COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Simulation of Stochastic Fluctuating Wind Field Using the Wave
Superposition Method with Random Frequencies
D. J. Han *, J. J. Luo
Department of Civil Engineering, South China University of Technology, Guangzhou, 510640 China
Email: ardjhan@scut.edu.cn, axljj@163.com
Abstract The wave superposition method is one of the effective methods to simulate the stochastic fluctuating wind
field. The simulated wind speed samples with single-indexing frequencies are periodic, with period of T0 = 2π ⋅ N ωu ,
in which N → ∞ and ωu being the upper cutoff frequency. While using the double-indexing frequencies,
theoretically the samples’ period could increase to nT0 ,where n is the number of multivariate. However, in fact, the
period is not increased. It still keeps the value of T0 . Two examples using the double-indexing frequencies indicate this
problem. In this paper, the reason of this problem is studied in detail. And an improved method to generate wind speed
samples with non-periodic characteristics is proposed. Formulation is given. In this formulation, the problem existing
is that of the enormous memory consumption and time consuming. Therefore, characteristics for the curves of the
factorized power spectral density functions changing with frequencies are further studied. And the cubic uniform
B-spline interpolation method and the cut-off part of cosine terms method are utilized to improve the simulation
efficiency. An example using proposed methods indicates that the simulation is successful. The simulated stochastic
fluctuating wind field has features of non-periodicity. And the algorithm is effective and accurate.
Key words: wave superposition method, double-indexing frequencies, random rrequencies, cubic uniform B-spline
interpolation method, cut-off part of cosine terms method
THE THEORY OF WAVE SUPERPOSITION METHOD
The Wave Superposition Method including the Constant Amplitude Wave Superposition (CAWS) [1] and
theWeighted Amplitude Wave Superposition (WAWS) [2] is one of the traditional methods which are used to simulate
the fluctuating wind velocity. The principle of this method is that a series of cosine terms superpose to form the
artificial wind velocity fluctuations. In practice, the fluctuating wind field should be treated as a one-dimensional
multivariate(1D-nV) stationary Gaussian stochastic process having mean value equal to zero. According to the
Deodatis’ theory [3], consider an n-variates stochastic vector process with the corss-spectral density matrix given
by S (ω ) , the components v(t ) = {v1 (t ) v2 (t ) … vn (t )}T are defined as
%
%
j
N
v j (t ) = 2Δω ∑∑ H jm (ω ) cos(ωt + Φ ml − θ jm (ω ))
j = 1, 2,..., n
(1)
m =1 l =1
In which, the range of ω is [ωmin , ωmax ] , where the ωmin is the lower cutoff circular frequency and the ωmax is the upper
one. Frequency increment Δω = (ωmax − ωmin ) / N , where N is a large enough positive integer. H jm (ω ) is an element
of lower triangular matrix H (ω ) which is decomposed from the power spectral density(PSD) matrix S (ω ) by the
%
%
Cholesky factorization method. Also, H jm (ω ) is a real function because S (ω ) is a semi-definite positive real matrix,
%
and Φ ml are independent random phase angles distributed uniformly over the interval [ 0, 2π ] . If the height difference
of points is not large, it is reasonable to assume θ jm (ω ) = 0 . While using the single-indexing frequencies given by
— 60 —
ωl = ωmin + l ⋅ Δω ( l = 1, 2,..., N )
(2)
the period of this wind speed sample is T0 = 2π / Δω . In order to increase the period, the double-indexing frequencies
[4] are introduced, defined as:
ωml = ωmin + (l − 1)Δω +
m
Δω ; l = 1, 2,..., N ; m = 1, 2,..., n
n
(3)
Then, periodicity of the sample becomes Tn = n ⋅ (2π / Δω ) theoretically.
A DISCUSSION ON THE PERIODICITY OF WIND VELOCITY FLUCTUATION SAMPLES
Two examples involving generating wind velocity fluctuations using Eq. (1) with double-indexing frequencies are
demonstrated to analyze the periodicity. Consider 10 points ( n = 10 , in Example 1) and 100 points ( n = 100 , in
Example 2). Assume that all the points are at the same height and that the interval of two adjacent points is 10 meters.
The Kaimal PSD function[5] is selected to model the longitudinal wind speed samples. Meanwhile, other parameters
are: the height of each point is z = 37.5m ; the mean wind speed is U ( z ) = 38.3m / s ; and the surface roughness is of
type II. The range of radius frequency is [ωmin , ωmax ] = [ 0, 4π ] rad / s , N = 1024 . The time step is Δt = 0.25s .
Theoretically period of the Example 1 is T1 = 10 ⋅ 2π ( 2 ⋅ 2π /1024 ) = 5120 s and the Example 2 is T2 = 51200 s . The
generated time history of wind speed and temporal auto-correlation function of Example 1 at Point 1 and Point 10 for half
of period T1 2 = 2560s are displayed in Fig.1 and those of Example 2 for 2560s at Point 1 and Point 100 in Fig. 2.
(a) Point 1
(b) Point 10
Figure 1: Simulated wind speed samples with n = 10
(a) Point 1
(b) Point 100
Figure 2: Simulated wind speed samples with n = 100
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It can be seen that:
(1) The peak values of each auto-correlation function appear at the intervals of T0 = 512s which coincide with period
of the wind speed samples using single-indexing frequencies.
(2) The maximum value of each auto-correlation function appears at t = 0 . After that, the peak values of Point 1 in two
examples decline gradually, but the peak values of other points vary randomly.
(3) In Example 1, the periodicity of each point is not apparent by observing the time history. On the contrary, in
Example 2, it’s easy to find out the period of Point 1 from the time history; however, the period of Point 100 is not
obvious.
The reason of these phenomena can be analyzed by the frequency content of the samples. Taking into account sample
function of Point 1 given by
N
v1 (t ) = 2Δω ∑ H11 (ω1l ) cos(ω1l t + Φ1l )
(4)
l =1
As for Example 1, the frequency content are Δω 10 , 11Δω 10 , 21Δω 10 ,……, 102301Δω 10 . Therefore, the
periodicity of the first cosine term, named as C1 , is T1 = 10 ⋅ T0 = 5120s . The periodicity of other cosine terms is not
longer than T0 = 512s . Hence the summation of these terms, named as P1 , is still periodic, with period T0 . As a result,
periodicity of the whole sample superposed by C1 and P1 is 10 times longer than T0 theoretically. But in fact, the
actual period of wind speed sample is still T0 , but not 10T0 , owing to the fact that P1 is superimposed by every 1 10
part of C1 . This can be proved in the case that peak values of auto-correlation function appear at the intervals of T0 .
Consider the condition of n points, the whole period of cosine term C1 is then divided into n parts. When n is not
large, the difference of amplitude between two adjacent parts is notable. Therefore, after C1 is combined by P1 , it is
difficult to observe the periodicity from wind velocity time history in Fig.1. As n increases, the change of amplitude
between two adjacent parts is not distinct gradually. So the periodicity is easy to be found out from time history in Fig.
2 at the intervals of T0 .
In addition, consider the simulated sample function of Point 10 in Example 1:
10
N
v10 (t ) = 2Δω ∑∑ H jm (ω ml ) cos(ω ml t + Φ ml )
(5)
m =1 l =1
It can be regarded as a superposition of ten series of cosine terms with respect of frequencies ωml (m = 1, 2,...10) and
different amplitudes. Each cosine term is a combination of Cm and Pm . On the other hand, v10 (t ) can be considered as
a composition of two parts: Csum superposed by Cm , with the period of 10T0 and Psum superposed by Pm , with the
period of T0 . Then periodicity of the sample can be analyzed by means of the method utilized for Point 1.
It’s necessary to mention that: in two examples, the amplitude of Csum varies more complex than that of C1 because
Csum consists of Cm with different amplitudes. So the more Cm joint to Csum , the more complex the amplitude of Csum
varies. After Csum is combined by Psum , periodicity of the wind speed sample is difficult to be found out from the time
histories in Fig.1.b) and Fig.2.b). Moreover, the peak values of auto-correlation function of other points vary
irregularly owing to the complication of the varying amplitude, too.
In brief, periodicity of simulated wind speed samples in terms of double-indexing frequencies is still T0 because the
interval of single-indexing frequencies Δω is divided into n parts equally, that leads to the pseudo-periodicity nT0 .
In fact, equally dividing the interval of frequencies is required by FFT technique[6] which can improve efficiency of
the summations of cosine terms dramatically. But it’s not reasonable to simulate the natural wind speed sample with
frequency content that are stochastic. Significantly, the application of auto-correlation function is a better way to find
out the actual periodicity hidden behind the wind speed time history.
SIMULATED WIND SPEED SAMPLE WITH STOCHASTIC FREQUENCIES
The observations of the natural wind show that the wind speed is a stochastic process without periodicity. The
frequencies of measurement sample are of randomness that leads to the fact that the peak value of auto-correlation
function only appears at t = 0 , and then the value of function tends to zero. Therefore, to simulate a natural wind speed
— 62 —
sample, the frequency content should be stochastic. According to Shinozuka’s theory [7], a 1D-nV stationary process
v(t ) = {v1 (t ) v2 (t ) … vn (t )}T is defined as:
%
12
j ⎧
⎫⎪
⎪ ⎛2⎞ N
(6)
v j (t ) = ∑ ⎨σ m ⎜ ⎟ ∑ γ jm (ωml ) × cos ⎣⎡ωml t + ϕ ml + θ jm (ωml ) ⎦⎤ ⎬ ( j = 1, 2,..., n)
m =1 ⎩
⎪ ⎝ N ⎠ l =1
⎭⎪
Where ωml (l = 1, 2,..., N ) are random variables identically and independently distributed with the density function
g m (ω ) = H mm (ω )
2
σ m2
(7)
with
∞
σ m2 = ∫ H mm (ω ) dω
2
(8)
−∞
the relative amplitudes γ jm (ωml ) are determined by
γ jm (ωml ) = H jm (ωml ) H mm (ωml )
(9)
θ jm (ωml ) are the phase angles equal to zero; and ϕml are sequences of independent random phase angles distributed
over the interval [ 0, 2π ] . In Eq. (6), ωml are independent random variables that can reflect the actual frequency
characters of natural wind. The simulated steps are given by:
(a) Decompose the PSD matrix S (ω ) using Cholesky factorization method in order to acquire the lower triangular
%
matrix H (ω ) , then attain σ m2 and g m (ω ) according to (8), (7), respectively.
%
(b) Generate the independent random frequencies ωml which conform to the random distribution with density function
g m (ω ) .
(c) Decompose the PSD matrix S (ωml ) again to obtain γ jm (ωml ) .
%
(d) Superimpose the cosine terms to synthesize wind velocity fluctuations using Eq. (6).
It can be seen from the simulated process that amount of computation is enormous for two reasons: Firstly, the
Cholesky factorizations are applied in two steps. In step (a),since there are no explicit expressions for H mm (ω ) , the
frequency range [ωmin , ωmax ] needs to be divided into k intervals equally.And, therefore, the Cholesky factorizations
are performed k times. In step c), after attaining the random frequencies ωml , Cholesky factorizations are utilized
n × N times for γ jm (ωml ) . So there are lots of matrix factorization processes to be performed. Secondly, FFT
technique can not be employed due to the adoption of random frequencies. The summations of cosine terms are then of
enormous time consuming. Therefore, measures should be sought to improve the simulated efficiency according to the
characteristics of sample function in Eq. (6).
1. Study on the elements of factorized power spectral density matrix In the PSD matrix S (ω ) , the diagonal
%
elements S mm (ω ) are named as auto-spectral density function of ω that are determined by the PSD of natural wind
speed sample. In this paper, Kaimal PSD function is selected to model the longitudinal wind velocity fluctuations:
ω S (ω )
u∗2
=
200 f
(1 + 50 f )
(10)
35
where ω = frequency, in rad / s ; u∗ = shear velocity of the flow, in m / s ; f = reduced frequency, given by
(ω 2π ) ⋅ z Vz , dimensionless form, with Vz = mean wind speed at the height , in
m/ s .
The off-diagonal elements S jm (ω )( j ≠ m) named as cross-spectral density function of ω are determined by the
auto-spectral density function and the spatial coherence function. The coherence function is an empirical formula
suggested by Davenport
— 63 —
2
2
2 12⎫
⎧
2
2
⎡ 2
⎤
⎪ 2ω ⎣⎢Cx ( x j − xm ) + C y ( y j − ym ) + Cz ( z j − zm ) ⎦⎥ ⎪
Coh ( x j , xm , y j , ym , z j , zm , ω ) = exp ⎨−
⎬
2π (U zj + U zm )
⎪
⎪
⎩
⎭
(11)
where ( x j , y j , z j ) , ( xm , ym , zm ) are the coordinate of point j and m, respectively; U zj , U zm = mean wind speed at height
Z j and Z m ; and C x , C y , Cz are non-dimension reduced constants given by C x = 16 , C y = 6 , C z = 10 [5],
respectively.
It is shown that the spatial coherence function associated with coordinate of points and ω can reflect the spatial
correlation of wind velocity fluctuations between two spatial points. Therefore, the cross-spectral density function
between Point j and m is expressed as
S jm = S jj (ω ) Smm (ω )Coh ( x j , xm , y j , ym , z j , zm , ω )
(12)
As for Example 2, after decomposing the PSD matrix S (ω ) , the lower triangular matrix H (ω ) is acquired and the
%
%
elements of H (ω ) are used to studied. There are four types of factorized spectral function curves such as
%
H11 (ω ) , H 21 (ω ) , shown in Fig. 3(a), and H 5050 (ω ) , H 5150 (ω ) , as shown in Fig. 3(b).
12
3
Target auto-PSD function of point1
Target cross-PSD function of point2 and point1
Fitting auto-PSD function of point1
Fitting cross-PSD function of point2 and point1
10
2
2
H(ω) (m /s)
2
H(ω) (m /s)
8
Target auto-PSD function of point50
Target cross-PSD function of point51 and point50
Fitting auto-PSD function of point50
Fitting cross-PSD function of point51 and point50
6
4
1
2
0
0
2
4
6
8
10
12
14
ω (Rad/s)
(a) Point 1
0
0
2
4
6
8
ω (Rad/s)
10
12
14
(b) Point 50
Figure 3: Fitting factorized PSD function curves versus corresponding targets
In Fig. 3(a), two types of factorized spectral function curves associated with Point 1 are plotted and they decline as ω
increases. The difference between them lies in that the auto-spectral curve approaches to a constant, but the
cross-spectral curve to zero. Furthermore, as the distance between Point 1 and other points increases, the factorized
cross-spectral function value (not shown in Fig.3) approaches to zero at a smaller frequency value.
Fig. 3(b) shows the factorized auto-/cross-spectral function curves of points except Point 1.The curve increases until a
peak appears at a certain frequency, then declines to a constant for the factorized auto-spectral function curve, but to
zero for the cross-spectral function ones. Moreover, the larger the distance of two points is, the earlier the
cross-spectral function approaches zero.
2. Optimal methods to obtain the wind velocity fluctuations According to the characteristics of factorized spectral
function curves, the interpolation method can be applied to attain the function values of random frequencies in order to
reduce computing time. It’s obvious that more interpolating points should be arranged at the turning areas of curve and
less points at smooth areas for the balance between accuracy and computing consumption. The interpolating process is
performed as:
(1) Find the PSD matrices S (ω ) at the interpolating frequency points.
%
(2) Find the factorized matrices H (ω ) by decomposing S (ω ) using Cholesky factorization method.
%
%
(3) Fit the whole curves of H (ω ) using interpolation algorithm, then the function values of random radius frequencies
%
can be obtained.
— 64 —
This measure can reduce both memory and time consumption owing to cutting back the times of Cholesky
factorization dramatically.
Since the cubic uniform B-spline interpolation method allows the interpolating points have different intervals, it is
suitable to fit the target curve. Suppose there are m interpolating points selected among N frequency points which
divide the curve into m − 1 intervals, then the interpolation function M k ( u ) ( k = 1, 2,..., m − 1) are expressed as [8]
M k (u ) = u ⋅ B ⋅ V
% % %
where u = {u 3 u 2
%
(13)
u 1} , u =
ω − ωk
(ω ∈ [ωk , ωk +1 ]) is the variable ranging between 0 and 1;
ω k +1 − ω k
V = [Vk Vk +1 Vk + 2 Vk + 3 ] is the control points matrix and B is coefficient matrix of 4 × 4 order. It should be
%
%
mentioned that two free boundary conditions H1′′ (0) = 0 and H m −1′′ (1) = 0 are employed to determine the control
points matrix with m + 2 unknown variables before using Eq. (13). Moreover, it’s noticed that the ratio of two
adjacent intervals at an interpolating point should be in the range of [1 3 , 3] inorder to fit a smooth uniform curve [8].
In this way, four types of fitting curves in terms of 19 interpolating points are plotted in Fig. 3(a) and Fig. 3(b). It can
be seen that the fitting curves are met the target curves very well.
On the other hand, as the distance of two points is getting larger, the coherence between them will become less. Then,
the factorized cross-spectral function tends to zero at a certain frequency. In practice, a series of random frequencies
are produced according to step (a) and (b). So the frequencies of each factorized cross-spectral function are omitted
when they are larger than a certain frequency. In Example 2, about 1 3 of the cosine terms can be cut off in this way.
Hence time of computation is reduced.
NUMERICAL EXAMPLE
Example 2 is selected to demonstrate efficiency of the above optimal methods. The computer conditions are CPU:
AMD Barton 2500+; memory: 512M; operating system: Windows2000; programming language: Fortran. Then 5000
seconds of artificial wind speed sample are generated, taking 550s CPU time and 12M memory consuming. The wind
speed time histories at Points 1, 10, and 100, denoted by v1 (t ) , v10 (t ) ,and v100 (t ) , are plotted in Fig. 4, respectively.
The correlation functions and PSD functions are shown in Fig. 5 and Fig. 6, respectively.
Figure 4: Wind speed time histories at three spatial points
— 65 —
It can be seen from Fig.4 that the wind speed samples of points are not of periodicity. Fig.5 shows the auto-correlation
functions of temporal average and ensemble average of 30 samples at Point 1 and 10, and the cross-correlation
function between them. It’s obvious that the peak values of correlation functions appear only at t = 0 , but not at
intervals of 512 s through the whole time histories. That means the generated samples are non-periodic. Fig.6 shows
that although the frequencies are generated randomly, frequency contents of the samples are sufficient. Moreover, the
simulated PSD functions are met with the target functions. So the wind speed samples with random frequencies can
efficiently avoid the periodicity produced by the double-indexing frequencies.
It should be pointed out that although the efficiency of cosine terms summation is lower than that of using
FFT technique, the former is more adaptable in three aspects. Firstly, the sample time history is of arbitrary
time length, but not of the whole period necessary for FFT technique. Secondly, the sample can begin at any
time, but not only at t = 0 . Thirdly, different time interval of points can be acquired if necessary [9].
20
20
temporal average
ensemble average
15
15
2
R1, 10 (m /s)
10
2
R10, 10 (m /s)
5
0
-5
5
0
0
1000
τ (s)
2000
-5
3000
(a) Auto-correlation of Point 1
temporal average
ensemble average
15
10
2
R1, 1 (m /s)
10
20
temporal average
ensemble average
5
0
0
1000
2000
τ (s)
-5
3000
(b) Auto-correlation of Point 10
0
1000
τ (s)
2000
3000
(c) Cross-correlation of Point 1,10
Figure 5: Auto-/cross-correlatin function of two points
Simulated curve
Target curve
10
10
10
2
0.1
0.01
1E-3
1E-4
1E-3
0.01
0.1
f (Hz)
1
2
2
2
PSD (m /s )
100
1
1
0.1
1
0.1
0.01
0.01
1E-3
1E-3
1E-4
1E-3
0.01
0.1
1
Simulated curve
Target curve
1000
100
PSD (m /s )
2
1000
100
2
PSD (m /s )
1000
Simulated curve
Target curve
1E-4
1E-3
0.01
0.1
1
f (Hz)
f (Hz)
(a) Point 1
(b) Point 10
(c) Point 100
Figure 6: Simulated PSD functions of three points versus corresponding targets
SUMMARY
The periodicity of wind speed sample simulated by Wave Superposition Method with double-indexing frequencies can
not increase to nT0 due to the fact that the frequencies are adopted in the way of deterministic values, but not of
random values. Two examples are applied to analyze the reason in detail. In order to improve the efficiency of
generating wind speed sample with random frequencies, characteristics for curves of the factorized PSD functions
changing with the radius frequencies are studied. Then the cubic uniform B-spline interpolation method are utilized to
fit the target function curves. And the random frequencies resulting in zero value of H (ω ) are cut off. Finally, an
%
example using the proposed methods indicates that the simulation is successful. The simulated stochastic fluctuating
wind field has features of non-periodical. And the algorithm is effective and accurate.
— 66 —
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44(4): 191-204.
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