0.2\left(2\right)\end{array}\right.;$

(11)

In step S1, described Evolutionary Spectral is specially:

According to " Evolutionary Spectral " theory of Pristley ^[12], a zero-mean nonstationary random process f (t) can be expressed as:

$f\left(t\right)={\&Integral;}_{-\&infin;}^{+\&infin;}A(\mathrm{\&omega;},t)e^{\mathrm{i\&omega;t}}\mathrm{dZ}\left(\mathrm{\&omega;}\right)$

(12)

In formula, A (ω, t) is non-uniformly modulated function; Z (ω) is an orthogonal process, and meets

E[dZ(ω)]＝0

(13)

E[dZ ^*(ω)dZ(ω′)]＝J(ω)δ(ω-ω′)dωdω′

(14)

Wherein * represents conjugation, and δ is Dirac function.The average of nonstationary random process is

$E\left[f\left(t\right)\right]={\&Integral;}_{-\&infin;}^{+\&infin;}A(\mathrm{\&omega;},t)e^{\mathrm{i\&omega;t}}E\left[\mathrm{dZ}\left(\mathrm{\&omega;}\right)\right]=0$

(15)

Related function is:

$\begin{array}{c}{R}_{\mathrm{ff}}(t,\mathrm{\&tau;})=E\left[f^{*}\left(t\right)f(t+\mathrm{\&tau;})\right]\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}{\&Integral;}_{-\&infin;}^{+\&infin;}{A}^{*(\mathrm{\&omega;},t)A({\mathrm{\&omega;}}^{\&prime;},t+\mathrm{\&tau;})e^{-\mathrm{i\&omega;t}}{e}^{i{\mathrm{\&omega;}}^{\&prime;}(t+\mathrm{\&tau;})}E\left[d{Z}^{*}\left({\mathrm{\&omega;}}^{\&prime;}\right)\right]}\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}{A}^{*}(\mathrm{\&omega;},t)A\left(\text{\&omega;,t+\&tau;}\right)e^{\mathrm{i\&omega;\&tau;}}J\left(\mathrm{\&omega;}\right)\mathrm{d\&omega;}\end{array}$

(16)

When τ=0,

$E\left[f^2\left(t\right)\right]={\&Integral;}_{-\&infin;}^{+\&infin;}{\left|A(\mathrm{\&omega;},t)\right|}^{2}J\left(\mathrm{\&omega;}\right)\mathrm{d\&omega;}$

(17)

Therefore, Evolutionary Spectral S (ω, t) can represent the non-uniformly modulated of power spectrum by a time-frequency function A (ω, t) ^[16],

S(ω,t)＝|A(ω,t)| ²J(ω)

(18)

According to formula (17) and formula (18), can set up the related function of nonstationary random process and the relation between time-varying power spectrum:

$\begin{array}{c}{R}_{\mathrm{ff}}(t,\mathrm{\&tau;})=E\left[f^{*}\left(t\right)f(t+\mathrm{\&tau;})\right]\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}{\left[S^{*}(\mathrm{\&omega;},t)\right]}^{1/2}{\left[S(\mathrm{\&omega;},t+\mathrm{\&tau;})\right]}^{1/2}{e}^{\mathrm{i\&omega;\&tau;}}\mathrm{d\&omega;}\end{array}$

(19)

S2, the AR model of stationary stochastic process is further considered to time varying characteristic sets up the TAR time-varying model of nonstationary random process

In step S2, the process of establishing of non-stationary pulsation wind speed TAR time-varying model is specific as follows:

In the simulation of stationary stochastic process, comparatively ripe method mainly contains spectral representation and linear filter method.In order further to set up the analogy method of nonstationary random process, Liang etc. ^[16]derived in detail and considered the spectral representation method of the nonstationary random process of time varying characteristic.In Simulation of Stationary Stochastic Processes, the simulation precision of spectral representation method is higher, but simulation precision is lower.For the simulation of wind of actual heavy construction, often need on spectral representation method basis, improve simulation precision.Spectral representation is further considered to time varying characteristic simulation nonstationary random process, and obviously its simulation precision is difficult to meet the application of Practical Project more.Linear filtering method, because calculated amount is little, speed is fast, has been widely used in the simulation of stochastic process, but mainly for the simulation of Stationary Gauss Random process.Li Jinhua and Li Chunxiang ^[17]consider the non-Gaussian feature of stochastic process, realized the non-Gaussian random process simulation based on AR, arma modeling in linear filter method.This section is further considered time varying characteristic by continuing to the AR model in linear filtering method, sets up the TAR time-varying model of nonstationary random process.

If TAR time-varying model exponent number is p, nonstationary random process analog sample is counted as N, and sampling time interval is Δ t, and the simulation of the nonstationary random process based on TAR (p) time-varying model formula can be expressed as:

$f\left(t\right)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t\right)f(t-\mathrm{i\&Delta;t})+L\left(t\right)w\left(t\right)$

(20)

In formula: f (t) is zero-mean nonstationary random process; A _i(t) for time become autoregressive model coefficient; L (t) is time-varying model coefficient undetermined; The white noise sequence that w (t) is 1 for variance, and meet $R_{\mathrm{ww}}(t,\mathrm{j\&Delta;t})=E\left[w\left(t\right)w(t-\mathrm{j\&Delta;t})\right]=\left\{\begin{array}{cc}1,& j=0\\ 0,& j\&NotEqual;0\end{array}\right.$ 。The foundation key of TAR time-varying model is to determine time-varying model coefficient A _iand L (t) (t).

Carve t=t ' time at a time, have

$f\left(t^{\&prime;}\right)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)f(t^{\&prime;}-\mathrm{i\&Delta;t})+L\left(t^{\&prime;}\right)w\left(t^{\&prime;}\right)$

(21)

For A _i(t ') determines, f (t '-j Δ t) taken advantage of in the right side in formula (21) both sides simultaneously, and get mathematical expectation, has

$E\left[f\left(t^{\&prime;}\right)f(t^{\&prime;}-\mathrm{j\&Delta;t})\right]=E\left[\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)f(t^{\&prime;}-\mathrm{i\&Delta;t})f(t^{\&prime;}-\mathrm{j\&Delta;t})\right]+E\left[L\left(t^{\&prime;}\right)w\left(t^{\&prime;}\right)f(t^{\&prime;}-\mathrm{j\&Delta;t})\right]$

(22)

According to the definition of related function, can obtain left item=R _ff(t ', j Δ t); ; Right binomial=L (t ') R _wf(t ', j Δ t),

$R_{\mathrm{ff}}(t^{\&prime;},\mathrm{j\&Delta;t})=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{ff}}(t^{\&prime;}-\mathrm{i\&Delta;t},\mathrm{j\&Delta;t}-\mathrm{i\&Delta;t})+L\left(t^{\&prime;}\right)R_{\mathrm{wf}}(t^{\&prime;},\mathrm{j\&Delta;t})$

(23)

For right binomial, R _wf(t ', j Δ t) can be regarded as the cross correlation function of w (t ') and f (t '-j Δ t).The input that w (t) is this system, and the output that f (t) is system.Current output only depends on current and input in the past, and irrelevant with input in the future, so w (t ') is independent mutually with f (t '-j Δ t), therefore

R _wf(t′,jΔt)＝0

(24)

By formula (24) substitution formula (23), can determine A _i(t ')

$R_{\mathrm{ff}}(t^{\&prime;},\mathrm{j\&Delta;t})=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{ff}}(t^{\&prime;}-\mathrm{i\&Delta;t},\mathrm{j\&Delta;t}-\mathrm{i\&Delta;t})$

(25)

Its expansion is

For determining of L (t '), w (t ') is taken advantage of in the right side in formula (21) both sides simultaneously, and get mathematical expectation,

Have

$R_{\mathrm{fw}}(t^{\&prime;},0)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{fw}}(t^{\&prime;}-\mathrm{i\&Delta;t},\mathrm{i\&Delta;t})+L\left(t^{\&prime;}\right)R_{\mathrm{ww}}(t^{\&prime;},0)$

(26)

Current input w (t ') is with to export in the past f (t '-i Δ t) independent mutually,

R _fw(t′-iΔt,iΔt)＝0

(27)

By formula (27) substitution formula (26), can obtain

R _fw(t′,0)＝L(t′)R _ww(t′,0)

(28)

Characteristic according to white noise, has R _ww(t ', 0)=1,

R _fw(t′,0)＝L(t′)

(29)

Again f (t ') is taken advantage of in the right side in formula (21) both sides simultaneously, and get mathematical expectation, have

$R_{\mathrm{ff}}(t^{\&prime;},0)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{ff}}(t^{\&prime;}-\mathrm{i\&Delta;t},-\mathrm{i\&Delta;t})+L\left(t^{\&prime;}\right)R_{\mathrm{wf}}(t^{\&prime;},0)$

(30)

Again because

R _wf(t′,0)＝E[w(t′)f(t′)]＝E[f(t′)w(t′)]＝R _fw(t′,0)

(31)

By formula (29) and (31) substitution formula (30), have

$R_{\mathrm{ff}}(t^{\&prime;},0)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{ff}}(t^{\&prime;}-\mathrm{i\&Delta;t},-\mathrm{i\&Delta;t})+L^2\left(t^{\&prime;}\right)$

(32)

Make A ₀(t ')=-1,

$L\left(t^{\&prime;}\right)=\sqrt{\underset{i=0}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t^{\&prime;}\right)R_{\mathrm{ff}}(t^{\&prime;}-\mathrm{i\&Delta;t},-\mathrm{i\&Delta;t})}$

(33)

Therefore, time-varying model coefficient A _i(t ') and L (t '), t '=1,2,3 ..., can according to formula (25) and (33), determine respectively.

S3, by described time-varying power spectrum, TAR time-varying model, downburst non-stationary pulsation wind speed is effectively simulated

In step S3, the equal wind speed simulation that flattens during downburst is specially:

Be positioned at 6 downburst Wind observation point P of sustained height ₁～P ₆, as shown in Figure 2.Initial time, observation station and downburst centre distance are r ₀=3500m, position angle is respectively θ ₀₁=0 °, θ ₀₂=15 °, θ ₀₃=30 °, θ ₀₄=45 °, θ ₀₅=60 °, θ ₀₆=90 °.Consider the movement of downburst self, the distance r of a certain height observation station and thunderstorm center _ttemporal evolution, the mean wind speed of observation station can be expressed as radially wind speed V _rwith downburst point-to-point speed V _ovector synthetic:

$V_p\left(t\right)=\sqrt{V_r^2+V_o^2+2V_r\&CenterDot;V_o\frac{r_0\cos \left({\mathrm{\&theta;}}_0\right)-V_o\&CenterDot;t}{\sqrt{{\left(r_0\sin \left({\mathrm{\&theta;}}_0\right)\right)}^2+{(r_0\cos \left({\mathrm{\&theta;}}_0\right)-V_o\&CenterDot;t)}^2}}}$

(34)

The vertical distributed model V of mean wind speed (z) of downburst adopts Vicroy model, and becoming mean wind speed during downburst can be expressed as:

$\tilde{U}(z,t)=V\left(z\right)\&CenterDot;\frac{V_p\left(t\right)}{\max \left|V_p\left(t\right)\right|}$

(35)

Consider the maximum wind velocity V that downburst vertically distributes in wind speed _max=80m/s, the residing height and position z of maximum wind velocity _max=67m; A certain At The Height maximum wind velocity V radially in wind speed field _{r, max}get 47m/s, and the horizontal range r between downburst center _maxget 1000m, radical length scale-up factor R _rget 700m, thunderstorm intensity Π over time, can be expressed as:

$\mathrm{\&Pi;}=\left\{\begin{array}{cc}t/5& 0\&le;t\&le;5\min \\ \exp [-(t-5)/11.542]& t>5\min \end{array}\right.$

(36)

The point-to-point speed of downburst is got V _o=8m/s.

Each observation station time become mean wind speed as shown in Figure 3, for each identical observation station of radial distance, time become mean wind speed fluctuation along with azimuthal increase, weaken.From formula (8)～(11), the power spectrum non-uniformly modulated function of non-stationary pulsation wind speed with time to become mean wind speed relevant, because the undulatory property of the equal wind speed that now flattens will further affect the time variation of non-stationary pulsating wind power spectrum.Consider observation station P1 time become mean wind speed and there is stronger undulatory property, herein the TAR time-varying model based on setting up is further simulated the non-stationary pulsation wind speed of this vertical difference in position.

In step S3, the downburst non-stationary pulsation wind speed simulation based on TAR time-varying model is specially:

Now adopt the TAR time-varying model of setting up to simulate the non-stationary pulsation wind speed of vertical (Z1=15m, Z2=25m, Z3=45m) three differences at 3500 meters of, initial distance thunderstorm center in downburst course.The upper limit cut-off circular frequency that simulation adopts is taken as 4rad/s, and the time span of simulation is 800s, TAR time-varying model exponent number P=15.The time-varying power spectrum of downburst non-stationary pulsation wind speed adopts Kaimal non-uniformly modulated function to modulate Kaimal spectrum.The downburst Kaimal non-uniformly modulated function at three difference places, as shown in Figure 4.

Coherence function between difference adopts Davenport coherency function model:

$\mathrm{Coh}\left(\mathrm{\&omega;}\right)=\exp (-\frac{\mathrm{\&omega;}}{2\mathrm{\&pi;}}\frac{C_z|z_i-z_j|}{\frac{1}{2}(U_i+U_j)})$

(37)

In formula, z _i, z _jbe respectively the vertical direction spatial coordinate that i, j are ordered, C _zfor the attenuation coefficient between vertical 2 of space, U _i, U _jbe respectively the mean wind speed that i, j are ordered, while being taken as, become the arithmetic mean of mean wind speed.

Downburst Wind Velocity History based on TAR modeling and its fluctuating wind are rapid-result to be divided as shown in Figure 5.From Fig. 5, can find to pulse wind speed amplitude with time to become mean wind speed size relevant, immediately become mean wind speed larger, the amplitude of pulsation wind speed is larger, this and actual wind field characteristic match, and the validity of non-uniformly modulated function has also been described.In order to illustrate that the downburst pulsation wind speed of simulation has target non-stationary characteristic, 100 groups of downburst pulsation wind speed samples of simulation having been carried out to time-varying power spectrum herein estimates as shown in Figure 6, the power spectrum that sample is estimated has obvious time varying characteristic, and matches with the time varying characteristic of target time varying spectrum.

In order to further illustrate the validity of simulation, Fig. 7,8 has shown respectively statistical average power spectrum, autocorrelation function and the target statistical average power spectrum of single sample downburst non-stationary pulsation wind speed ${\stackrel{\&OverBar;}{S}}_{\mathrm{ff}}\left(\mathrm{\&omega;}\right)=\frac{1}{T}{\&Integral;}_0^T{S}_{\mathrm{ff}}(\mathrm{\&omega;},t)\mathrm{dt}$ , autocorrelation function ${\stackrel{\&OverBar;}{R}}_{\mathrm{ff}}\left(\mathrm{\&tau;}\right)=\frac{1}{T}{\&Integral;}_0^T{R}_{\mathrm{ff}}(t,\mathrm{\&tau;})\mathrm{dt}$ Contrast, single sample statistics average power spectra, autocorrelation function and target are coincide mutually.In addition, in Fig. 5, the fluctuation of the downburst non-stationary of differing heights pulsation wind speed has obvious correlativity, and correlativity is along with the distance between diverse location increases and weakens as shown in Figure 9, and this and actual wind field characteristic match.

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Shortcoming and defect than prior art, the present invention has following beneficial effect: in order to simplify, prior art is assumed to non-uniformly modulated function only relevant with the time even modulating function conventionally, the simulation of nonstationary random process is converted into the modulation of even modulating function to stationary stochastic process, has avoided according to the difficult point of time-varying power spectrum simulation nonstationary random process.And, for the simulation of non-stationary pulsation wind speed, conventionally by time the even modulating function that equal function of wind speed is assumed to steady pulsation wind speed that flattens do not have correlation theory foundation.The present invention does not avoid according to the difficulties of time-varying power spectrum simulation nonstationary random process, first by downburst time become the mean wind speed model non-uniformly modulated function relevant with temporal frequency with power spectrum modulated to obtain the downburst non-stationary wind speed time-varying power spectrum of pulsing; Then by for simulate the AR model of stationary stochastic process further consider time varying characteristic set up simulation nonstationary random process TAR time-varying model, thereby solve according to the difficulties of downburst time-varying power spectrum simulation nonstationary random process, finally realize effective simulation of downburst non-stationary pulsation wind speed.

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any modifications of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., within all should being included in protection scope of the present invention.

Claims (6)

1. a method for numerical simulation for downburst non-stationary pulsation wind speed, is characterized in that comprising the following steps:

S1, theoretical according to Evolutionary Spectral, by downburst time become the mean wind speed model non-uniformly modulated function relevant with temporal frequency with power spectrum modulated to obtain the downburst non-stationary wind speed time-varying power spectrum of pulsing;

S2, the AR model of stationary stochastic process is further considered to time varying characteristic sets up the TAR time-varying model of nonstationary random process;

S3, by described time-varying power spectrum, TAR time-varying model, downburst non-stationary pulsation wind speed is effectively simulated.

2. the method for numerical simulation of downburst non-stationary pulsation wind speed as claimed in claim 1, is characterized in that, in step S1, described non-uniformly modulated function is Kaimal modulating function:

$A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{U}_j}}{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}}$

In formula: ω is circular frequency; z _jvertical ground height for space point; for some place, space time become mean wind speed; statistical average wind speed for some place, space non-stationary wind speed.

3. the method for numerical simulation of downburst non-stationary as claimed in claim 2 pulsation wind speed, it is characterized in that, described downburst non-stationary pulsation wind speed time-varying power spectrum comprises the non-uniformly modulated function after Davenport spectrum, Harris spectrum, the modulation of Simiu spectrum.

4. the method for numerical simulation of downburst non-stationary pulsation wind speed as claimed in claim 3, is characterized in that, in step S1, the non-uniformly modulated function after the modulation of described Davenport spectrum is:

$A(\mathrm{\&omega;},t)={\left(\frac{1+{\left(\frac{1200\mathrm{\&omega;}}{2\mathrm{\&pi;}{U}_{10}}\right)}^2}{1+{\left(\frac{1200\mathrm{\&omega;}}{2\mathrm{\&pi;}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{2/3};$

Non-uniformly modulated function after the modulation of described Harris spectrum is:

$A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_{10}\left(t\right)}{U_{10}}{\left(\frac{2+{\left(\frac{1800\mathrm{\&omega;}}{2\mathrm{\&pi;}{U}_{10}}\right)}^2}{2+{\left(\frac{1800\mathrm{\&omega;}}{2\mathrm{\&pi;}{\tilde{U}}_{10}\left(t\right)}\right)}^2}\right)}^{5/6}}$

Non-uniformly modulated function after the modulation of described Simiu spectrum is:

$\left\{\begin{array}{cc}A(\mathrm{\&omega;},t)=\sqrt{\frac{{\tilde{U}}_j\left(t\right)}{U_j}{\left[\frac{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{U}_j}}{1+50\frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}}\right]}^{5/3}},& \frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}\&le;0.2\left(1\right)\\ A\left(t\right)={\left(\frac{{\tilde{U}}_j\left(t\right)}{U_j}\right)}^{4/3},& \frac{\mathrm{\&omega;}{z}_j}{2\mathrm{\&pi;}{\tilde{U}}_j\left(t\right)}>0.2\left(2\right)\end{array}\right.;$

In various above, ω is circular frequency; z _jvertical ground height for space point; for some place, space time become mean wind speed; statistical average wind speed for some place, space non-stationary wind speed; be 10 meters of eminences time become mean wind speed; it is the statistical average wind speed of 10 meters of eminence non-stationary wind speed.

5. the method for numerical simulation of downburst non-stationary pulsation wind speed as claimed in claim 1, is characterized in that, in step S2, described AR model is autoregressive model:

$f\left(t\right)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_{i}f(t-\mathrm{i\&Delta;t})+\mathrm{Lw}\left(t\right);$

In formula, A _irepresent autoregressive model coefficient, L represents model coefficient undetermined.

6. the method for numerical simulation of downburst non-stationary pulsation wind speed as claimed in claim 5, is characterized in that, in step S2, described TAR time-varying model function definition is:

$f\left(t\right)=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t\right)f(t-\mathrm{i\&Delta;t})+L\left(t\right)w\left(t\right);$

In formula, time become autoregressive model coefficient A _i(t) available functions is defined as:

$R_{\mathrm{ff}}(t,\mathrm{j\&Delta;t})=\underset{i=1}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t\right)R_{\mathrm{ff}}(t-\mathrm{i\&Delta;t},\mathrm{j\&Delta;t}-\mathrm{i\&Delta;t}),$

Time-varying model coefficient L (t) by function definition is:

$L\left(t\right)=\sqrt{\underset{i=0}{\overset{p}{\mathrm{\&Sigma;}}}{A}_i\left(t\right)R_{\mathrm{ff}}(t-\mathrm{i\&Delta;t},-\mathrm{i\&Delta;t})}$

In above formula, t=1,2,3, F (t) is zero-mean nonstationary random process; R _ff(t, j Δ t) be f (t) at t related function constantly; A _i(t) for time become autoregressive model coefficient; L (t) is time-varying model coefficient undetermined; The white noise sequence that w (t) is 1 for variance, and meet related function $R_{\mathrm{ww}}(t,\mathrm{j\&Delta;t})=E\left[w\left(t\right)w(t-\mathrm{j\&Delta;t})\right]=\left\{\begin{array}{cc}1,& j=0\\ 0,& j\&NotEqual;0\end{array}\right.$ , p is model order.