RESEARCH ON RANDOM FOURIER WAVE-NUMBER SPECTRUM OF FLUCTUATING WIND SPEED

Transcription

1 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan RESEARCH ON RANDOM FORIER WAVE-NMBER SPECTRM OF FLCTATING WIND SPEED Qi Yan 1, Jie Li 1 Ph D. andidate, Department of Building Engineering, Tongji niversity Shanghai 9, China, sevenyan_tj@hotmail.om Professor, Department of Building Engineering, Tongji niversity The State Key Laboratory of Disaster Redution in Civil Engineering, Tongji niversity Shanghai 9, China, lijie@tongji.edu.n ABSTRACT In the paper, a random Fourier spetrum model of flutuating wind speed with solid physial basis is proposed. Firstly, definition of normalized random Fourier wave-number spetrum of flutuating wind speed is presented. Based on physial sene of atmospheri turbulene, relationship between vortex sales and energy spetrum sub-ranges is presented. Aording to the different exponential laws of Fourier Spetrum in energy-ontaining sub-range and inertial sub-range, a bilinear fitting is employed to fit the Fourier spetrum. The boundary wave-length l of the two sub-ranges an be identified and it should be pointed out that, in the atual physial proess, the boundary wave-length orresponds to the boundary value of the vortex sales of energy-ontaining range and equilibrium range. Essentially, l is a variable and the main physial fator affeting it is main flow shear-ratio. Probability distribution of boundary wave-length orresponding to different shear-ratio is alulated by Probability Density Evolution Method (PDEM) and probability distribution model of boundary wave-length is established. Finally, taking ground roughness length and 1-min mean wind speed as two basi variables and onsidering the probability distribution model of l, the normalized random Fourier wave-number spetrum with the bilinear harateristi an be obtained. The theoretial results aord with the measured results well. KEYWORDS: RANDOM FORIER SPECTRM, FLCTATING WIND SPEED, PROBABILITY DENSITY EVOLTION METHOD, ENERGY-CONTAINING SB-RANGE, PHYSICAL MODELING Introdution Flutuating wind is essentially the flutuating part of atmospheri turbulene. In wind engineering, power spetral density funtion is ommonly adopted to desribe energy distribution of flutuating wind speed. However, the power spetral density is the seond-order numerial harateristi of a stohasti proess in nature. Beause of the omplexity of atmospheri turbulene, the power spetral density is far from haraterizing the probability information adequately. Furthermore, reliability analysis an t be arried through due to that only the seond-order moment of response is available. Random Fourier spetrum model tries to build a dynami exitation model by starting with the point of view of random funtion and ombining the understanding of the physial mehanism of the random exitations as presented in the literatures of [Li (6), Li and Zhang (7) and Zhang et al. (8)]. Along with probability density evolution method (PDEM) proposed in the literature of [Chen (6) and Li and Chen (6)], dynami reliability of nonlinear system an be alulated. Starting from the physial sene of atmospheri turbulene and ombining the modeling of physial parameters, this paper established a physial model of flutuating wind speed spetrum. The model is based on the physial mehanisms and thus an reflet the probability information of flutuating wind speed.

2 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan Random Fourier funtion Random Fourier amplitude spetrum There are two basi methods for modeling stohasti dynami exitations: phenomenology-based modeling and physis-based modeling, stated in the literature of [Li (6)]. Due to the diffiulties in establishing the finite-dimensional probability distributions, the orrelation funtion or the power spetral density funtion is ommonly used in the phenomenology-based modeling. In essene, the method is based on statistial moments. In ontrast, the physis-based modeling is fous on giving a random funtion model of dynami exitations onsidering the real physial bakground. These models an not only give a omplete mathematial desription for the stohasti proessed or random fields, but also make the experimental verifiation of the random proess possible beause of their physial signifiane. For the flutuating wind speed u( t ), the random Fourier frequeny spetrum (amplitude spetrum only) is defined in the literature of [Li and Zhang (7)] 1 T πint F ( η, n) = u( η, t) e dt (1) T where η is basi random variable or random vetor whih affets the proess of flutuating wind speed, n is natural frequeny and T is time duration. It should be pointed out that η must be observable. Normalized Fourier wave-number spetrum Wave-length l, the reiproal of wave-number k, ould give a straight view of the eddy sale. Therefore, in turbulene theory wave-number spetrum is more often employed than frequeny spetrum. Aording to Taylor s frozen hypothesis, wave-number k is expressed below: π k = n () where is mean wind speed in time duration T and n is the natural frequeny, the same in Eq. (1). Based on frequeny spetrum F( η, n), random Fourier wave-number spetrum of flutuating wind speed is defined as F( η, k) = F( η, n) (3) π In order to eliminate the effet of the value of total energy, the wave-number spetrum is divided by the standard deviation σ and the normalized Fourier wave-number spetrum Fnor ( η, k) is finally defined as T πint Fnor ( η, k) = F( η, k) = F( η, n) = u( η, t) e dt σ σ π σ πt (4) The normalized spetrum Fnor ( η, k) satisfies following energy riterion Fnor ( η, k) dk = 1 () Modeling on normalized random Fourier wave-number spetrum Physial sene of atmospheri turbulene The existing of eddies is one of the main haraters of turbulene. The eddy sale in atmosphere is in a wide range from more than one thousand meters to several millimeters.

3 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan Casade, whih means gradually energy-transfer from big eddies to small eddies, exists in turbulene beause of some kinds of nonlinear interation. A physial sene of atmospheri turbulene is drawn by sientists as in the work of [Hu (199)]: big eddies with energy-ontaining sale gain kineti energy by interating with main flow and they ontain most of the turbulene energy; then big eddies ollapse and thus smaller and smaller eddies are reated while energy is transferred to the smaller eddies; the small eddies in a ertain sale range transfer as muh energy to smaller eddies as they reeive from larger eddies and therefore the range is alled equilibrium range ; finally, energy is transferred into dissipation range where turbulene kineti energy was onverted to internal energy by the ation of fluid visosity. Based on the sene stated above, it is reognized that turbulent eddy motion in the atmospheri boundary layer exhibits three spetral regions: energy-ontaining sub-range, inertial sub-range, and dissipation range. In wind engineering, the first two sub-ranges are mostly onerned. For the energy spetrum of inertial sub-range, -/3 power law has been reported both theoretially and in atmospheri boundary-layer measurements. But the analysis of large-sale turbulene is more omplex beause of their strong anisotropy. Despite the omplexity, several laboratory and field experiments reported -1 power law at prodution sales for the longitudinal veloity spetrum in boundary-layer flows [Katul and Chu (1998)]. Employing wind speed reords of our researh group, we also find the -1 power law in energy-ontaining sub-range of both three-dimensional spetrum and longitudinal spetrum. Bilinear fitting A speified wind speed reord is the ombined effet of many eddies with different sales. Fourier transform, whih is known as the tools of transforming time domain signals to frequeny domain, ould separate the energy distribution of eddies with different sales. From the setion above, we have reason to believe that in energy spetrum the energy-ontaining sub-range orresponding to the energy-ontaining sale obeys -1 law and inertial sub-range orresponding to the equilibrium sale behaves -/3 law. Therefore, a boundary position k between energy-ontaining range and inertial range appears in the wave-number energy spetrum of flutuating wind speed. There is no harm to all this position k boundary wave-number and the reiproal of k marked as l is alled boundary wave-length. In the atual atmospheri turbulene proess, l represents the boundary value between the sales of eddies in energy-ontaining range and in equilibrium range. Eddies bigger than l belong to energy-ontaining range and eddies smaller than l (of ourse, bigger than dissipation sale) belong to equilibrium range, as illustrated in Figure 1. Beause of the omplexity of atmospheri turbulene, l must be a variable affeted by some other fators. Employing power format to desribe the energy distribution of energy-ontaining sub-range with index of -1 and inertial sub-range with index of -/3 ould not only reflet the physis-based energy distribution but also mark out the boundary of the two sub-ranges. In double logarithmi oordinate, this format performs as two lines so it ould be known as bilinear format. Therefore, a bilinear funtion is seleted to fit the random Fourier spetrum. The expetation of the square of the assembly of Fourier spetrum is equal to the energy spetrum: E F ( η, n) = S( n) (6) Taking into aount what is showing in Eq. (4), the bilinear funtion for normalized wave-number Fourier spetrum is defined as

4 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan 1/ Ak ( k k ) Fnor ( η, k) = (7) /6 Bk ( k > k ) where A and B are dependent parameters related to amplitude. The only independent parameter in Equation (7) is k, the boundary wave-number of two sub-ranges. It an be identified from the measured Fourier spetrum of flutuating wind speed. l e l i Energy (Logarithmi oordinate) -1 law l -/3 law k (Logarithmi oordinate) Figure 1: Relationship between Eddy Sales and Spetral Sub-ranges in Atmosphere Now we ll see whether the bilinear funtion is proper for the Fourier Spetrum of flutuating wind speed. Our researh group installed a set of anemometers over typial ountry terrain at east China in 6 [Yan et al. (9)], by whih simultaneous wind speed at four heights (1m, m, 8m and 43m) an be reorded. 1-min averaged wind speed at the four heights is used to fit logarithmi law of wind profile, see Figure. Height (m) Wind profile fitted by Log-law Measured 1-min mean wind speed 1 1 Mean Wind Speed (m/s) Figure: Mean Wind Speed at Four Heights and Log-law Fitting The logarithmi expression of wind profile is u* z ( z) = ln κ z (8)

5 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan where κ is von Karman s onstant (κ~.4), z is ground roughness length and u * is frition speed. z and u * an be identified and in the sample orresponding to Figure, z =.16m and u * =.66m/s. Differentiating the wind profile in Eq. (8), main flow shear-ratio at height z an be expressed as d ( z) u* = (9) dz κ z Figure 3 displays the flutuating wind speed and bilinear fitting of Fourier spetrum of the four heights. Least square riterion is adopted. We an see that the bilinear fitting is reasonable for the fitting. The small ross in the right figures in Figure 3 is the boundary positions m Flutuating wind speed (m/s) Fourier spetrum (m 1/ ) 1 m m m Time (s) Figure 3: Simultaneous Flutuating Wind Speed and Normalized Fourier Spetrum along with Bilinear Fitting The identified value of k and the shear-ratio alulated aording to Eq. (9) are listed in Table 1. Aording to the information in Table 1, we find that the position of k is affeted by the shear-ratio. Generally speaking, the interation between turbulene and main flow ours in a broader range while the shear-ratio inreases as stated in the work of [Hinze (197)]. Thus the wave-number eiling of energy-ontaining sub-range inreases, that is, k inreases. Table 1: Parameters Identified from the Bilinear Fitting Position 1 m m 8 m 43 m Shear-ratio(s -1 ) k (m -1 ) l (m) k (m -1 ) Probability desription of the relationship between shear-ration and l The shear-ratio is different at different height of one wind profile and it also varies with time at the same height due to the hange of profile. Moreover, differing from laboratory turbulene, atmospheri turbulene is influened by more fators like terrain, temperature and

6 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan the stationarity of atmosphere. Therefore, the relationship between shear-ratio and l must be stohasti and a probability desription is needed. Study on measured data shows that different sets of wind speed lead to different z, u * and l. They are essentially random variables. The key point of the physial modeling is to establish the probability desription of shear-ratio and l whih represents how parameters affet the energy distribution of flutuating wind speed. So, 4 sets (four samples per set ) of Fourier spetrum are fitted and the points of l are plotted in Fig 4. Then probability distribution of l at ertain shear-ratio is evaluated by PDEM (see details in the literature of [Chen (6)]). Different distribution funtions are used to fit the distribution alulated before and at last lognormal distribution is seleted. For the two parameters in lognormal distribution, μ an be expressed as l l ( / ) D μ = C d dz (1) where C=3.1 and D= -.8 and the other parameter σ l =.μ l. The probability density funtion at ertain shear-ratio is ( ) 1 ln x μ l l( x μl, σ ) exp l = (11) xσ σ l π l Figure displays the omparison between PDEM results and the distribution of Eq. (1). Figure 4: Relationship between shear-ratio and l.1.8 PDEM Lognormal.1.8 ShearRatio=.s -1 PDEM Lognormal.6.6 ShearRatio=.1s -1 PDF.4. SheaRatio=.1s -1 SheaRatio=.s -1 PDF l (m) l (m) Figure: Comparison of probability distribution between PDEM results and lognormal model Verifiation of proposed model Probability of basi variables Shear-ratio is the main fator affeting the energy distribution, but the basi random

7 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan variables are roughness length z and 1-min mean wind speed at h-meter high ( h ). Derived from Log-law of wind profile, shear-ratio an be expressed as d ( z) ( h) = (1) dz z ln ( h / z ) The same proedure of fitting l is used to obtain the distribution of z and ( h ). The results show that z an be haraterized by lognormal distribution and ( h ) an be haraterized by Gumbel Distribution. The probability density funtion of z is 1 ( ln x μ z ) z ( x μz, σ ) exp z = (13) xσ σ z π z where μ z and σ z are the mean and standard deviation of the natural logarithm of measured z, respetively. The probability density funtion of ( h ) is 1 ( h) = ( x α, β) = exp ( ( x α) / β) exp( exp ( ( x α) / β) ) (14) β where α is position parameter and β is sale parameter. They an be evaluated by equations below: β = σ 6/ π α = μ γβ (1) where μ and σ are the mean and standard deviation of the mean wind speed at the referene height, and γ is a onstant equivalent to about.77. Figure 6 displays the omparison between the measured probability and the suggested probability of z and ( h ). The measured probability is alulated by 187 sets of 1-min data and the parameters are presented in Table. For larity onsideration, the probability distribution of wind speed at m is not plotted here. PDF PDEM Lognormal z (m) PDF m 8m PDEM Gumbel Distribution 43m 1 1 1min Mean Wind Speed (m/s) Figure 6: Fitness of basi random variables Table : Parameters of the Basi Random Variables Variable z ( h ) Parameter μ σ α β 1m m m m Comparison of model and measurements Considering the distribution model of z and ( h ) presented above, the mean urve of normalized random Fourier wave-number spetrum an be alulated by funtion below E F z, h, k = F z, h, n p h p z d hdz ( ) nor ( ) ( ) ( ) nor (16) Ω

8 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan where Ω is integral area. The measured mean urves orresponding to the data in Figure6 are also alulated. Figure 7 displays the omparison of mean spetra and we an see that the model mean spetra are almost idential with measured mean spetra at all 4 positions. The spetra of m and 43m are shifted upward by a deade for larity. Fourier Spetrum(m 1/ ) m k(m -1 ) Measurements Model m Fourier Spetrum(m 1/ ) Figure7 Comparison of mean spetrum 8m k(m -1 ) Measurements Model 43m Conlusion Shear-ratio is the main fator affeting the boundary loation of energy-ontaining sub-range and inertial sub-range in Fourier spetrum of atmospheri turbulene. This paper established the probability relationship between mean flow shear-ratio and boundary loation of the two sub-ranges. Considering bilinear fit of Fourier spetrum and the distribution of basi random variables, mean urves of random spetrum model are alulated and orrespond to measured mean urves well. The normalized random Fourier spetrum of flutuating wind speed proposed here an be used as dynami exitation model. Aknowledgement The support of the Natural Siene Foundation of China for Innovative Researh Groups (Grant No. 616), the Natural Siene Foundation of China (Grant No. 86 and 77813) and Fok Ying Tung Eduation Foundation for Young Experts (Grant No.1141) are gratefully aknowledged Referenes Chen, J. B. and Li, J. (6), Probability Density Evolution Method for Probability Density Funtion Estimation for Random Variables, Journal of Tongji niversity (Natural Siene), 34(4): Hinze, J. Q. (197), Turbulene, New York: MGraw-Hill. Hu, F. (199), Turbulene, Intermitteny and Atmospheri Boundary Layer, Beijing: Siene Press (in Chinese) Katul, G. and Chu, C. R. (1998), A Theoretial and Experimental Investigation of Energy-ontaining Sales in the Dynami Sublayer of Boundary-layer Flows, Boundary-Layer Meteorology, 86: Li, J. (6), A Physial Approah to Stohasti Dynamial Systems, Siene Paper Online, 1():9-14 (in Chinese). Li, J. and Chen, J. B. (6), Generalized Density Evolution Equation in Stohasti Systems, Progress in Natural Siene, 16(6): (in Chinese). Li, J. and Zhang, L.L. (7), Random Fourier Spetrum of the Measured Wind Field, Journal of Vibration Engineering, (1): 66-7 (in Chinese). Yan, Q., Xie, Q. and Li, J. (9), Long-turn Observation and Data Analysis of Wind Field, Journal of Arhiteture and Civil Engineering, 6(1): 37-4 (in Chinese). Zhang, L. L., Li, J. and Peng, Y. B. (8), Dynami response and reliability analysis of tall buildings subjet to wind loading, Journal of Wind Engineering and Industrial Aerodynamis, 96: -4.