Correlation detection of turbulence and turbulent-induced...

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Correlation detection of turbulence and turbulent-induced forces using continuous and discrete wavelet transforms Le Thai Hoa Bridge and Wind Engineering Laboratory Graduate School of Engineering, Kyoto University Yoshida-honmachi, Sakyo-ku, Kyoto-shi 606-8501, Japan. [email protected] Tóm tt : Các phép biến đổi wavelet được phát trin gn đây và đang ng dng trong nhiu lĩnh vc như xlý tín hiu s, phân tích s, mã hoá và nén tín hiu, hình nh, nhn dng hthng động lc và dòng chy… Ưu đim ca phép biến đổi wavelet so vi phép biến đổi Fourier thông thường là khnăng phân tích các tín hiu n định, không n định, phi tuyến hay gián đon trong cmin tn svà thi gian. Trong kthut gió, vic đánh giá tác động gió ngu nhiên và tính toán phn ng ca công trình căn cvào mô hình quan hgia vn tc gió và lc phát sinh. Hin nay, mô hình quan hgia thành phn biến thiên ca vn tc gió và lc gió phát sinh vn da trên lý thuyết gin định. Tuy nhiên, lý thuyết này còn tn ti nhiu ri ro do bn cht bên trong mi quan hgia vn tc và lc gió chưa được làm rõ, và liên quan ti các nguyên nhân như tính không n định ca tín hiu vn tc và lc gió, sgián đon năng lượng trên min thi gian, hay sphân bnăng lượng tín hiu trên các di tn s. Do vy các nghiên cu vmi tương quan này sgóp phn gii quyết các vn đề liên quan ti tính toán phn ng công trình và mô phng trong thí nghim hm gió. Báo cáo này áp dng các công ctrong các phép biến đổi wavelet liên tc và gián đon để nhn dng tương quan gia vn tc và lc gió trong min thi gian và tn s. Các công cnhư hàm phchéo wavelet, hàm coherence, hàm lch pha wavelet được trình bày trong phép biến đổi wavelet liên tc, hơn na phân tích độ phân gii đa cp sdng các hàm wavelet trc giao sđược nghiên cu trong phép biến đổi wavelet ri rc. Sliu phân tích và kho sát ly tkết quđo trc tiếp, đồng thi vn tc và lc phát sinh nhthí nghim mô hình trong hm gió. 1. Introduction Certainly, the Fourier transform (FT) has been most popularly used to represent a given signal in the frequency domain. When the Fourier approach is combined with a time-shifted window function, known as the short-time Fourier transform (STFT) or Gabor transform, a spectral-temporal representation of the signal can be obtained. Hidden events inside the signal, however, are difficult to be tracked by the STFT because of equal resolution analysis in a time-frequency plane. Wavelet transforms (WT) is recently developed basing on a convolution operation between a signal and a wavelet function which allows to represent in the time-scale (frequency) domains with a flexible resolution, also called as a time-frequency analysis (Daubechies, [4]; Boggess and Narcowich, [1]; Bruns, [2]). The WT, furthermore, is advantageous over the FT and the STFT as an analyzing tool for non-stationary, non-linear and intermittent signals. The WT uses the wavelet functions, which can dilate (or compress) and translate basing on two parameters: scale (frequency) and translation (time shift) to apply short windows at low scales (high frequencies) and long windows at high scales (low frequencies). Basing on a discretized manner of the time-scale plane and characteristics of wavelets, the WT can be branched by continuous wavelet transform (CWT) and discrete wavelet transform (DWT). Since first invention in the late of 80’s years, the WT have been applied widely

Transcript of Correlation detection of turbulence and turbulent-induced...

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Correlation detection of turbulence and turbulent-induced forces using continuous and discrete wavelet transforms

Le Thai Hoa

Bridge and Wind Engineering Laboratory Graduate School of Engineering, Kyoto University

Yoshida-honmachi, Sakyo-ku, Kyoto-shi 606-8501, Japan. [email protected]

Tóm tắt: Các phép biến đổi wavelet được phát triển gần đây và đang ứng dụng trong nhiều lĩnh vực như xử lý tín hiệu số, phân tích số, mã hoá và nén tín hiệu, hình ảnh, nhận dạng hệ thống động lực và dòng chảy… Ưu điểm của phép biến đổi wavelet so với phép biến đổi Fourier thông thường là khả năng phân tích các tín hiệu ổn định, không ổn định, phi tuyến hay gián đoạn trong cả miền tần số và thời gian. Trong kỹ thuật gió, việc đánh giá tác động gió ngẫu nhiên và tính toán phản ứng của công trình căn cứ vào mô hình quan hệ giữa vận tốc gió và lực phát sinh. Hiện nay, mô hình quan hệ giữa thành phần biến thiên của vận tốc gió và lực gió phát sinh vẫn dựa trên lý thuyết giả ổn định. Tuy nhiên, lý thuyết này còn tồn tại nhiều rủi ro do bản chất bên trong mối quan hệ giữa vận tốc và lực gió chưa được làm rõ, và liên quan tới các nguyên nhân như tính không ổn định của tín hiệu vận tốc và lực gió, sự gián đoạn năng lượng trên miền thời gian, hay sự phân bố năng lượng tín hiệu trên các dải tần số. Do vậy các nghiên cứu về mối tương quan này sẽ góp phần giải quyết các vần đề liên quan tới tính toán phản ứng công trình và mô phỏng trong thí nghiệm hầm gió. Báo cáo này áp dụng các công cụ trong các phép biến đổi wavelet liên tục và gián đoạn để nhận dạng tương quan giữa vận tốc và lực gió trong miền thời gian và tần số. Các công cụ như hàm phổ chéo wavelet, hàm coherence, hàm lệch pha wavelet được trình bày trong phép biến đổi wavelet liên tục, hơn nữa phân tích độ phân giải đa cấp sử dụng các hàm wavelet trực giao sẽ được nghiên cứu trong phép biến đổi wavelet rời rạc. Số liệu phân tích và khảo sát lấy từ kết quả đo trực tiếp, đồng thời vận tốc và lực phát sinh nhờ thí nghiệm mô hình trong hầm gió.

1. Introduction

Certainly, the Fourier transform (FT) has been most popularly used to represent a given signal in the frequency domain. When the Fourier approach is combined with a time-shifted window function, known as the short-time Fourier transform (STFT) or Gabor transform, a spectral-temporal representation of the signal can be obtained. Hidden events inside the signal, however, are difficult to be tracked by the STFT because of equal resolution analysis in a time-frequency plane. Wavelet transforms (WT) is recently developed basing on a convolution operation between a signal and a wavelet function which allows to represent in the time-scale (frequency) domains with a flexible resolution, also called as a time-frequency analysis (Daubechies, [4]; Boggess and Narcowich, [1]; Bruns, [2]). The WT, furthermore, is advantageous over the FT and the STFT as an analyzing tool for non-stationary, non-linear and intermittent signals. The WT uses the wavelet functions, which can dilate (or compress) and translate basing on two parameters: scale (frequency) and translation (time shift) to apply short windows at low scales (high frequencies) and long windows at high scales (low frequencies). Basing on a discretized manner of the time-scale plane and characteristics of wavelets, the WT can be branched by continuous wavelet transform (CWT) and discrete wavelet transform (DWT). Since first invention in the late of 80’s years, the WT have been applied widely

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in such various fields as the digital signal processing, the numerical analysis, the signal, image coding and compressing, the flow and dynamic system identification, and their applications still are increasingly evolving. The first idea of using the WT to study the fluid turbulence was introduced by Farge, [6], in which some new concepts such as the local intermittency, the wavelet power spectrum, the energy decomposition were reviewed with potential applications of the CWT and the DWT for the turbulence. Applications of the CWT for meteorological records and geophysical turbulence were presented by Torrence and Compo,[12]; Kumar and Foufoula-Georgiou,[9]. Camussi and Guj [3] used orthonormal wavelets and the DWT to identify time information of coherent structures in wind turbulence. Hajj [7], moreover, decomposed the atmospheric turbulence and to investigate the energy intermittency of prominent scales.

In the field of the wind engineering, random responses of engineering structures due to the atmospheric turbulence has been based on typical models between turbulence and turbulent-induced forces. Because of the nature of random processes, fluctuating components of velocities have been described in terms of statistical parameters such as turbulence densities, integral scale, correlation and power spectrum density functions (Panofsky, [10]). Up to now, the classical approach to relation between velocity fluctuations of atmospheric turbulence and turbulent-induced forces (also known as buffeting forces) is basing on the “quasi-steady theory” (Davenport, [5]). Accordingly, it is assumed that the buffeting forces are proportional to the instantaneously velocity fluctuations. Uncertainly, the quasi-steady theory is consistent only if there are correspondences of the signal in the time domain and of the temporal-spatial structures in the frequency domain between the velocity fluctuations and wind forces. The spectral presentation, moreover, using the Fourier transform is only valid for stationary signals. Some authors (Tieleman,[11]; Hajj, [7]) presented the natural existence of the non-stationary characteristics of turbulence and the nonlinear cross-bicoherence between velocity fluctuations and unsteady surface pressure. Thus cross correlation between the velocity fluctuations and forces needs to detect in the time-frequency domain in order to obtain better knowledge, that is essential to treat with various problems from analytical computations to physical simulations in the field of wind engineering.

This paper will present the applications of the continuous and discrete wavelet transforms to detect the cross correlation between the wind turbulence (velocity fluctuations) and the turbulent-induced forces (buffeting forces) in the time-frequency domain. Experimental data have been obtained by the direct measurements on 2D sectional model in the wind tunnel.

2. Continuous wavelet transform

The continuous wavelet transform (CWT) of the given signal x(t) is defined as the convolution operation between the signal f(t) and the wavelet function )(, tsτψ :

∫∞

∞−

== dtttxxsW ssx )()(,),( *,, ττ

ψ ψψτ (1)

where ),( τψ sWx : the CWT coefficients at translation τ and scale s in the time-scale plane; the brackets denote the convolution operator; the asterisk * means complex conjugate; )(, tsτψ : wavelet function at translation τ and scale s of the basic wavelet function )(tψ that is also known as the mother wavelet function

⎟⎠⎞

⎜⎝⎛ −

=s

ts

tsτψψ τ

1)(, (2)

The mother wavelet functions, shortly called wavelets, satisfy such conditions as oscillatory functions with fast decay toward zero, zero mean value, normalization and admissibility condition:

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∫∫∞

∞−

∞−

== ;1|)(|;0)( 2 dttdtt ψψ ∞<= ∫∞

∞−

ωωωψ

ψ dC2|)(ˆ| (3)

The CWT coefficients ),( τψ sWx can be considered as a correlation coefficient and a measure of similitude between wavelet and signal in the time-scale plane. The higher coefficient is, the more the similarity. It is noted that the wavelet scale is not a Fourier frequency, but revealed as an inverse of frequency. Accordingly, a relationship between the wavelet scale (s) and the Fourier frequency (fF) can be easily determined basing on a central frequency of wavelet (f0) and a sampling frequency (fs):

sfff s

F0= (5)

Because the signal is sampled discretely, thus a discretized manner in the CWT computation can be used. The CWT coefficient at a certain scale and a time shift (position of wavelet) on signal is computed discretely:

)()(1)(1

0 spTqTqTx

ssW

N

q

xp

−= ∑

=

ψ (6)

where q: time index; p: time shift index; N: number of samples on discrete signal; T: time interval of samples or sampling period.

2.1. Morlet wavelet

The Morlet complex wavelet is the most applicable for the CWT, due to its containing of harmonic components and analogs to the Fourier transform (see Figure 1 and Figure 2):

( ) ( )2/expexp)( 20

4/1 ttit −= − ωπψ (7a)

( )( )20

24/1 2exp)(ˆ ωωππωψ −= − ss (7b) where 0ω , s: central circular frequency and wavelet scale of the Morlet wavelet.

Figure 2: Morlet wavelet at time shift τ=2s and at scales s=1,2,4 (solid & dashed lines: real & imaginary parts)

2.2. Measures of cross correlation detection

Traditionally, the FT-based quantities such as the cross power spectrum, the coherence and phase difference have been used to identify the cross correlation and the interrelation of two given signals in the frequency domain. Thus, one would like to develop corresponding WT-based tools in the same manner with FT’s ones that can detect the cross correlation of two signals in both frequency and time domains. Accordingly, the wavelet cross spectrum at time shift index p and scale s of two signals x(t), y(t), having their respectively CWT coefficients )(sW x

p , )(sW yp is defined as follows:

)()()( * sWsWsWCS yp

xp

xyp = (8)

Figure 1: Morlet wavelet and its Fourier transform

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.8

-0.6

-0.4

-0.2

0

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Am

plitu

de

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.8

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plitu

de

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plitu

de

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.8

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0.8Morlet wavelet

Time (s)

-4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Fourier transform

st

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where denotes smoothing operation in both time and scale directions; the wavelet auto spectrum

)(sWPS xxp of x(t) is defined in the same manner as )()()( * sWsWsWPS x

px

pxxp = .

With respect to the Fourier coherence, the squared wavelet coherence of x(t), y(t) is defined as the absolute value squared of the smoothed wavelet cross spectrum, normalized by the smoothed wavelet auto spectra:

|)(||)(|

|)(|)(

11

212

sWPSssWPSs

sWCSssWCO

yyp

xxp

xypxy

p −−

= (9)

where denotes the absolute operator; s-1 is used to convert to an energy density; by meaning

1)(0 2≤≤ sWCO xy

p in which wavelet coherence come to unit, two signals x(t), y(t) are full-correlated.

Then, the wavelet phase difference is given by:

])(Re[

])(Im[tan)(

1

11

sWCSs

sWCSss

xyp

xypxy

p −

−−=Φ (10)

where Im, Re denote the smoothed imaginary and real parts.

2.3. Time and scale smoothing, end effect

Averaging in both time and scale directions must be required, especially in computing the wavelet spectrum and wavelet coherence. The averaging techniques of the wavelet power spectrum in time and scale at the time-shifted index p can be expressed by such following formula (Torrence, [12]):

∑=

=2

1

22 |)(|1)(p

ppp

ap sWPSNsWPS (Time smoothing) (11a)

∑=

=2

1

22 |)(|)(

j

jj j

jpp s

sWPSC

tjsWPSδ

δδ

(Scale smoothing) (11b) where p assigned between p1 and p2; Na: number of averaged points( 112 +−= ppNa ); tj δδ , : factor of window width and sampling period ; Cδ: constant (for Morlet wavelet: 776.0,6.0 == δδ Cj )

Because the CWT deals with finite-length signals, errors and bias values usually occur at two ends of signals, known as the end effect. One effective solution to eliminate the end effect is to truncate number of results at two ends of signals after the CWT is completed. Removed number, however, depend on the wavelet scale, thus so-called cone of influence should be estimated for more accuracy.

3. Discrete wavelet transform

In the CWT, given signal x(t) is sampled continuously on all over time-scale plane and computed at every scale, thus it creates a redundant information and less effective computation. The time and the scale parameters should be discretized in a mutual dependence and with respect to the Nyquist’s sampling rule. A binary logarithmic disretization (dyadic grid) is commonly used due to octave by octave computational procedure:

mm ns 2,2 == τ (m,n=1,2,3…; refer to scale level and location of wavelets) (12)

The CWT is carried out in this dyadic grid of the time-scale plane, the wavelet function is written: )2(2)( 2/ ntt mm

mn −= −− ψψ (13)

If the wavelets form an orthonormal basis ( ∫∞

∞−= ln)()( δδψψ kmmnkl dttt ), signal )(tx can be

decomposed from the convolution operation (1) under such following sums (Daubechies,[4]):

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∑ ∑∑ ∑∞

−∞=

−∞=

−∞=

−∞=

==m n

mnmnm n

mnx txtnmWtx )(,)(),()( ** ψψψ (14)

3.1. Multiresolution analysis

The discrete wavelet transform (DWT) carries out a multiresolution analysis for decomposition and reconstruction of the given signal. The ),( nmW x this can be considered as digital filters, in which the signal passed through low-pass filters to decompose in low-frequency components and high-pass filters to analyze in high-frequency ones. Thus, the DWT uses two functions: wavelet function

)(tmnψ (or mother wavelet) and additionally associated one )(tmnφ , known as a scaling function (or father wavelet). The scaling function is used for the low-pass filtering, whereas the wavelet one for the high-pass filtering. The scaling function is expressed by the same way as the wavelet one:

)2(2)( 2/ ntt mmmn −= −− φφ (15)

The scaling and wavelet functions at lower resolution )(),( tt mm ψφ can be expressed by a weighted sum of shifted version of corresponding functions at higher resolution )2(),2( tt mm ψφ :

∑∞

−∞=

−=n

mm ntngt )2(][)( φφ ; ∑∞

−∞=

−=n

mm ntnht )2(][)( ψψ (16)

where g[n], h[n]: scaling and wavelet coefficients, respectively; linkage of g[n] and h[n] is obtained: ]1[)1(][ ngnh n −−= (17)

With the wavelet and scaling functions, the DWT’s decomposition of )(tx in (14) can be represented at certain level m0 by two summations as follows:

∑ ∑∑∞

=

−∞=

−∞=

+=0

00)(,)(,)(

mmmn

nmn

nnmnm txtxtx ψψφφ (18)

∑ ∑∑∞

=

−∞=

−∞=

+=0

00)(][)(][)(

mmmn

nm

nnmm tndtnatx ψφ (19)

As the result in (10), the first summation refers to the low resolution or coarse approximation of )(tx at the level m0 that corresponds to the low-frequency band components, whereas the second

one is sum of high-resolution details of )(tx , corresponds to the high-frequency band ones. The DWT’s decomposition at level M can be expressed by the summation of approximation coefficient at final level M (AM) and sum of detail coefficients at lower level Mm ≤ (Dm):

∑=

+=M

mmM DAtx

1)( (20)

Figure 3 describes three levels of the DWT’s decomposition of x(t) (here M=3), called as the filter-bank tree. Haar’s scaling and wavelet functions (also known as Daubechies family’s D1) used in this study is expressed in Figure 4.

Figure 3: Scheme of three-level filter-bank tree Figure 4: Haar’s scaling and wavelet functions

Low pass filters

High pass filters

(D1)

(D2)

(D3)

(A3)

(A1)

(A2)

0 0.5 1 1.50

0.2

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0.8

1

1.2

1.4haar wavelet: scaling function

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5haar wavelet: mother function

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3.2. Measure of cross energy correlation

Energy of the DWT’s decomposition coefficients plays important role than their amplitude. The energy of decomposition coefficients is determined in the same way as that of discrete signal. A concept of cross energy correlation of two given signals is defined as a product of energy of decomposition coefficients of these signals at the same level. Thus, the cross correlation of turbulence and forces can be detected thanks to such cross energy correlation.

4. Experimental apparatus

Data were measured in the open-circuit wind tunnel at the Bridge and Wind Engineering Laboratory, Kyoto University (BELCEKU) (See Figure 5). Wind turbulence was generated by grid device at the mean wind velocity U=3m/s, moreover, the velocity fluctuating components (u and w) of the wind turbulence were measured by the X-type hot-wire anemometer (Model 0252, Kanomax Co., Ltd.). Direct measurements of turbulent-induced forces (lift and moment) were simultaneously carried out on a motionless sectional model thanks to the dynamic multi-component load cells (Nissho Electronic Co., Ltd). Electric signals of velocity fluctuations and forces were passed through the 100Hz low-pass filters (E3201, NF Design Block Co., Ltd.) before discretized by the A/D converter (Thinknet DF3422, Pavec Co., Ltd.) with the sampling rate at 1000Hz during 10 seconds.

Figure 5: Configuration of open-circuit wind tunnel and experimental set-ups

Figure 6: Time series of u-, w-components (upper) and wind-induced lift, moment (lower)

5. Results and discussion

The CWT coefficients ( )(sW x ) of the u-, w-components and lift (L), moment (M) are computed in the first step. Then, the wavelet auto spectra ( )(sWPS xx ), wavelet cross spectra ( )(sWCS xy ) are carried out in the next one, and the wavelet coherence and the wavelet phase difference lastly computed. Figure 7 illustrates the wavelet auto-spectral maps, whereas the wavelet coherence maps of the u-, w-components and lift, moment are expressed by Figure 8.

5 10 15 20 25 30 35-1.5

-1

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0

0.5

1

1.5

Time (s)

u component (m

/s)

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1

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wcom

ponent (m

/s)

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Time (s)

Mom

ent (N)

F a n

S i l e n c e r

φ18 00

1 8 5 0 1 300655 01 3 0 0 3 000

1 40 0 0

Turn Table

Honeycomb Small Test Section

Large Test Section

Motor

Mesh

Wind 1st Entrance Cone

2nd Entrance Cone

Adjustable Wall

Grid Model

Support and Load C ll

2000 4200 2000

Plan View

Model

Grid

w

u

L

M

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Figure 7: Auto wavelet spectra maps and corresponding auto Fourier power spectra of u-,w-components and lift, moment

Figure 8: Wavelet coherence maps between u-,w-components and lift, moment

01

020

30

40

50

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en

cy (

Hz)

PSD of u-component(m2/s)

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en

cy (

Hz)

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qu

en

cy (

Hz)

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2)

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qu

en

cy (

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PSD of moment(N2.s)

)( fWPS uu )( fWPS ww

)( fWPS LL )( fWPS MM

)(2 fWCOuL )(2 fWCOuM

)(2 fWCOwL )(2 fWCOwM

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As can be seen from Figure 7, there is correspondent of spectral components between the wavelet auto-spectra and the Fourier auto-spectra, especially in the cases of lift and moment when peak energy occurs at certain spectral position. The Fourier spectrum focuses on certain frequencies, however, the wavelet auto spectrum contributes on the frequency bands. But no time information of spectral components is given by the Fourier spectrum, whereas the happening time of high-energy spectral components can be observed by the wavelet spectrum, moreover, the energy intermittency is seen as the nature of signals of velocities and forces. Taking the wavelet auto spectrum of moment for example, the high energy spectrum (associated with frequency band between 30÷40Hz) occurs intermittently in time domain at such time points and intervals as roughly 6s, 11s, 15s, 18s, 22s, 27÷33s, whereas high energy spectra of w-signal distribute more discretely on broader frequency band (see Figure 7). It is noted that the high energy contributions of turbulence u, w and induced forces are completely different in the spectral-temporal space.

The spectral-temporal information of the cross correlation between velocity fluctuations and lift, moment can be observed by the CWT’s wavelet coherence maps in the low frequency band between 0÷50Hz (see Figure 8). This cross correlation distributes intermittently and discretely on the time-frequency plane. The wavelet coherence maps of u and lift, u and moment, w and lift, w and moment exhibit differently on the time-frequency domain. On the other hand, the contribution of the turbulent components u, w on turbulent-induced lift and moment differs without rule from one case to another, and their relations are correlated at differently and randomly spectral-temporal points in the time-frequency plane.

The DWT’s 7-level resolution analysis in a means of energy using the Haar wavelet which consists of 8 decomposition coefficients D1, D2, D3, D4, D5, D6, D7, A7 has been carried out with measured u-, w-components and lift, moment of 10-second long records. Self energy and cross energy of DWT’s decomposition coefficients are computed. Figure 9 shows the energy contribution (in percentage) of decomposition coefficients for total energy of w-component and lift force. Frequency bands between 0÷500Hz (maximum frequency here is half of sampling rate) corresponding to each decomposition coefficients are given in a Table 1. As can be seen from Figure 9, the energy contribution of decomposition coefficients and dominant spectral components as well obviously differ between turbulence and force. Coefficients D4 (spectral component 31.25÷62.52Hz), D5 (15.62÷31.25Hz), D6 (7.81÷15.61Hz) contribute 21.8%, 31.9%, 21.7%, respectively on the whole energy of w-component, whereas coefficients D6 (spectral component 7.81÷15.62Hz), D7 (3.91÷7.81Hz), A7 (0÷3.91Hz) hold 23.8%, 19.2%, 37.1%, respectively on entire energy of lift. As a result, the lowest frequency band 0÷15.62Hz dominates over totally 80.1% energy on the lift, whereas the frequency band 7.81÷62.5Hz is prominent over 75.4%f of energy of the w-component signal. It seems that the linkage of corresponding decompositions of high energy spectra between turbulence and induced forces (here between w-component and lift) is less consistent, except coefficients D6 (7.81÷15.61Hz) in w signal and lift which jointly contribute considerable energy.

Figure 10 expresses the cross energy of the DWT’s corresponding decompositions between u-, w-components and lift, moment, respectively. High cross energy correlation can be detected associated with the time domain and spectral bands. It seems that the cross correlations between u and lift, u and moment, w and lift, w and moment are considerable at the low spectral bands among 0÷31.25Hz with D5, D6, D7, A7, and are small at the high spectral ones with D1, D2, D3, D4. Their cross energy correlations, moreover, differ on certain spectral bands. For example, u-component and lift are more correlated at spectral bands of A7(0÷3.91Hz) and D6 (7.81÷15.62Hz), whereas u and moment at spectral bands of A7(0÷3.91Hz) and D5(15.62÷31.25Hz). High cross correlations also occur at differently temporal points in these spectral bands (see Figure 10).

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Figure 9: Energy contribution of DWT decomposition on w-component(left) and lift(right)

Table 1: DWT decomposition’s frequency bands at levels Level Coefficients Freq. Band (Hz) Level Coefficients Freq. Band (Hz)

1 D1 500 ÷ 250 5 D5 31.25 ÷ 15.62 2 D2 250 ÷ 125 6 D6 15.62 ÷ 7.81 3 D3 125 ÷ 62.5 7 D7 7.81 ÷ 3.91 4 D4 62.5 ÷ 31.25 A7 3.91 ÷ 0

Figure 10: Cross energy of DWT decomposition between u,-w-components and lift, moment

-101

u(m

/s)

Cross energy of decomposition

-0.20

0.2

Lift(

N/m

)

0

0.020.04

A7

00.020.04

D7

00.020.04

D6

00.020.04

D 5

00.020.04

D4

00.020.04

D3

0

0.020.04

D 2

0 1 2 3 4 5 6 7 8 9 100

0.020.04

D1

Time (m)

-101

u(m

/s)

Cross energy of decomposition

-0.020

0.02

0

4E-3

A7

0

4E-3

D7

0

4E-3

D6

0

4E-3

D 5

0

4E-3

D4

0

4E-3

D3

0

4E-3

D 2

0 1 2 3 4 5 6 7 8 9 100

4E-3

D1

Time (m)

Mom

ent(N

)

-101

w(m

/s)

Cross energy of decomposition

-0.20

0.2

Lift(

N/m

)

0

0.020.04

A7

00.020.04

D 7

00.020.04

D6

00.020.04

D5

00.020.04

D 4

00.020.04

D 3

0

0.020.04

D2

0 1 2 3 4 5 6 7 8 9 100

0.020.04

D1

Time (m)

-101

w(m

/s)

Cross energy of decomposition

-0.020

0.02

Mom

ent(N

)

0

2E-3

A7

0

2E-3

D 7

0

4E-3

D6

0

4E-3

D5

0

5E-3

D 4

0

2E-3

D 3

0

E-3

D2

0 1 2 3 4 5 6 7 8 9 100

4E-4

D 1

Time (m)

D1 D2 D3 D4 D5 D6 D7 A70

5

10

15

20

25

30

35

Per

cent

age

(%)

Energy of decomposition

D1 D2 D3 D4 D5 D6 D7 A70

5

10

15

20

25

30

35

40

Per

cent

age

(%)

Energy of decomposition

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6. Conclusion

The wavelet transforms has been applied to detect the cross correlation between the velocity fluctuations and the induced lift, moment in the spectral-temporal representation. It can be concluded as follows:

(1) Dominant spectral bands that contribute considerably on total energy are different from the turbulence to the induced forces. There is no consistently correspondent between high energy spectrum of turbulence and that of induced forces. At the dominant spectral bands, the energy distribution, moreover, characterizes intermittently and discretely in the time domain. This may imply that signals of turbulence and induced forces exhibits as the non-stationary and non-linear characteristics.

(2) Cross correlation between turbulence and induced forces has been detected in the time-frequency plan, in the time domain and the spectral bands as can be observed from the wavelet coherence maps and the cross energy of decompositions. Their relations are correlated randomly in the spectral-temporal representation. Thus, the Fourier-transform-based classical model of relationship between the velocity fluctuations and the induced forces may contain uncertainties.

Acknowledgements

Author would like to express many thanks to the Professor M.Matsumoto, Associate Professor H.Shirato and Dr. T.Yagi of the Bridge and Wind Engineering Laboratory (BWECEKU), Graduate School of Engineering, Kyoto University for their continuous supports during fulfilling this work. Thanks also express to Messrs K.Yamane, T.Furukawa and Y.Sumikura for their collaboration in experiments in the wind tunnel.

Reference

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