- Email: [email protected]
Characterization of turbulence scales in the atmospheric surface layer with the continuous wavelet transform
JOURNAL OF
windengineering ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 69 71 (1997) 709 716
~~i~N~
Characterization of turbulence scales in the atmospheric surface layer with the continuous wavelet transform D.A. J o r d a n a, M.R. Hajj b'*, H.W. T i e l e m a n b aSchool of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, USA bDepartment of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061-0219, USA
Abstract Turbulence scales of the velocity components of wind in the atmospheric surface layer are characterized by using the continuous wavelet transform. The Morlet wavelet is used to study time variations of the energy of the different scales. By applying wavelet analysis, many of the shortcomings of Fourier analysis are overcome. The wavelet energy density is used to measure the intermittency levels and associated energy content. An intermittency factor is defined and used to quantify the intermittent character of turbulence scales. The percentage of energy in these events is also quantified. These parameters are important for wind tunnel simulation of the atmospheric surface layer conducted for the purpose of prediction of pressure peaks on low-rise structures.
Keywords. Low-rise structures; Turbulence scales; Wavelets; Interrnittency
1. Introduction Characterization of turbulence scales in the atmospheric surface boundary layer is required in the simulation or evaluation of wind forces on structures. In general, the velocity components of the wind are characterized by the typical random nature of turbulence. More specifically, these components usually show unsteady characteristics and exhibit high levels of fluctuations. To date, Fourier decomposition has been the main tool used to characterize turbulence dynamics in incident wind. The underlying process in Fourier decomposition is the representation of a complex time series by a set of complex sinusoids. There are several advantages for such a representation. First, turbulence scales can be represented by frequency components and their energy is obtained from the power spectrum. Second, higher-order moments can be used to quantify nonlinear coupling and energy transfer among spectral components of velocity and/or pressure signals. Third, estimates of spectral energy in * Corresponding author. E-mail: [email protected]. 0167-6105/97/$17.00 ~.(~ 1997 Elsevier Science B.V. All rights reserved. PII S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 1 9 9 - 2
710
D.A. Jordan et al./J. Wind Eng. Ind. Aeroclvn. 69 71 (1997) 709 716
particular frequency bands are statistically independent of energy estimates in other bands. Finally, for large sets of data, frequency decomposition can be obtained rapidly with F F T techniques. The above-mentioned advantages make use of frequency domain analysis to characterize atmospheric turbulence very attractive. However, as stressed in Ref. [1], such analysis is of limited importance in characterizing wind flows for the purpose of predicting wind loads on structures. First, the autocorrelation function properties of wind velocity components vary with segment lengths. Moreover, spectral curves,iS(f) versus frequency, tend to be flat. Consequently, the use of Fourier decomposition to obtain integral length scales is not satisfactory. Second, in order to reduce their variance, Fourier domain characteristics of wind velocity components must be averaged over time periods on the order of few hours. On the other hand, peak suction pressures take place over time periods of few seconds. Relating peak suction pressures to averaged characteristics can thus be misleading. Third, the representation of intermittent events, which are often observed in time series of incident wind, by infinite sinusoids may result in wrong conclusions and misunderstanding about incident wind characteristics. These shortcomings are due to the fact that, in frequency domain representation, all temporal information is eliminated. If provided, such information can contribute significantly to our understanding and characterization of turbulence in atmospheric wind. The objective of this work is to present time-scale domain analysis of the u- and v-velocity components of atmospheric wind. In particular, we will examine how the energy content of these scales varies with time and show how their intermittent characteristics can be quantified using the continuous wavelet transform. The complex Morlet wavelet is used to examine the temporal characteristics of the u- and v-velocity components of two wind records obtained at the Wind Engineering Research Field Laboratory (WERFL) at Texas Tech University.
2. Fourier-domain characteristics of velocity fluctuations in atmospheric wind Analytical models of normalized spectra of the velocity components of atmospheric wind have been developed to characterize energy content at all frequencies. The majority of these models are interpolation expressions between low-frequency and high-frequency asymptotes E2]. Fig. 1a and Fig. 1b show the time series of the u- and v-velocity components of two records, namely M15N085 and M15N086 obtained at the W E R F L . The sampling rate for these records is 10 Hz and the measurements were taken at an elevation of 13ft. More information about the terrain and experimental setup is given in Refs. F3,4]. The power spectral density functions of u- and v-velocity components of Fig. la and lb are given in one of our previous articles [-5]. Those results show that the observed spectra do not match the model. This mismatch is due to the fact that, in our estimation of the spectra, no segment averaging was performed. Because only one segment was used to obtain estimates of the spectra, the variance is equal to the estimated value. We should note here that when segment averaging over eight records was performed, the estimated spectra became smooth and approached
D.A. Jordan et al./,Z Wind Eng. Ind. Aerodyn. 69 71 (1997) 709 716
40
.
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.
.
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.
.
.
I
. . . .
[
. . . .
40
.
.
.
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.
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.
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.
.
.
711
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. . . .
20
20
10
o
.... 200
i .... 300
i .... 400 T'an¢ (s)
i 500
....
20
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go
-20 t 21)0
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Ttme (s) .
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
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-10
(a)
10 200
600
3170
400 T ~ e (s)
501)
-20
600
2111)
300
41)0 rtme (s)
(b)
Fig. 1. T i m e r e c o r d s o f u- a n d v - v e l o c i t y c o m p o n e n t s
from WERFL:
500
1500
(a) M I 5 N 0 8 5 ; (b) M 1 5 N 0 8 6 .
the model given in Ref. [23. However, such averaging requires long velocity records measured over a period of a few hours, especially when the energy content of the low-frequency components is to be determined. Such averaged spectra cannot be of any use in a wind tunnel experiment that aims at simulating full-scale pressure peaks that take place over a time period of few seconds. These peaks can only be related to time variations of the energy of turbulent fluctuations.
3. W a v e l e t
analysis
of velocity
fluctuations
in a t m o s p h e r i c
wind
The time-varying characteristics of any signal can be examined using short-time Fourier transform or the G a b o r transform. However, it is proposed to use wavelets in the analysis of atmospheric wind for two reasons. First, the wavelet analysis has the advantage of providing a better time-scale resolution. A short-time Fourier transform uses a single analysis window. Consequently, the time-frequency resolution is fixed over the entire time frequency plane. In contrast, the wavelet transform uses short windows at high frequencies and long windows at low frequencies. Because wind velocity components can contain low-frequency components over long durations and relatively high-frequency components of short duration, wavelet analysis is a more suitable technique than short-time Fourier domain analysis. Second, while frequency analysis is performed by projecting a signal onto a number of sinusoids which are infinite in extent, wavelet analysis is performed by projecting the signal onto a set of highly localized basis functions. These basis functions are called wavelets and are obtained from a single "mother" wavelet by dilations and translations. Thus, in wavelet analysis, the notion of a scale replaces that of frequency which leads to a time-scale representation. Because it is localized in time, a scale representation is more suitable than a frequency representation for examining temporal characteristics of turbulence.
D.A. Jordan et al./J. Wind Eng. Ind. Aeroclvn. 69 71 (1997) 709 716
712
Given a time signal, r i 0 , its continuous wavelet transform is defined as
~'(a,r)= {f(t),Oa.~) = i f(t)0*.~(t)dt, --
(1)
oc:
with
a
1
where 0(t) is a function called a wavelet, a a dilation parameter, r a translation or shift parameter and the * denotes complex conjugate when the wavelet is complex. The wavelet transform coefficients, ~#/'(a,r), represent the contribution of scales a to the signal at time, r. In this work, we use a complex Morlet wavelet mother function which is given by 0(t) -
exp(io)~,t) exp(-1ti2/2),
(3)
where coq, is a constant that forces the wavelet to be admissible, i.e. it possesses an inverse transform. Wavelet energy is defined from the wavelet coefficients as ~/f~#"*. F r o m Ref. [6], a wavelet energy density (energy per scale size) can be defined as 5t~~'°*/a. When integrated over time, the wavelet energy density yields the global wavelet energy spectrum which gives the energy content at that scale. The digital implementation of the continuous wavelet transform is a discrete convolution between the sampled time series and sampled versions of the analyzing wavelet at all scales. All scaled versions of the complex Morlet wavelet were sampled with enough samples to avoid aliasing and the convolutions were performed in the frequency domain using the FFT. A relationship can be established between the wavelet scale and the peak frequency of the scaled wavelet bandpass filter. In this work, this relationship is given by fp =
2.90 a
(4)
Forty-seven values of a were spaced logarithmically to cover a frequency range from 2.9 to about 0.02 Hz. The large scale cut-offwas determined by the number of points in the sampled time series. More information about the application of wavelet analysis to the study of turbulence is given in Ref. [6]. Contour plots of the wavelet energy density of the u- and v-velocity components of the incident wind are shown in Fig. 2a and Fig. 2b. The axes are time in seconds and the natural log of a dimensionless scale, a. The plots show that scales corresponding to frequencies below 0.15 Hz contribute significantly but intermittently to the signal. Such contribution is seen in the high amplitude at various times. By comparison with the time series it can be determined that these peaks correspond to jumps in the velocity signal. The plots also show the presence of smaller scales indicated by the streaks. These scales correspond to the frequency range between 0.25 and 1.0 Hz. The dominant scale will have the highest level of energy in the global wavelet energy
D.A. Jordan et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 709-716
(a)
713
uoss
_=
0
250
500
750
500
750
500
750
500
750
Time (s) v085
_=
0
250 Time (s)
(b)
u086
0
250 Time(s) v086
J
0
250 Time (s)
0
25
50
75
100
125
WW*/a
Fig. 2. Wavelet energy density, ~/'~t"*/a, for u- and v-velocity time series: (a) MI5N085; (b) M15N086.
714
D.A. J o r d a n et al./J.
W i n d Eng. Ind. A e r o d y n . 69 7l (1997) 709
716
spectrum. Integrating the energy density shown in Fig. 2a and Fig. 2b shows that the highest energy content is in the scales around In a 4.1 which corresponds to a peak frequency of about 0.05 Hz or time scale about 20 s.
4. Intermittency quantification By examining time records of the velocity components in the atmospheric wind it can be easily noticed that the fluctuations are highly intermittent. By intermittency, we mean that the energy is not evenly distributed in time or space [6] rather than the fraction of turbulence against non-turbulence as defined by others. Following Ref. [7], we define an intermittency factor as the percentage of time the measuring device sees the variable in its higher amplitude state. Because turbulence fluctuations contain different time scales, we have opted to examine the intermittency of scales rather than the intermittency of the whole time record. In order to estimate the percentage of time
I
InterrnitterlcyI
0.4
_e..-~
.~;.~..~._~.~_.w.~.q:.~.
........ X
0.5
! ..........
~rf\
.................................................
0.2
i ......
z
0.1 ,, ..........
I 0.4
~" 0.3
lna
3
i
1
> :i, ......... i
\ f
[
-,,
1
0.5
-0.5
-1
0
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0.5
!
2
3
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0.4
1
.•
0.5
0.3
0.2
0.1
F i g . 3. I n t e r m i t t e n c y
\
-0.5
0
0.2
(b)
,
0.1
5
.... '
~.
0 4
................ •
:
0
Interrni"encYlI --%Ener~l ....... _
1
. . . . . . . . . . . . . . . .
0.2
J .......
0
, , , ,
0.3
=
~. E
-0.5
0
(a)
0.4
1
0.3
Intermiuencyl J --%Ener~'/I
I
[ - - % EnergY I
lna
4
5
factor and percent energy content
0.I
0
-0.5
....
ii~,
2
~a
ill,
3
o f u- a n d t , - v e l o c i t y c o m p o n e n t s :
4
(a) M I 5 N 0 8 5 ; (b)
D.A. Jordan et al./J. Wind Eng. InK Aerodyn. 69 71 (1997) 709 716
715
where these scales are in their higher amplitude state, it is necessary to define a threshold value and time of integration of energy. Here, the threshold is defined as twice the average energy content of that scale. The time of integration of a scale is taken to be equal to the inverse of the peak frequency of the magnitude of the Fourier transform of the wavelet. It must be recognized that the measured intermittency factor is dependent on the choice of threshold. However, varying the threshold value between 1.5 and 2.5 times the average did not cause significant variations in the measured intermittency factor. Once intermittent peaks in the wavelet transform energy were located, the total energy in these peaks were calculated. Fig. 3a and Fig. 3b show the intermittency factor and percentage of energy content in the peaks of wavelet transform energy of the u- and v-velocity components for records M15N085 and M15N086. The results show that all scales are intermittent, i.e. energy is not distributed evenly in time. For M 15N086, the measured intermittency factor, for both u- and v-components, varies around 0.15 for the smaller scales (lna ~< 3.9) and is slightly higher, up to 0.2, for the larger scales. Another feature is the percent of energy contained in these intermittent peaks. For the larger scales, the amount is 5 ~ 6 0 % . These results imply that up to 60% of the total energy of these scales appears over less than 20% of the time. Similar trends are shown in the u- and v-components of M15N086 except for a drop in intermittency factor and percent of energy near In a = 4.5. However, it remains noticeable that one c o m m o n feature of all the plots of u- and v-components is that at least one large component (In a ~> 4.0) has an intermittency factor around 0.2, where the intermittent peaks contain about 60% of the total energy.
5. Conclusions In this work, shortcomings of using Fourier decomposition for wind tunnel simulation are pointed out. As an alternative, a continuous wavelet transform was used to characterize turbulence scales of the velocity components of wind in the atmospheric surface layer. Because it resolves temporal information, wavelet analysis overcomes many of the shortcomings of Fourier analysis. By applying the Morlet wavelet, the time variation of the energy of the different turbulence scales was determined from the wavelet energy density. It was shown that these scales are highly intermittent. Fifty to sixty percent of the energy of these scales could appear in less than 20% of the time. These intermittent peaks might need to be simulated in wind tunnel experiments for the purpose of simulating or predicting low-pressure peaks.
Acknowledgements The financial support of the National Science Foundation under G r a n t # CMS9412905 is greatly acknowledged. The author also wish to thank Dr. K. Mehta and his staff for providing the W E R F L data.
716
D.A. Jordan et aL /J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 709-716
References [1] M.R. Hajj, H.W. Tieleman, Application of wavelet analysis to atmospheric wind in relevance to loadings on low-rise structures, J. Fluids Eng., accepted. [2] H.W. Tieleman, Universality of velocity spectra, J. Wind Eng. had. Aerodyn. 56 (1995) 55 69. [3] M.L. Levitan, K.C. Mehta, Texas Tech field experiments for wind loads, Part I: building and pressure measuring system, Presented at the 8th Int. Conf. on Wind Engineering, London, Ontario, Canada, 8 12 July, 1991. [4] M.L. gevitan, K.C. Mehta, Texas Tech field experiments for wind loads, Part II: building and pressure measuring system, Presented at the 8th Int. Conf. on Wind Engineering, London, Ontario, Canada, 8 12 July, 1991. [5] M.R. Hajj, H.W. Tieleman, M. Bikdash, Characterization of turbulence scales in atmospheric wind by orthonormal wavelets, Proc. l lth ASCE/EMD Specialty Conf., vol. 2, 1996, pp. 971 974. [6] M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24 (1992) 395 457. [7] A.A. Townsend, The Structure of Turbulent Shear Flows, Cambridge University Press, Cambridge, 1956.
windengineering ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 69 71 (1997) 709 716
~~i~N~
Characterization of turbulence scales in the atmospheric surface layer with the continuous wavelet transform D.A. J o r d a n a, M.R. Hajj b'*, H.W. T i e l e m a n b aSchool of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, USA bDepartment of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061-0219, USA
Abstract Turbulence scales of the velocity components of wind in the atmospheric surface layer are characterized by using the continuous wavelet transform. The Morlet wavelet is used to study time variations of the energy of the different scales. By applying wavelet analysis, many of the shortcomings of Fourier analysis are overcome. The wavelet energy density is used to measure the intermittency levels and associated energy content. An intermittency factor is defined and used to quantify the intermittent character of turbulence scales. The percentage of energy in these events is also quantified. These parameters are important for wind tunnel simulation of the atmospheric surface layer conducted for the purpose of prediction of pressure peaks on low-rise structures.
Keywords. Low-rise structures; Turbulence scales; Wavelets; Interrnittency
1. Introduction Characterization of turbulence scales in the atmospheric surface boundary layer is required in the simulation or evaluation of wind forces on structures. In general, the velocity components of the wind are characterized by the typical random nature of turbulence. More specifically, these components usually show unsteady characteristics and exhibit high levels of fluctuations. To date, Fourier decomposition has been the main tool used to characterize turbulence dynamics in incident wind. The underlying process in Fourier decomposition is the representation of a complex time series by a set of complex sinusoids. There are several advantages for such a representation. First, turbulence scales can be represented by frequency components and their energy is obtained from the power spectrum. Second, higher-order moments can be used to quantify nonlinear coupling and energy transfer among spectral components of velocity and/or pressure signals. Third, estimates of spectral energy in * Corresponding author. E-mail: [email protected]. 0167-6105/97/$17.00 ~.(~ 1997 Elsevier Science B.V. All rights reserved. PII S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 1 9 9 - 2
710
D.A. Jordan et al./J. Wind Eng. Ind. Aeroclvn. 69 71 (1997) 709 716
particular frequency bands are statistically independent of energy estimates in other bands. Finally, for large sets of data, frequency decomposition can be obtained rapidly with F F T techniques. The above-mentioned advantages make use of frequency domain analysis to characterize atmospheric turbulence very attractive. However, as stressed in Ref. [1], such analysis is of limited importance in characterizing wind flows for the purpose of predicting wind loads on structures. First, the autocorrelation function properties of wind velocity components vary with segment lengths. Moreover, spectral curves,iS(f) versus frequency, tend to be flat. Consequently, the use of Fourier decomposition to obtain integral length scales is not satisfactory. Second, in order to reduce their variance, Fourier domain characteristics of wind velocity components must be averaged over time periods on the order of few hours. On the other hand, peak suction pressures take place over time periods of few seconds. Relating peak suction pressures to averaged characteristics can thus be misleading. Third, the representation of intermittent events, which are often observed in time series of incident wind, by infinite sinusoids may result in wrong conclusions and misunderstanding about incident wind characteristics. These shortcomings are due to the fact that, in frequency domain representation, all temporal information is eliminated. If provided, such information can contribute significantly to our understanding and characterization of turbulence in atmospheric wind. The objective of this work is to present time-scale domain analysis of the u- and v-velocity components of atmospheric wind. In particular, we will examine how the energy content of these scales varies with time and show how their intermittent characteristics can be quantified using the continuous wavelet transform. The complex Morlet wavelet is used to examine the temporal characteristics of the u- and v-velocity components of two wind records obtained at the Wind Engineering Research Field Laboratory (WERFL) at Texas Tech University.
2. Fourier-domain characteristics of velocity fluctuations in atmospheric wind Analytical models of normalized spectra of the velocity components of atmospheric wind have been developed to characterize energy content at all frequencies. The majority of these models are interpolation expressions between low-frequency and high-frequency asymptotes E2]. Fig. 1a and Fig. 1b show the time series of the u- and v-velocity components of two records, namely M15N085 and M15N086 obtained at the W E R F L . The sampling rate for these records is 10 Hz and the measurements were taken at an elevation of 13ft. More information about the terrain and experimental setup is given in Refs. F3,4]. The power spectral density functions of u- and v-velocity components of Fig. la and lb are given in one of our previous articles [-5]. Those results show that the observed spectra do not match the model. This mismatch is due to the fact that, in our estimation of the spectra, no segment averaging was performed. Because only one segment was used to obtain estimates of the spectra, the variance is equal to the estimated value. We should note here that when segment averaging over eight records was performed, the estimated spectra became smooth and approached
D.A. Jordan et al./,Z Wind Eng. Ind. Aerodyn. 69 71 (1997) 709 716
40
.
.
.
.
.
.
.
.
I
. . . .
[
. . . .
40
.
.
.
.
.
.
.
.
.
.
.
711
.
I
. . . .
20
20
10
o
.... 200
i .... 300
i .... 400 T'an¢ (s)
i 500
....
20
20
0
go
-20 t 21)0
300
400
500
600
Ttme (s) .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
~" -10
-10
(a)
10 200
600
3170
400 T ~ e (s)
501)
-20
600
2111)
300
41)0 rtme (s)
(b)
Fig. 1. T i m e r e c o r d s o f u- a n d v - v e l o c i t y c o m p o n e n t s
from WERFL:
500
1500
(a) M I 5 N 0 8 5 ; (b) M 1 5 N 0 8 6 .
the model given in Ref. [23. However, such averaging requires long velocity records measured over a period of a few hours, especially when the energy content of the low-frequency components is to be determined. Such averaged spectra cannot be of any use in a wind tunnel experiment that aims at simulating full-scale pressure peaks that take place over a time period of few seconds. These peaks can only be related to time variations of the energy of turbulent fluctuations.
3. W a v e l e t
analysis
of velocity
fluctuations
in a t m o s p h e r i c
wind
The time-varying characteristics of any signal can be examined using short-time Fourier transform or the G a b o r transform. However, it is proposed to use wavelets in the analysis of atmospheric wind for two reasons. First, the wavelet analysis has the advantage of providing a better time-scale resolution. A short-time Fourier transform uses a single analysis window. Consequently, the time-frequency resolution is fixed over the entire time frequency plane. In contrast, the wavelet transform uses short windows at high frequencies and long windows at low frequencies. Because wind velocity components can contain low-frequency components over long durations and relatively high-frequency components of short duration, wavelet analysis is a more suitable technique than short-time Fourier domain analysis. Second, while frequency analysis is performed by projecting a signal onto a number of sinusoids which are infinite in extent, wavelet analysis is performed by projecting the signal onto a set of highly localized basis functions. These basis functions are called wavelets and are obtained from a single "mother" wavelet by dilations and translations. Thus, in wavelet analysis, the notion of a scale replaces that of frequency which leads to a time-scale representation. Because it is localized in time, a scale representation is more suitable than a frequency representation for examining temporal characteristics of turbulence.
D.A. Jordan et al./J. Wind Eng. Ind. Aeroclvn. 69 71 (1997) 709 716
712
Given a time signal, r i 0 , its continuous wavelet transform is defined as
~'(a,r)= {f(t),Oa.~) = i f(t)0*.~(t)dt, --
(1)
oc:
with
a
1
where 0(t) is a function called a wavelet, a a dilation parameter, r a translation or shift parameter and the * denotes complex conjugate when the wavelet is complex. The wavelet transform coefficients, ~#/'(a,r), represent the contribution of scales a to the signal at time, r. In this work, we use a complex Morlet wavelet mother function which is given by 0(t) -
exp(io)~,t) exp(-1ti2/2),
(3)
where coq, is a constant that forces the wavelet to be admissible, i.e. it possesses an inverse transform. Wavelet energy is defined from the wavelet coefficients as ~/f~#"*. F r o m Ref. [6], a wavelet energy density (energy per scale size) can be defined as 5t~~'°*/a. When integrated over time, the wavelet energy density yields the global wavelet energy spectrum which gives the energy content at that scale. The digital implementation of the continuous wavelet transform is a discrete convolution between the sampled time series and sampled versions of the analyzing wavelet at all scales. All scaled versions of the complex Morlet wavelet were sampled with enough samples to avoid aliasing and the convolutions were performed in the frequency domain using the FFT. A relationship can be established between the wavelet scale and the peak frequency of the scaled wavelet bandpass filter. In this work, this relationship is given by fp =
2.90 a
(4)
Forty-seven values of a were spaced logarithmically to cover a frequency range from 2.9 to about 0.02 Hz. The large scale cut-offwas determined by the number of points in the sampled time series. More information about the application of wavelet analysis to the study of turbulence is given in Ref. [6]. Contour plots of the wavelet energy density of the u- and v-velocity components of the incident wind are shown in Fig. 2a and Fig. 2b. The axes are time in seconds and the natural log of a dimensionless scale, a. The plots show that scales corresponding to frequencies below 0.15 Hz contribute significantly but intermittently to the signal. Such contribution is seen in the high amplitude at various times. By comparison with the time series it can be determined that these peaks correspond to jumps in the velocity signal. The plots also show the presence of smaller scales indicated by the streaks. These scales correspond to the frequency range between 0.25 and 1.0 Hz. The dominant scale will have the highest level of energy in the global wavelet energy
D.A. Jordan et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 709-716
(a)
713
uoss
_=
0
250
500
750
500
750
500
750
500
750
Time (s) v085
_=
0
250 Time (s)
(b)
u086
0
250 Time(s) v086
J
0
250 Time (s)
0
25
50
75
100
125
WW*/a
Fig. 2. Wavelet energy density, ~/'~t"*/a, for u- and v-velocity time series: (a) MI5N085; (b) M15N086.
714
D.A. J o r d a n et al./J.
W i n d Eng. Ind. A e r o d y n . 69 7l (1997) 709
716
spectrum. Integrating the energy density shown in Fig. 2a and Fig. 2b shows that the highest energy content is in the scales around In a 4.1 which corresponds to a peak frequency of about 0.05 Hz or time scale about 20 s.
4. Intermittency quantification By examining time records of the velocity components in the atmospheric wind it can be easily noticed that the fluctuations are highly intermittent. By intermittency, we mean that the energy is not evenly distributed in time or space [6] rather than the fraction of turbulence against non-turbulence as defined by others. Following Ref. [7], we define an intermittency factor as the percentage of time the measuring device sees the variable in its higher amplitude state. Because turbulence fluctuations contain different time scales, we have opted to examine the intermittency of scales rather than the intermittency of the whole time record. In order to estimate the percentage of time
I
InterrnitterlcyI
0.4
_e..-~
.~;.~..~._~.~_.w.~.q:.~.
........ X
0.5
! ..........
~rf\
.................................................
0.2
i ......
z
0.1 ,, ..........
I 0.4
~" 0.3
lna
3
i
1
> :i, ......... i
\ f
[
-,,
1
0.5
-0.5
-1
0
M15N086.
0.5
!
2
3
lna
IntermatencYl I --%Energy I
0.4
1
.•
0.5
0.3
0.2
0.1
F i g . 3. I n t e r m i t t e n c y
\
-0.5
0
0.2
(b)
,
0.1
5
.... '
~.
0 4
................ •
:
0
Interrni"encYlI --%Ener~l ....... _
1
. . . . . . . . . . . . . . . .
0.2
J .......
0
, , , ,
0.3
=
~. E
-0.5
0
(a)
0.4
1
0.3
Intermiuencyl J --%Ener~'/I
I
[ - - % EnergY I
lna
4
5
factor and percent energy content
0.I
0
-0.5
....
ii~,
2
~a
ill,
3
o f u- a n d t , - v e l o c i t y c o m p o n e n t s :
4
(a) M I 5 N 0 8 5 ; (b)
D.A. Jordan et al./J. Wind Eng. InK Aerodyn. 69 71 (1997) 709 716
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where these scales are in their higher amplitude state, it is necessary to define a threshold value and time of integration of energy. Here, the threshold is defined as twice the average energy content of that scale. The time of integration of a scale is taken to be equal to the inverse of the peak frequency of the magnitude of the Fourier transform of the wavelet. It must be recognized that the measured intermittency factor is dependent on the choice of threshold. However, varying the threshold value between 1.5 and 2.5 times the average did not cause significant variations in the measured intermittency factor. Once intermittent peaks in the wavelet transform energy were located, the total energy in these peaks were calculated. Fig. 3a and Fig. 3b show the intermittency factor and percentage of energy content in the peaks of wavelet transform energy of the u- and v-velocity components for records M15N085 and M15N086. The results show that all scales are intermittent, i.e. energy is not distributed evenly in time. For M 15N086, the measured intermittency factor, for both u- and v-components, varies around 0.15 for the smaller scales (lna ~< 3.9) and is slightly higher, up to 0.2, for the larger scales. Another feature is the percent of energy contained in these intermittent peaks. For the larger scales, the amount is 5 ~ 6 0 % . These results imply that up to 60% of the total energy of these scales appears over less than 20% of the time. Similar trends are shown in the u- and v-components of M15N086 except for a drop in intermittency factor and percent of energy near In a = 4.5. However, it remains noticeable that one c o m m o n feature of all the plots of u- and v-components is that at least one large component (In a ~> 4.0) has an intermittency factor around 0.2, where the intermittent peaks contain about 60% of the total energy.
5. Conclusions In this work, shortcomings of using Fourier decomposition for wind tunnel simulation are pointed out. As an alternative, a continuous wavelet transform was used to characterize turbulence scales of the velocity components of wind in the atmospheric surface layer. Because it resolves temporal information, wavelet analysis overcomes many of the shortcomings of Fourier analysis. By applying the Morlet wavelet, the time variation of the energy of the different turbulence scales was determined from the wavelet energy density. It was shown that these scales are highly intermittent. Fifty to sixty percent of the energy of these scales could appear in less than 20% of the time. These intermittent peaks might need to be simulated in wind tunnel experiments for the purpose of simulating or predicting low-pressure peaks.
Acknowledgements The financial support of the National Science Foundation under G r a n t # CMS9412905 is greatly acknowledged. The author also wish to thank Dr. K. Mehta and his staff for providing the W E R F L data.
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