portional to the wavenmber in the region of the larger eddies while the spectrm drops off mch faster for the largest eddies. Frthermore, some experime

Transcription

1 The Seventh International Colloqim on Blff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, 01 ARTICLES Statistical spectrm model of wind velocity at Beiing Meteorological Tower Yang Qingshan,Tian Yi, Li Bo and Chen Bo School of Civil Engineering, Beiing Jiaotong University, Beiing , China Abstract: The wind velocity spectra at Beiing Meteorological Tower are calclated sing Hilbert-Hang transform and Forier transform, respectively. A innovative model of wind velocity spectrm, which is accordant with the characteristics in both the inertial sbrange and the large eddies range, is presented in this paper. The method of least sqares is adopted to obtain the parameters in the model. Then the differences between the FFT spectrm and the HHT spectrm are compared. It is indicated that the vales of the HHT spectrm in the energy containing range are slightly larger than those of the FFT spectrm while the vales of the HHT spectrm in both inertial sbrange and dissipation sbrange are very close to that of Forier spectrm. It is conclded that the HHT spectrm describes elaborately and accrately the spectrm vales in the low freqencies and the fitted wind velocity model provides a reference for reconstrcting the near-grond wind field of Beiing city in wind tnnel test and for nmerical simlation. Keywords: trblent spectrm, wind velocity spectrm, inertial sbrange, HHT, FFT 1 Trblent spectrm theory The theory of the trblent spectrm originated from Richardson's view of the energy cascade [1], that is to say, the trblence is composed of eddies of different sizes. When the energy enters into the atmosphere, the inetic energy of the molecles increases and then the large eddies flow, brea p and transfer their energy to somewhat smaller eddies. These smaller eddies ndergo a similar brea-p process, and transfer their energy to the smallest eddies. This is called energy cascade, in which energy is transferred to sccessively smaller and smaller eddies ntil the inetic energy is dissipated by the moleclar viscosity. In trblent flow at sfficiently high Reynolds nmber, the energy of the small-scale motions is determined by the inematic viscosity and the energy transfer rate. This is called Kolmogorov's first hypothesis. In the region of small eddies, the energy of the relatively larger eddies is niqely determined by the energy transfer rate, independent of the inematic viscosity. This is called Kolmogorov's second hypothesis []. According to the two hypotheses, the atmospheric molecles absorb energy and flow in the form of the large eddies; the energy containing range is composed of these large-scale eddies. The trblent motions of the small-scale eddies are statistically isotropic and form a niversal eqilibrim range, in which the energy transfer range is named the inertial sbrange while the region of the smallest-scale eddies is designated as the dissipation sbrange. From the second hypothesis it follows that the trblent spectrm in the inertial sbrange is dedced from the dimensional analysis and is expressed as E ( ) C 3 53 (1) where is the wavenmber, is the energy transfer rate, and C is a niversal constant. In Eqation (1), the trblent spectrm is nominated as the Kolmogorov 53 law. Since 1940s, many researchers have devoted their attention to stdying the trblent spectrm and many spectrm models have been established. It is assmed that the trblent energy is transferred from large to small eddies in terms of an eddy viscosity [3]. This logical assmption indicates that the transfer term in the trblent spectrm dynamic eqation is represented by the same form of the viscos dissipation term. Kolmogorov s 53 law in the inertial sbrange is confirmed after some other assmptions are introdced while the spectrm decay qite rapidly with the -7 th power of the wave nmber in the region of the smallest eddies. The same conclsions were drawn by some other predecessors [4, 5]. The decay of the spectrm of isotropic trblence was investigated when the external forces casing the trblent motion were removed [6]. The approximate soltion of the spectrm dynamic eqation gives that the spectrm is pro- 183

2 portional to the wavenmber in the region of the larger eddies while the spectrm drops off mch faster for the largest eddies. Frthermore, some experimental reslts show that the spectrm vales of the largest eddies are not the maximm and that the spectrm of the largest eddies decays with the -4 th power of the wave nmber [7]. Traditionally, the energy spectrm of the homogeneos and isotropic trblent flow is the Forier transform of its ato-correlation fnction, i.e., Spectrm L 4-5/3-7 1 E( ) Q( )exp( i) d () where is the wavenmber, E ( ) is designated as the one-side spectrm of th (, v, w) trblent component, where vw,, are the symbols of longitdinal, horizontal and vertical components, respectively, and Q ( ) is the spatial correlation fnction of th component. Considering the even fnction Q ( ) in homogeneos trblence, Eqation () can tae the form of the correlation coefficient q ( ) and it follows that E( ) q ( )exp( ) 0 i d (3) where is the variance of the th trblent component, and q ( ) is the spatial correlation coefficient, q ( ) Q( ). When the wavenmber tends to zero at the two sides of Eqation (3), it is obtained that E (0) L (4) where L is the integral length scale of th component ( L q ( ) 0 d ). It is indicated that the spectrm vale of zero wavenmber is dependant on the integral length scale and the standard deviation of the flctating wind. Under the condition of roghness terrain, the integral length scale increases with the increase of height while the standard deviation decreases with the increase of height; hence, the trblent spectrm varies with the increase of height. When the wavenmber is eqal to zero, the derivative of E ( ) is zero, i.e., de ( ) d 0 (5) The aforementioned research reslts of trblent spectrm are illstrated in Figre 1, in which the energy spectrm crve varies with the wavenmber. 0 Wavenmber Energy containing range Universal eqilibrim range Largest Larger Energy Inertial Energy eddies eddies containing sbrange dissipation eddies sbrange Figre 1 Illstration of trblent spectrm and wavenmber Wind velocity spectrm model From Taylor's hypothesis or the frozen-trblence approximation, the wavenmber can be expressed as the freqency, n U, where n is the freqency and U is the mean wind velocity, and the wavenmber spectrm of trblence is represented by the freqency spectrm, E( ) US( n), where S ( n ) is the one-side power spectrm of th trblent component in the freqency domain. Considering Eqations (4) and (5), the power spectrm S ( n ) mst satisfy the following two conditions, i.e., S (0) 4 L U (6) ds ( n ) dn 0 (7) n 0 Based on the Kolmogorov s second hypothesis, the power spectrm in inertial sbrange was presented by Batchelor in 1953 [8], which is given by ns ( n) 3 f (8) where is the friction velocity, is a constant, and f is the Monin similarity coordinate, f nz U and f 0.. Simi and Scanlan [9] dedced the constant in Eqation (8) of longitdinal velocity ctation of wind trblence and 0.6 while Tieleman [10] gave the constants of the three components and 0.7, v 0.36 and w In 1946, Von Karman [11] presented an empirical power spectrm model of the longitdinal trblent flow observed in grid generated trblence in a wind tnnel as 184

3 The Seventh International Colloqim on Blff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, 01 where ns ( n) 4f L z 5/ f L z (9) is the trblence intensity coefficient,, and z is the height. It is obvios that Karman spectrm S ( n ) in Eqation (9) satisfies the two conditions shown in Eqations (6) and (7); meanwhile, it satisfies other three reqirements, as presented by Simi [9], for wind velocity spectrm; the third reqirement is that the Karman spectrm S ( n ) in Eqation (9) is consistent with 3/ Eqation (8) in the inertial sbrange if L 0.3 z ; the forth reqirement is that S ( n ) is a monotonically descending fnction; the fifth reqirement is that the integral of S ( n ) is the variance. In fact, the Karman spectrm is an interpolated expression of the spectrm in the inertial sbrange and the spectrm vale at zero freqency. Karman spectrm was fitted statistically from the data of the homogenos and isotropic trblence, therefore it is nsitable for the wind spectrm of the roghness terrain in the near-grond range [9]. It is evident that Karman spectrm agrees well with Eqation (8) in free stream conditions while it deviates slightly from the spectrm crve at the energy containing range and the dissipation sbrange. In the energy containing range, Karman spectrm crve is an approximately horizontal line; it is implied that the large eddies with different freqencies contain the similar or identical amonts of inetic energy. This overestimates the spectrm vales at the energy containing range. Therefore, the wind-resistant design of lower freqency strctres, sch as an oil platform or a cable-stayed bridge, assres more safety when Karman spectrm is sed. The fondation of stdying the wind velocity spectrm in wind engineering is established on the basis of Eqations (8) and (9). As stated by Solari [1], the spectral formlas in engineering practice are actally dedced from three procedres. The first procedre is the direct se of Eqation (9) in practice, in which parameters L and are determined by the actal trblence observed at the rogh sites[13-15]. The second procedre consists of the polynomial spectral eqations valid in the energy containing range and lined smoothly to Eqation (8) to represent the spectrm at the inertial sbrange. The third procedre is to define a single expression, sch as Karman spectrm, to describe the spectrm at zero freqency and in the inertial sbrange. A niversal spectrm formla for homogenos and isotropic trblence was presented by Fichtl [18], which has the form ns ( n) A f 1 5/(3 ) B f (10) where A, B and are the three parameters to be determined. It is clear that the reqirement of Eqation (6) is satisfied if A 4 L z and the other for reqirements are satisfied. Based on different observations, some expressions of spectrm lie the form of Eqation (10) have been presented [16, 19-5] since 1960s, in which parameter is sally specified as 1, or 53. To avoid the determination of the friction velocity and the coefficient dring the statistical process for the actal wind flow, Karman spectrm in Eqation (9) is rewritten as ns ( n) 4f /6 f (11) where f is a dimensionless coordinate, and f nl U. After the three parameters U, L and are specified by their empirical formlas on any height, Karman spectrm on this height can be expressed definitely. Eqation (11) is directly sed by Japanese Wind Code (AIJ-RLB-004) and Astralian Code (AS/NZS ). An empirical spectrm model presented by Solari [19] is adopted by American Wind Code (ANSI/ASCE 7-10), i.e., ns ( n) f /3 f (1) The spectrm expression in Eropean Wind Code (BS EN :005) is similar to Eqation (1) and it is ns ( n) 6.8f /3 f (13) Both Chinese Wind Code (GB ) and Canadian Code (NBCC-005) directly se Davenport spectrm [6], that is, ns n x 1 x ( ) 3 4/3 (14) where x 100nU10 and U 10 is denoted as the 10-minte mean wind speed over a flat, open terrain at a height of 10m. Davenport spectrm was fitted on the basis of different data observed in different contries, at different terrains and different heights. It cannot reflect the variation of the spectrm at different heights. The vale of Davenport spectrm at zero freqency is zero, and less than the theoretical vale shown in Eqation (6). Compared with the three spectra models presented respectively by Karman [11], Simi [17] and Solari [19], the vales of Davenport spec- 185

4 trm are mch smaller in the low freqency range and larger in the high freqency range. 3 HHT and Hilbert energy spectrm In the history of wind engineering, Forier transform is the efficient tool for transforming the time histories of wind velocity and pressre into the spectra in the freqency domain. In fact, a time history is represented by the smmation of many sine and cosine complex fnctions via Forier transform. These trigonometric fnctions are symmetrical and stationary, therefore a crcial restriction of the Forier transform is that the measred signals mst be strictly periodic or stationary. In the process of Forier spectrm analysis, the transient characteristics of a time history are lost since Forier transform is the integral with respect to time; on the other hand, Forier transform needs many high-freqency harmonic waves to fit non-stationary data. These sprios harmonics might mae mathematical sense, bt do not really mae physical sense at all and can disorder the energy spectrm distribtion. Conseqently, the vales of energy spectrm at the low-freqency range are sally nderestimated in engineering practice [7]. The other signal processing methods based on Forier transform, sch as short-term Forier transform, Wigner-Ville distribtion, sffer all the limitations of the Forier analysis. As the measred time histories of wind flow manifest strong nonlinear and nonstationary featres, it is necessary to select the relatively stationary records of wind velocity as the samples of Forier transform. The advent of wavelet transform in 1980s provided an attractive method to analyze nonstationary process. Some researchers in wind engineering tried to se wavelet transform to analyze the measred signals of wind velocity [8-30]. Wavelet transform, whose basis fnction is also priori as it is for Forier transform, decomposes a time history into a series of wavelet components, which have the nonlinear and nonstationary characteristics. However, wavelet transform is a regional presentation of the data in time-freqency plane and is sefl for characterizing gradal freqency changes, bt does not provide the instantaneos freqencies. In 1998, Hilbert-Hang transform (HHT) [31] was presented to serve as an innovative signal processing method to analyze adaptively nonlinear and nonstationary data. HHT is based on the empirical mode decomposition (EMD) method, which generates a collection of intrinsic mode fnctions (IMFs). Expressed as IMFs, the data have well-behaved Hilbert transform, from which the instantaneos freqencies can be estimated at a proper time. Ths, any event on the time as well as the freqency axis can be localized. The IMFs can serve as the basis of an expansion of the data which can be linear or nonlinear, stationary or nonstationary, and they are complete, nearly orthogonal and adaptive. In the traditional statistical analysis of trblence, the power spectrm density reslted from Forier transform of correlation fnction describes the flctating wind energy distribtion in the freqency domain. Similarly, this concept can be extended to the definition of the flctating wind energy spectrm via HHT, i.e., the Hilbert energy spectrm, which describes detailedly the flctating wind energy distribtion in the time-freqency plane. Assming t () is the flctating wind velocity of longitdinal component, the analytical signal of t () is as follows () t () t i() t c () t ic () t a i () t A (, t n) e (15) where c () t is the th IMF of t () ; c () t is denoted as the Hilbert transform of c () t ; A (, tn ) is the amplitde of th analytical component in time-freqency plane and A(, tn) c() t c () t ; () t is the instantaneos phase; n is the instantaneos freqency, and n d () t dt. It is recognized that the energy contained in analytical signal a () t is two times as mch as that of the original signal t (), i.e., A (, t n) (t). The dration T of the flctating wind history is divided into N 1 intervals and the time step is t while the freqency range f max, where f max is the sampling freqency, is divided into N intervals and the freqency step is n. Then the Hilbert energy spectrm of t () can be defined as the energy density distribtion in a time-freqency space divided into eqal size bins of t n with the vale in each bin designated as A (, tn) ( NN 1 ) at the proper time and the corresponding instantaneos freqency, that is, S A (, tn) (, t n) NN t n 1 (16) It is clear that the doble integrals of Hilbert energy spectrm S (, t n ) with respect to time and freqency reslt in the variance of the flctating wind history. If integrating the Hilbert energy spectrm S (, t n ) with respect to time only, the Hilbert marginal spectrm with respect to freqency is defined simply as A (, t n) T S ( n) dt (17) 0 NN t n 1 The definition of the Hilbert marginal spectrm in Eqation (17) wold facilitate comparison with the Forier power spectrm. The marginal spectrm reflects the wind energy 186

5 The Seventh International Colloqim on Blff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, 01 distribtion at each instantaneos freqency whereas the Forier power spectrm represents the wind energy distribted in the global freqency. 4 Wind spectrm at Beiing Meteorological Tower The Beiing Meteorological Tower (Figre ), which is sitated inbetween the Northern 3 rd Ring Road and the Northern 4 th Ring Road of Beiing city, was set p in 1978 by the Institte of Atmospheric Physics (IAP), Chinese Academy of Sciences (CAS). Before 1990, there was no any high strctre near the tower, ths the tower was the best station to observe meteorology and measre the sbrb bondary layer of Beiing city. Since 1990s, the scale of Beiing city has largely expanded with the tremendos rbanization. MDan Yan and GanCheng Yan residential bildings with height less than 60m were bilt in 1990s in the north and soth of the tower. GanCheng Yan residential bildings, located closely to the tower, are the primary rogh elements inflencing the near-grond wind flow. The tower is now srronded by residential bildings of 30m to 60m in height and is located in a prosperos city zone completely. Hence, the meteorological tower becomes the best observation station to stdy the rban bondary layer (UBL) of Beiing city. In 000, three spersonic anemometers named UAT_1, whose sampling freqency is 10Hz, were installed at the 47m, 10m and 80m above grond level, respectively to record the trblence flow. Since then, more and more measred records of trblence flow have been accmlated. The wind velocity spectrm is investigated in this paper sing the measred trblent records by the tower from 005 to 007, pls partial records of 00. All of the records were provided by IAP, CAS. (a) 1980s (b) 010s Figre Beiing Meteorological Tower According to the previos research, the northwest wind dominates in the spring, atmn and winter. Ths all the 10-minte, northwest wind records of the three seasons in 00, 005, 006 and 007 were classified by the mean speeds at 47m. The nmber of samples in each speed bloc is listed in Table 1. It is nown from Eqations (6), (8)~(14) that the trblent spectrm vales are dependant on the integral length scale and the mean speed. Therefore, both the integral length and the mean speed varying with the height mst be determined before the wind spectrm is fitted. The integral length scales of the wind samples in each bloc are calclated. It is indicated that the mean integral length scales increase with the increase of the wind speeds while the slope of the integral length scale crve becomes smaller and smaller. The varied integral length scales exhibit a stable vale when the wind speed is larger than 8m/s at 47m. Then the integral length scales of the 378 samples of the last for blocs in Table 1 are reanalyzed. It is shown that the probability distribtion of the integral length scales of the 378 samples scatters in a very large range, in which the probabilities of occrrence of the largest or the larger integral length scales are very small. After the singlar samples in the last for blocs shown in Table 1 are discarded, the integral length scales and the mean speeds are calclated and listed in Table. The fitted profiles of the integral length scales sing the available samples are as follows: Z , 30m Z 450m L 90, Z 30m Z , 30m Z 450m Lv 45, Z 30m L w Z , 30m Z 450m 15, Z 30m (18) (19) (0) After Hilbert-Hang transform is condcted, the analytical signals of the available samples are obtained. Then the Hilbert energy spectra defined in Eqation (16) and the marginal spectra defined in Eqation (17) are calclated. Figres 3~5 give the dimensionless marginal spectra and their mean spectra (the thic solid line) of the longitdinal components for the wind speed blocs. It can be seen from the spectra crves in doble logarithm coordinate system that the nine mean spectra are nearly straight lines if the dimensionless freqency nl U is larger than 0.. Therefore it is reasonable to regard the point of nl U 0., where the corresponding freqency n is abot 0.016Hz, as the lower bond of the inertial sbrange. After the nine mean spectra crves in the inertial sbrange ( n ~ 0.5Hz ) are fitted with linear model, the slope of the fitted line varies from to with 95% confidence and the mean slope is , which is less than the theoretical vale -/3 (shown in Eqation (8)) of the homogenos and isotropic trblence becase the wind flow at 187

6 Beiing Meteorological Tower is inflenced by the rogh srface. Considering this phenomenon, it is assmed approximately that the dimensionless spectrm ns( n) of the longitdinal component is proportional to the dimensionless freqency f 35 ( f nl U ) in the inertial sbrange and the conterparts of the horizontal and vertical components tae the same forms. Table 1 Nmber of 10-minte, northwest samples in each bloc Mean speed at 47m -3m/s 3-4m/s 4-5m/s 5-6m/s 6-7m/s 7-8m/s 8-9m/s 9-10m/s 10-11m/s 11-1m/s Nmber of samples Table Integral length scales and mean speeds of available samples Mean speed at 47m 8-9m/s 9-10m/s 10-1m/s 47m 10m 80m v w v w v w Nmber of samples Integral length scale (m) Mean speed (m/s) Nmber of samples Integral length scale (m) Mean speed (m/s) Nmber of samples Integral length scale (m) Mean speed (m/s) According to the niversal formla of the empirical wind velocity spectrm in Eqation (10), the following expression is presented to fit the flctating wind velocity spectrm at Beiing Meteorological Tower, that is, where ns ( n) a f 1 8/(5 ) b f is the variance of the th component and (1) a, b and are three parameters to be estimated. Eqation (1) is an interpolation fnction of the spectrm in the inertial sbrange and the theoretical spectrm vale at zero freqency if a =4, and it mainly describes the shape of spectrm in the inertial sbrange, in which S ( n ) is propor- 8/5 tional to f, and does not represent well the shape of the spectrm in energy containing range and the dissipation sbrange. (a) At the height of 47m (b) At the height of 10m (c) At the height of 80m Figre 3 Longitdinal spetra and mean spectra for wind speed at 8-9m/s 188

7 The Seventh International Colloqim on Blff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, 01 (a) At the height of 47m (b) At the height of 10m (c) At the height of 80m Figre 4 Longitdinal spetra and mean spectra for wind speed at 9-10m/s (a) At the height of 47m (b) At the height of 10m (c) At the height of 80m Figre 5 Longitdinal spetra and mean spectra for wind speed at 10-1m/s To fit better in the energy containing range, another expression of the flctating wind velocity spectrm is presented in this paper, i.e., ns 4 1 ( n) f a f () 1 13/(5 ) b f It is evident that the reqirement of Eqation (6), S(0) 4 L U, is satisfied at zero freqency while the derivative at zero freqency is a constant, ds ( n) dn 4 L a U, which means that the S ( n ) is a linear fnction with the slope of 4 La U, agreeing with the conclsion of Heisenberg [6] in the region of the larger eddies. In the inertial sbrange, the dimensionless spectrm ns ( n) is approximately proportional to 3/5 f if f is larger than 0.. This agrees with the mean spectra shapes, shown in Figres 3~5, in the inertial sbrange. In a word, Eqation () is consistent with the characteristics of spectrm shape in large eddies range and the inertial sbrange. With the increase of the mean wind speed, the nmber of the available samples decreases. If the total samples of the speed bocs in Table are collected to fit the spectrm, the contribtions of the high speed samples can not be reflected becase the nmber of high speed samples is mch less than the nmber of the low speed samples. To overcome this drawbac, the nine mean spectra crves shown in Figres 3~5 are collected and considered as the fitted target. The fitted reslts sing Eqations (1) and () are plotted in Figres 6~8. By sing the fast Forier transform (FFT), the same process is condcted and the fitted spectra are indicated in Figres 9~11. Compared with the calclated spectra sing HHT, the deviatability of the FFT spectra in the high freqency range is mch larger in the niversal eqilibrim range. Figres 1~14 show the comparison of the fitted spectra sing HHT and FFT. It is apparent that the fitted spectrm vale sing HHT is larger than the conterpart sing FFT in the energy containing range, and the pea freqency of the HHT spectrm is less than that of the FFT spectrm. This phenomenon can be explained by FFT itself: Becase many high freqency harmonic waves in FFT are introdced to fit the pea of a time history, the high freqency spectrm becomes larger and correspondingly the low freqency spectrm becomes smaller. It is nown that the HHT spectrm behaves better than the FFT spectrm in the energy containing range. In the inertial sbrange, the measred trblent spectra at Beiing Meteorological Tower vary approximately with the -8/5 power of the freqency and agree closely with the Kolmogorov 53 law of the homogenos and isotropic trblence. Hence, the fitted spectra can be compared with Karman spectrm (Figre 1). The FFT spectrm is very close to Karman spectrm both in the niversal eqilibrim range and in the large or largest eddies range while the Karman spectrm is slightly larger in the region of energy 189

8 containing eddies than the FFT spectrm. The HHT spectrm is larger than Karman spectrm in the energy containing range while it is approximately identical to Karman spectrm in the niversal eqilibrim range. Since Karman spectrm is based on FFT, its vale in the energy containing range may be less than the accrate vale. Figre 6 Longitdinal spectra sing HHT Figre 7 Horizontal spectra sing HHT Figre 8 Vertical spectra sing HHT Figre 9 Longitdinal spectra sing FFT Figre 10 Horizontal spectra sing FFT Figre 11 Vertical spectra sing FFT Figre 1 Comparison of longitdinal spectra Figre 13 Comparison of horizontal spectra Figre 14 Comparison of vertical spectra 5 Conclsions The research history of the trblent spectrm and the practical spectra models in the wind engineering is smmarized briefly in this paper. Then an innovative wind spectrm model is presented to reflect additionally the characteristic of wind spectrm in the region of the large eddies. The wind velocity spectra of the measred trblence at Beiing Meteorological Tower are calclated sing HHT and FFT, respectively, and are fitted sing the method of least sqares to estimate the parameters in the empirical models. The fitted reslts show that the presented model can fit the measred spectra well as the niversal spectrm model does while the spectrm vales of the presented model in the region of the large eddies are more accrate and the theoretical vale at zero freqency is satisfied. Dring the process of calclating the wind velocity spectra of the measred trblence sing FFT, many high-freqency harmonic waves are introdced to fit the peas of a nonstationary history. This confses the energy distribtion in the freqency domain and leads to the low vales of the low-freqency spectrm. The wind velocity spectra reslted from HHT can reflect the wind spectra at the proper time and the corresponding instantaneos freqency, and can obtain the accrate spectra vales in the region of the large eddies. Hence, the fitted spectrm sing HHT is slightly larger than the FFT spectrm while the spectrm in the niversal eqilibrim range is very close to the FFT spectrm. 1830

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