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Stochastic and statistical characteristics of artificially generated turbulent flow following karman spectrum in a wind-tunnel experiment

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Abstract

Wind gusts at the pedestrian level around buildings are caused by both effects of turbulent flow generated by surrounding buildings and the turbulent characteristic of the approaching flow. Although numerous researchers have stochastically investigated the contribution of buildings to gusts, the probabilistic characteristics of approaching flow have not been studied adequately. Therefore, the aim of this study was to investigate the statistical quantities such as high-order statistics, extreme wind speeds, and probability density functions (PDFs) of an artificially generated flow according to typical empirical equations. The approaching flow was generated by wind-tunnel experiments. In addition, we propose a PDF based on the Gram–Charlier series (GCS) to describe approaching flow. The determination of high-order statistics showed that these can be used as indices to validate whether the GCS can be applied to the PDFs of the approaching flow. Moreover, the current approaching flow was described effectively by the PDFs based on the GCS by considering the mean, standard deviation, skewness, and kurtosis. Furthermore, the mean, skewness, and kurtosis were correlated strongly with the percentile velocity components. This study demonstrates the importance of considering stochastic information of approaching flow when characterizing urban wind environments.
1
Stochastic and statistical characteristics of artificially generated turbulent flow following
1
Karman spectrum in a wind-tunnel experiment
2
3
Muhd Azhar bin Zainol1, Naoki Ikegaya*1,2, Mohd Faizal Mohamad3
4
1 Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Japan
5
2 Faculty of Engineering Sciences, Kyushu University, Japan
6
3 Universiti Teknologi MARA, Malaysia
7
8
*Corresponding author:
9
Naoki Ikegaya
10
Kasuga-koen 6-1, Kasuga-shi, Fukuoka 816-8580, Japan
11
+81-092-583-7644
12
13
Abstract
14
Wind gusts at the pedestrian level around buildings are caused by both effects of turbulent flow
15
generated by surrounding buildings and the turbulent characteristic of the approaching flow.
16
Although numerous researchers have stochastically investigated the contribution of buildings
17
to gusts, the probabilistic characteristics of approaching flow have not been studied adequately.
18
Therefore, the aim of this study was to investigate the statistical quantities such as high-order
19
statistics, extreme wind speeds, and probability density functions (PDFs) of an artificially
20
generated flow according to typical empirical equations. The approaching flow was generated
21
by wind-tunnel experiments. In addition, we propose a PDF based on the Gram–Charlier series
22
(GCS) to describe approaching flow. The determination of high-order statistics showed that
23
these can be used as indices to validate whether the GCS can be applied to the PDFs of the
24
approaching flow. Moreover, the current approaching flow was described effectively by the
25
PDFs based on the GCS by considering the mean, standard deviation, skewness, and kurtosis.
26
Furthermore, the mean, skewness, and kurtosis were correlated strongly with the percentile
27
velocity components. This study demonstrates the importance of considering stochastic
28
information of approaching flow when characterizing urban wind environments.
29
30
Keywords: Turbulent inflow; high-order statistics; percentile wind speed; probability density;
31
Karman spectrum; Gram–Charlier Series, Wind tunnel
32
Preprint published in Journal of Wind Engineering and Industrial Aerodynamics. Please refer as
“M.A. Zainol, N. Ikegaya, M.F. Mohamad, Stochastic and statistical characteristics of artificially
generated turbulent flow following Karman spectrum in a wind-tunnel experiment, JWEIA 229,
105148, 2022
2
1. Introduction
33
To maintain safety and conformable environment in urban areas, pedestrian-level wind
34
(PLW) environments have been a main issue in the field of the outdoor environmental studies
35
(e.g. Blocken, 2014, 2015; Blocken et al. 2011, 2012, 2016; Moonen et al. 2012; Stathopoulos,
36
1997, 2002, 2006). This is because of the importance of understanding the aerodynamic effect
37
of urban geometry on PLW environments.
38
A significant concern in PLW studies in the past decades has been the clarification of
39
the effect of urban morphology on urban ventilation efficiency to dilute the excess heat and
40
scalar concentration (Ng, 2009) for better urban environments. The relationships between
41
urban geometry and PLWs in terms of mean wind speeds have been studied by wind tunnel
42
experiments (WTEs; e.g. Kubota et al. 2008) and computational fluid dynamics (CFDs: e.g.
43
Abd Razak et al. 2013; Hang et al. 2012; Hertwig et al. 2011, 2017a, 2017b; Yuan and Ng,
44
2012). According to these studies, it is well established that the mean PLW speeds decrease
45
monotonically with an increase in building packing densities. This results in low urban
46
ventilation (Ikeda et al. 2015; Ikegaya et al 2017a). In contrast, the height variation of buildings
47
in urban areas can enhance the introduction of flow at the pedestrian level. This would cause
48
an increase in PLW to alleviate ineffective ventilation situations in dense conditions (Abd
49
Razak et al. 2013; Ikegaya et al. 2017a; Kubota et al. 2008). These relationships between PLW
50
and urban geometries are also modeled by simple empirical equations to assess urban
51
ventilation efficiency (Ikegaya et al. 2017a; Kubota et al. 2008; Yuan and Ng, 2012).
52
In addition to these studies on urban ventilation, the turbulent characteristic of PLWs
53
in complex urban areas has been studied using statistical analysis. For example, Ikegaya et al.
54
(2017b) showed the probability density functions (PDFs) of three velocity components at the
55
pedestrian level to identify infrequent albeit influential wind phenomena (i.e. infrequent wind
56
events). This study was extended by Kawaminami et al. (2018) to identify the relationship
57
between the PDFs of the velocity components and scalar concentrations of a contaminant
58
released from the bottom surface of simplified urban areas. Their study showed that a positive
59
correlation exists between the mean and strong PLW speeds, whereas the occurrence frequency
60
of a low-wind-speed event is inversely proportional to the mean wind speed. In addition,
61
extreme scalar concentrations are correlated negatively with mean and strong PLWs. Although
62
their studies employed idealized and generic block arrays, it is noteworthy that they introduced
63
a statistical approach based on the PDFs of the PLWs to quantify the magnitude of wind events
64
with a low occurrence frequency in terms of urban ventilation.
65
Another main concern in PLW studies is the evaluation of strong wind-speed
66
phenomena in terms of pedestrian safety. With regard to the prediction and evaluation of strong
67
PLW speeds, stochastic evaluations are more important. This is because gust events occur as a
68
result of the random process of the airflow caused by the strong shear owing to buildings in
69
urban areas. For example, He and Song (1999) conducted a large-eddy simulation (LES) to
70
investigate gust events. Based on their observations, they proposed an effective peak gust using
71
the mean and standard deviation of the PLWs for simplified buildings in a city model. This
72
study showed the importance of considering the unsteadiness of urban airflow while discussing
73
the gust events of PLWs. In addition, Ahmad et al. (2017) recently reported an unsteady
74
simulation over a realistic district in Tokyo using lattice Boltzmann LES to determine the
75
maximum PLW speeds. The study showed that the maximum wind speed decreases
76
monotonically with urban building packing density, by directly analyzing the strong wind
77
events from instantaneous phenomena.
78
It is necessary to obtain the probability information to understand the stochastic
79
phenomena of shear-driven flow in a more representative manner. For example, Ikegaya et al.
80
(2020) employed LES to scrutinize the effect of rare wind speeds and their occurrence
81
frequency for an isolated building exposed to turbulent inflow. They verified that the PDFs of
82
3
the velocity components can be approximated by Gaussian distributions, except for regions
83
where the wind speeds become marginal. Tominaga and Shirzadi (2021a, 2021b) conducted
84
WTEs to identify how high-rise buildings modified the PDFs of the PLW. Furthermore, H’ng
85
et al. (2022) conducted WTEs to classify the PDFs around a realistic building model, to clarify
86
their characteristics.
87
It is essential to investigate the stochastic characteristics of turbulent winds (Carta et
88
al. 2009; Efthimiou et al. 2017; He et al. 2010; Karthikeya et al. 2016; Masseran et al. 2013;
89
Morgan et al. 2011). Efthimiou et al. (2017) used turbulent flow datasets in the atmospheric
90
surface layer of a realistic district to perform a probabilistic analysis of PLWs. Their objective
91
was to demonstrate the superiority of the beta distribution over the commonly used Weibull
92
distribution. Accordingly, Wang and Okaze (2022) proposed a noteworthy procedure to predict
93
the peak factor by using mean, standard deviation, and skewness as representative statistics. It
94
is based on the assumption that PLWs are modeled by Weibull distributions. Although the
95
model still has limitations in describing the PDFs of a bi-modal shape, the study opened new
96
approaches to evaluating the rare wind speeds based on the modeling of PDFs and then,
97
quantifying the rare albeit strong PLWs.
98
To summarize, an understanding of the stochastic characteristics of PLWs is important
99
for assessing wind environments for both urban ventilation and strong wind predictions. In
100
most studies that addressed the external airflow around a building or buildings, the approaching
101
flow was verified to reproduce the turbulent characteristics regardless of CFD approaches or
102
WTEs in terms of turbulent statistics (Ikegaya et al 2019; Okaze et al. 2017b, 2021). This was
103
because the characteristics of an approaching flow dramatically alter the resultant PLWs
104
(Tominaga et al. 2008; Yoshie et al. 2007). The difficulty in reproducing approaching flow also
105
hinders the practical use of LES. This has resulted in the development of several methods for
106
artificially generating turbulent inflows (e.g., Lund et al. 1998; Okaze and Mochida 2017a; Xie
107
and Castro 2008).
108
Conventionally, inflow characteristics are discussed based on the mean and turbulence
109
intensity profiles, power spectral density, and turbulence length scale (AIJ, 2019). This is
110
because these are known to influence the resultant turbulent flow fields in terms of the
111
evaluation of turbulent statistics. However, the characteristics of the PDFs and rare wind speeds
112
of turbulent approaching flows have not been discussed thoroughly. It is essential to identify
113
such characteristics of turbulent approaching flows (which satisfy the fundamental statistical
114
criteria based on the conventional approach) to consider the current trends by applying a
115
stochastic approach for PLW evaluation, and to separately understand the original stochastic
116
characteristic of the approach flow and building aerodynamic effect. In addition, appropriate
117
modeling of the PDFs of approaching flows is required to understand the building effect on the
118
PDFs of PLWs.
119
To address these issues, this study aimed to 1) generate an artificial approaching flow
120
using empirical equations (namely, power-law profiles for the mean streamwise velocity
121
component, standard deviation, integral length scale, and Karman-type spectrum) to investigate
122
statistical quantities such as high-order statistics to predict and understand the characteristics
123
of the PDFs of the approaching flow and 2) consider the correlation between high-order
124
moments and rare wind speeds. In addition, we illustrated the framework for displaying the
125
stochastic information of the inflow when we consider PLW based on stochastic analysis. In
126
Section 2, theoretical and experimental methods for expressing the PDF of an approaching flow
127
are described. In Section 3, the results are discussed. Section 4 concludes this study.
128
129
130
131
132
4
2. Methods
133
2.1 Experimental description
134
The experiments were performed at Tokyo Polytechnic University in 2018 to measure
135
the vertical profile of the streamwise and vertical velocity components according to the
136
empirical equations for atmospheric velocity fields. The empirical equations were explained in
137
Section 2.3. The wind tunnel has a width, height, and length of 2.2 m, 1.8 m, and 19.1 m,
138
respectively (see Fig. 1). The inlet and outlet of the wind tunnel were exposed to a large
139
experimental room. Here, the flow from the outlet returned to the inlet. Although the inlet and
140
outlet were exposed to the room, these conditions can maintain a constant temperature within
141
the tunnel, which is suitable for velocity measurements using hot-wire anemometry (HWA).
142
The coordinates of
!
,
"
, and
#
are defined as the streamwise, spanwise, and vertical
143
directions, respectively. In addition, the velocity components in the
!
,
"
and z coordinates
144
are denoted as
$
,
%
, and
&
, respectively. Several meshes, honeycombs, and a contraction part
145
were installed at the upwind position of the test section to ensure uniformity of inflow. Spires,
146
a barrier, and roughness elements were installed in the upstream area of the test section as
147
shown in the Fig. 1 to generate the artificial approaching flow. The roughness element was
148
installed only in the test section, and no roughness element was installed in the measurement
149
area.
150
An x-type (0249R-T5, KANOMAX) hot-wire probe was used to determine the two
151
velocity components
$
and
&
. The hot-wire probe was calibrated with a probe calibrator
152
(KANOMAX, MODEL 1065). It can adjust both wind speed and angle of the flow approaching
153
the wire to determine the wire angle. The calibration coefficients are sensitive to temperature.
154
These were determined for each experiment in the wind tunnel. The mean wind speeds
155
determined by the i-type and x-type HWAs were compared to verify the validity of the x-type
156
HWA. The difference in streamwise velocity component between these probes was less than
157
0.2% at z = 0.1 [m], where the vertical velocity components were less than 1% of the
158
streamwise velocity component.
159
Measurements to obtain the vertical profile of the two velocity components were
160
conducted at the spanwise center 15.0 m downstream of the location where the spires and
161
barrier were installed. The measurement duration was 180 s at 1000 Hz. The measurements
162
were repeated three times at each measurement point. In a preliminary measurement, we
163
verified that the integral time scale under the present condition is approximately 0.05–0.06 s.
164
This indicates that the vortex turnover time in the present measurement period is sufficiently
165
large as 3000 times of the integral time scale in each measurement. This is notwithstanding our
166
repetition of the same experiment three times to verify the expected variations in the higher-
167
order statistics at each position.
168
169
170
Roughness
Barrier & Spire
2.2m
15.0m
HWA probe
5
Fig. 1 Photograph of the interior of the wind tunnel and the arrangement of roughness, barrier,
171
and spires. The vertical profile of
$
and
&
was measured by an x-type hot wire anemometer
172
15 m downstream from the barrier and spires.
173
174
2.2 Probability density and high-order moment
175
To discuss the relationship between the PDFs and high-order moments, we define the
176
mean and deviation of the velocity component
$
as follows:
177
$ ' (
)*$+,-
!., ' * /+$-$
"
#" .$0
(1)
$$' $ 1$
(2)
where
)
is the averaging duration,
$2
and
$3
are the mean and the deviation of
$
from the
178
mean, respectively; the overbar indicates the temporal averaging operation; and
/+$-
is the
179
PDF of the velocity component
$
. In a general expression, the nth-order statistic (
4 5 (
is an
180
integer) is defined as
181
$$%'(
)*$$+,-%
!., ' * /+$-+$ 1$-%
"
#" .$6
(3)
From Eq. (2),
$7 ' 8
and
$$&
are equivalent to the variance of
$
. In this study, the standard
182
deviation of
$
is denoted by
$'()
(
'
9
$$&
). The higher-order statistics normalized by
$'()
183
are defined as normalized statistics
:%' $$% ;$'()
%
. By definition,
:*' 8
;
:&' (
; and
184
:+
and
:,
correspond to the skewness
<-
and kurtosis
=.
, respectively.
185
Meanwhile, we can consider that the PDF at each height of the approaching flow
186
follows modified Gaussian distributions. This is because although
<-
and
=.
do not satisfy
187
the Gaussian values of
<-' 8
and
=.' >
, they do not deviate significantly from these values
188
(we recall these values in Section 3.1). When these statistics do not deviate significantly from
189
the Gaussian distribution values, we can formulate the PDFs using the Gram–Charlier series
190
(Hald 2000). Herein, the distribution function is modified to satisfy the given higher-order
191
statistics. When the standardized random variable is defined as
? ' +$ 1 $-;$'()
, the PDF
192
incorporating the modification by
<-
and
=.
is expressed as
193
/+?-' @(A <-
B?+?&1>-A=.1>
CD +?,1B?&A>-EF#/012!
GCH
(4)
Eq. (4) is the GCS considering the fourth-order normalized statistics of
<-
and
=.
.
194
Based on Eq. (4), the even- and odd-order normalized statistics are formulated
195
explicitly using
<-
and
=.
as follows:
196
:&3#* '+CI1(-J<-
>KC3#*+I1C-J
(5)
:&3 '+CI-J
C3@(
IJA=.1>
B+I1C-JE
(6)
Here,
I
is an integer starting from two (
4 ' CI 1(
for odd moments and
4 ' CI
for even
197
moments). This is because Eqs. (5) and (6) provide a higher-order moment than
:+
and
:,
,
198
respectively. These relationships also imply that nth-order normalized statistics can be
199
expressed by the general function
L42
M
+4-
for odd statistics and
L5
N
+40=.-
for even statistics as
200
follows:
201
:&3#* ' :&63#*7#* L4
N+4-0
(7)
:&3 ' :&
6
3#*
7
L5
N+
40=.
-
6
(8)
These relationships imply that statistics of higher than that of
<-
or
=.
are expressed
202
by the recurrence formula with lower-order statistics. For example, for the Gaussian
203
distribution,
<-' 8
and
=.' >
yield
:&3#* ' 8
and
:&3 '
+
CI1(
-
:&63#*7
. This
204
6
demonstrates that the higher-order statistics of the random variable following a Gaussian
205
distribution are determined recurrently by the order of the statistics. Because these parameters
206
are normalized by
$'()
, this also indicates that
$'()2
is the only statistical parameter that
207
determines the higher-order statistics in the Gaussian distribution. Similarly, when the PDF of
208
the velocity component can be expressed by Eq. (4), this assumption indicates that higher-order
209
statistics should satisfy the recurrence relationship in Eqs. (7) and (8). Therefore, higher-order
210
statistics (
4 O P
) must be expressed by
<-
and
=.
.
211
These formulations are derived mathematically from the GCS. Consequently, Eq. (4)
212
reveals the relationships between the high-order statistics of the PDFs. However, it is
213
noteworthy that these formulations clearly demonstrate the importance of investigating high-
214
order statistics. Mean and standard deviation are important parameters in typical wind
215
engineering studies because these are considerably influential for turbulent flow fields.
216
Accordingly, skewness and kurtosis (or third- and fourth-order statistics) can be considered in
217
certain cases when the intermittency of velocity components is considered (Hagishima et al.
218
2009). In contrast, the significance of determining higher-order statistics (
4 O P
) appears to be
219
obscure. Hence, to our knowledge, no information on higher-order statistics has been presented
220
in previous studies. However, according to the discussion, higher-order statistics can be used
221
as indices to clarify whether the PDFs of velocity components follow a certain PDF distribution.
222
To summarize, we can examine whether the PDFs are bell-shaped or modified bell-shaped
223
functions by calculating these statistics. We recall this aspect in the following section by
224
scrutinizing the values of high-order statistics and PDFs derived from time-series data.
225
226
2.3 Generated artificial turbulent flow
227
In this section, we describe the fundamental characteristics of the turbulent inflow
228
generated in the present experimental setup. In general, the atmospheric boundary layer profiles
229
of the mean and turbulent intensities can be expressed using power law. Here, we reproduced
230
the profiles based on the 1 / 4th power law for the mean streamwise velocity and turbulent
231
intensity:
232
!
!"#$ "#$
$"#$%%
(9)
!"&'
!"&'(#$
$"#$%(%()*)+
(10)
Here,
$'58
is the reference wind speed of 11.6 m/s measured at the reference height
#'58
=
233
1.125 m at the experimental scale.
234
According to the standard surface geometry classification in Japan (AIJ, 2019),
Q '235
86C
and
#'58 'DP8
m for full-scale velocity profile category III. The scale employed in the
236
experiment was 1 / 400. Therefore, all the length and velocity components were normalized by
237
#'58
and
$'58
, respectively, to clarify the relative length and strength of the entire boundary
238
layer depth and wind speed. In addition, the turbulent length scale
R9
was formulated by the
239
following empirical equation above
#;#'58 O 868C
in category III (AIJ, 2019):
240
),"(&&*$
+&,)*+
(11)
These vertical profiles were targeted a standard atmospheric flow in this study. Fig. 2
241
shows the vertical profiles of the velocity components
$
and
&
, turbulence intensities
242
$'();$
and
&'();$
, and integral length scales
R9
and
R:
. The integral length scales can
243
be calculated as
R9' $S92
and
R:' $S:
based on Taylor’s frozen hypothesis (Stull, 1988).
244
Here,
S9
and
S:
represent the integral time scales based on the autocorrelation functions of
245
$
and
&
, respectively. The blue- and red-shaded areas show the standard deviations of the
246
three trials. The gray dashed lines indicate the targeted empirical profiles in Eqs. (9)–(11).
247
7
The results indicate that the streamwise velocity and turbulence intensity are
248
consistent with those expressed by Eqs. (1) and (2), respectively, at the largest height. However,
249
there is a certain difference between the equations and measured values when
#;#'58 T 86(
.
250
That is, the streamwise velocity was larger than that in Eq. (1), whereas the turbulence intensity
251
was smaller than that in Eq. (2). This is probably because of the roughness arrangement in the
252
aligned array in the spanwise directions, resulting in insufficient velocity reduction and
253
turbulent generation especially in the lower ranges. For the vertical velocity components, the
254
mean values were less than 2% of
$'58
, and the turbulent intensity was comparable with
$'()
.
255
Meanwhile,
R9
was larger than that in Eq. (11) at
#;#'58 T 86>
. However, it showed good
256
agreement with Eq. (11) for
#;#'58 O 86>
The integral length scale of the vertical velocity
257
component
R:
was similar in order with
R9
, although
R:
was always smaller than
R9
.
258
Although we can admit differences in the targeted and generated profiles especially in the lower
259
ranges, they do not cause substantially effects in the following analyses.
260
In addition to the vertical profiles of the statistics, we verified the power spectral
261
densities. The power spectral densities of the streamwise and vertical components are denoted
262
as
U9
and
U:
, respectively. In typical atmospheric turbulent flows, the spectrum follows the
263
Karman spectrum (Von Karman, 1948, Zhang et al. 2015). They can be expressed as
264
-,.
!"&'
-"/.
0
1
(23(.
0
-
4
+./'
(12)
-0.
5"&'
-"/.0*(2(66'/17.04-,
1(23(.0-411./ '
(13)
Here,
/
V
' /R92;$
. Note that
$2
is also used for the normalized frequency in
U:
because the
265
dominant advection is in the streamwise direction.
266
Fig. 3 shows both components at
2#;#'58 ' 86(
with their standard deviations
267
represented by the blue and red shaded areas in the three trials. The experimental data were
268
smoothed by applying a Hanning filter 500 times to discuss the overall tendency in the spectra
269
of the peak location and energy cascade. The results show that the power spectral densities
270
agree well with the Karman-type spectrum at a reference height from
/
V
W(8#,
to
/
V
W(8/
. In
271
addition, the peaks of
U9
and
U:
indicates that most energy-containing eddies have time
272
scales of
(8S9
or
(8S:
in each velocity component, respectively. This is because the peaks
273
occur at
/
V
W(8#*
. The power spectral density appears to become larger than that of the Karman
274
spectrum at
/
V
WPX(8/
for
U:
and approximately
/
V
WCX(8*
for
U9
. This is
275
notwithstanding the smaller relative magnitudes of the peak values of the spectrum.
276
277
8
278
Fig. 2 Vertical profiles of streamwise and vertical velocity components. (a) Mean velocity
279
components
$
and
&
, (b) turbulence intensities
$'();$
and
&'();$
, and (c) integral
280
length scales
R9
and
R:
. The integral length scales were determined from the integral time
281
scales
S9
and
S:
as
R9' $S9
and
R:' $S:
by assuming Taylor’s frozen hypothesis.
282
The blue and red shaded areas show their standard deviations for the three trials. The grey
283
dashed lines indicate the targeted empirical profiles.
$'58
and
#'58
are the reference wind
284
speed and height, respectively.
285
286
287
Fig. 3 Power spectral densities for streamwise and vertical velocity components,
U9
and
U:
,
288
respectively. These values are scaled by their standard deviation,
$'()
and
&'()
. The
289
frequency f in the horizontal axis is normalized by
S9' R9;$
and
S:' R:;$
. The blue and
290
red shaded areas show their standard deviations for the three trials. The grey dashed line
291
indicates the Karman-type spectrum by Eqs. (12) and (13).
292
(a) (b)
!/!!"# !/!!"#
#/#!"# ,$/#!"#
Power law
(% = 0.2)
Power law
(% = 0.2)
(c)
!/!!"#
Power law
#!$%/#, $!$%/# *&/!!"#, *'/!!"#
#!$%
$!$%
+&!"#
+'!"#
#
$
+&
+'
*&
*'
+($
+(%
9
3. Results
293
3.1 Profiles of high-order statistics
294
In this section, we scrutinize the vertical profiles of high-order statistics of the
295
streamwise and vertical velocity components. Similar to Eqs. (5) and (6), the normalized
296
statistics tend to increase with the order of statistics. To consider only the vertical distributions
297
of the statistics, we introduce the following scaling based on the order indices
I
of the n-th
298
order statistics (
4 ' CI 1(
for odd moments and
4 ' CI
for even moments):
299
Y&3#* ' :&3#* >K C3#*+I 1 C-J
+CI1(-J
(14)
Y&3 ' :&3 C3IJ
+CI-J
(15)
By definition,
Y+' <-
, and
Y,' =.;>
.
300
Figs. 4 and 5 show the vertical profiles of
Y%
from
4 ' >
to n =
(8
for the
301
streamwise and vertical velocity components, respectively. To confirm random errors in the
302
measurements, an independent measurement was repeated three times, and the standard
303
deviations of
Y%
were calculated for each measurement position. The standard deviations of
304
Y%
are indicated by the blue shaded areas in Figs. 4 and 5. In addition, the red dashed lines are
305
added to the graphs for the even-order moments. These show the values of
Y%
when the
306
random variable follows a Gaussian distribution (e.g.,
Y&3 ' (
).
307
In Figs. 4 and 5, the standard deviations of
Y%2
increase gradually with the increase in
308
the order of the statistics because high-order statistics are determined by the power of the
309
deviation part of the velocity components. In addition, the statistics of the vertical velocity
310
component in Fig. 5 are larger than the streamwise velocity component. This is probably
311
because of the smaller velocity magnitude in the vertical direction compared with the
312
streamwise component. However, these deviations of
Z;!32
and
Z;!345
are smaller than the
313
variation of
Y[4
. Therefore, we can discuss the general tendency of the high-order statistics in
314
Figs. 4 and 5 by considering the expected error ranges.
315
The odd-order statistics of the streamwise velocity components
Y+\Y<
are shown in
316
Fig. 4 (a)–(d). The vertical profile of
Y+
, or skewness
<-
, implies that the PDFs of the
317
streamwise velocity component below
86C#'58
are skewed positively and those above
318
86>#'58
are skewed negatively.
Y+
is approximately zero in the middle range of
86C#'58 T319
# T 86>#'58
. These tendencies are commonly observed while developing turbulent boundary
320
layers. This is because sweep-dominant events (high-momentum downflow,
$3 O 8
and
&3 T321
8
) contribute mainly to the momentum transfer within the lower part of the boundary layers,
322
whereas ejection-dominant events (low-momentum upwind,
$3 T 8
and
&3 O 8
) occur in the
323
upper part of the boundary layers. For example, Raupach (1981) compared the skewness of the
324
streamwise and vertical velocity components over various types of rough and smooth surfaces.
325
This revealed that the values of skewness are positive and large near the surface, are
326
approximately zero in the middle range of the boundary layers, and decrease with height and
327
peak near the boundary layer depth. In addition, Hagishima et al. (2009) showed vertical
328
profiles of skewness over block arrays. They explained that the contribution of ejection events
329
prevails within the outer layers. The similar tendencies observed in
Y+
indicate that the current
330
approaching flow generated artificially using large spires and barriers (whose sizes can cover
331
the entire boundary layer depth (i.e.,
W#'58
)) also follows the typical turbulent characteristics
332
within the turbulent boundary layers.
333
The higher-order statistics of
Y1
Y<
in Fig. 4 (b–d) show a noteworthy characteristic:
334
the gradient of the statistics decreases gradually when the order of the statistics increases. In
335
addition,
Y1
,
Y=
, and
Y<
are nearly zero in the range between
# ' 86(#'58
and
86P#'58
. This
336
10
is wider than the range wherein
Y+
is approximately zero. In contrast, these statistics remain
337
positive below
86(#'58
although the slopes of the profiles reduce with increase in order. These
338
tendencies of the odd-order statistics indicate that the PDFs at the middle height follow non-
339
skewed, symmetric distributions, whereas those in the range of the lower and upper parts of
340
the boundary layer are skewed because of the interaction with the floor surface or upper free-
341
stream airflow.
342
The even-order statistics from
Y,
to
Y*/
are shown in Fig. 4 (e)–(h), respectively.
343
Y&32
increases with the increase in order because even-order statistics normalized using Eq. (15)
344
still contain the order index
I
. Note that the abscissa in Figs. 4 (e, f) and Fig. 4 (g, h) shows
345
different ranges.
Y&3
becomes 1.0 when
=.' >
, as shown in the figures. The vertical profile
346
of
Y,
is less sensitive to the height than that of
Y+
. However, a marginal increase could be
347
verified below
868P#'58
. Above this height,
Y,
is approximately constant and marginally
348
smaller than
(68
. This tendency implies that the PDFs of the streamwise velocity component
349
tend to be flattened marginally compared with the Gaussian distributions. The higher-order
350
statistics
Y>\Y*/
are also constant above
86(#'58
, whereas these increase near the surface.
351
Fig. 5 shows the same profiles for the vertical velocity components. The Reynolds
352
stress within the boundary layer is generally negative because of the downward momentum
353
transport. Consequently, the odd order statistics of
&
generally has a sign opposite to that of
354
$
. Although
Y+\Y<
for
&
are smaller than those for
$
, these values are nearly zero below
355
0.3zref and increase gradually with height. In addition, the higher-order statistics show less
356
sensitivity to height, similar to those of
$
. With regard to the even-order statistics,
Y>\Y*/
of
357
w are larger than those of
$
. This indicates that the PDFs of the vertical velocity component
358
are sharpened compared with the Gaussian distributions. This trend is the converse of that of
359
the PDFs of
$
.
360
The higher-order statistics of the velocity components have not been discussed
361
adequately (except for skewness and kurtosis). However, their investigation can help
362
understand whether the PDFs can be expressed using parameters represented by the skewness
363
and kurtosis. As shown in Figs. 4 and 5, the vertical profiles of the higher-order statistics should
364
be consistent with the lower-order statistics if these represent the probability distributions.
365
11
366
Fig. 4 Vertical profiles of the coefficient determined by high-order moments for streamwise
367
velocity component. (a)–(d) odd-order moments from
Y+
to
Y<
, and (e)–(h) even-order
368
moments from
Y,
to
Y*/
. The blue shaded area shows their standard deviations for the three
369
trials. The red dashed lines in (e)–(h) indicate the values for
Y%' (
.
370
12
371
Fig. 5 Identical to Fig. 4 except for the vertical velocity component.
372
373
3.2 Relationship between statistics and PDF
374
To quantify the relationship between the high-order statistics shown in the vertical
375
profiles, Fig. 6 shows the variations in the statistics with the order indices
4
(
4 ' CI
for
376
even-order and
4 ' CI 1(
for odd-order statistics) for the streamwise and vertical velocity
377
components. According to Eqs. (5) and (6),
:,\:*/
can be expressed by
<-
,
=.
, and
I
378
when the velocity components follow the modified Gaussian distribution by the GCS. The
379
shaded areas in Fig. 6 show the ranges of
:%
expressed by Eqs. (5) and (6) assuming
C6] T380
=.T D68
for the even-order statistics and
186P T <-T 86P
for the odd-order statistics. These
381
ranges of the statistics can vary depending on the random variables to be considered. However,
382
here, we determined these based on the vertical profiles of the velocity components in Figs. 4
383
and 5 to investigate whether the high-order statistics can follow the prediction by Eqs. (5) and
384
(6), respectively. When the PDFs follow a Gaussian distribution, the even-order statistics
385
follow
+CI-J;+C3IJ-
indicated by the gray line in Fig. 6 (a, c), and the odd-order statistics in
386
Fig. 6 (b, d) become zero (for example,
=.' >
and
<-' 8
in Eqs. (5) and (6), respectively).
387
13
The symbols in Fig. 6 are denoted by open squares when any of the statistics deviates from the
388
shared areas (which indicates that the PDFs cannot be expressed by Eq. (4)).
389
As is evident from the figure, the statistics at most heights fall within the shaded areas,
390
whereas the values at
# T 868>#'58
depart from the areas owing to the influence of the strong
391
shear caused by the bottom surface. These results imply that the PDFs of the velocity
392
components are probably expressed by modified Gaussian distributions based on the GCS in
393
Eq. (4) at the most heights. As shown in Section 3.1, we can qualitatively discuss the PDFs
394
based on the skewness and kurtosis values. However, the applicability of Eq. (4) is uncertain
395
when these values are considered. Therefore, it is appropriate to scrutinize the trends of higher-
396
order statistics with respect to the order indices (as shown in Fig. 6) to assess whether the
397
distributions follow Eq. (4) or not.
398
399
400
Fig. 6 Relationships between the high-order statistics and the orders for (a) even-order and (b)
401
odd-order statistics for
$
and (c, d) for
&
. The yellow lines in (a, c) indicate the statistics of
402
the variable following the Gaussian distribution. The blue-shaded areas denote the expected
403
ranges by Eq. (6) when
C6] T =.T D68
, and by Eq. (5) when
186P T <-T 86P
.
404
405
To verify the relationships between the PDFs and statistics in Fig. 6, Figs. 7 shows the
406
PDFs at the corresponding heights of the statistics in Fig. 6 for the streamwise and vertical
407
velocity components. Fig. 7 shows the original PDFs with respect to
$;$'58
and
&;$'58
,
408
respectively. Fig. 7 (a) shows that as the streamwise velocity component increases in the
409
14
vertical direction, the PDFs of
$
shift gradually in the positive direction with the increase in
410
#
. To summarize, the PDFs are skewed positively at lower heights, whereas these are skewed
411
negatively at higher positions. For the PDFs of
&
, the distribution ranges broaden when the
412
positions become higher because of the increase in the turbulence scales (in Fig. 2).
413
To demonstrate whether these skewed PDFs can be represented by Eq. (4), Fig. 8
414
shows the PDFs with respect to the standardized random variable
? '
+
$1$
-
;$'()
for
$
415
and
? '
+
&1&
-
;&'()
for
&
, respectively. The PDFs are categorized into two groups based
416
on
<-T 8
(Fig.8 (a) and (c)] and
<-O 8
(Fig. 8 (b) and (d)]. In addition, two distribution
417
functions are plotted: a Gaussian distribution (grey solid line), and a modified Gaussian
418
distribution in Eq. (4) with the expected ranges of
<-
and
=.
. The ranges of
<-
and
=.
are
419
consistent with those shown in Fig. 6. The ordinate is displayed on the logarithmic axis to
420
emphasize the prediction accuracy at a low probability. In addition, the PDFs at the
421
measurement positions
868(#'58
and
868>#'58
for
$
, and
868(#'58
,
868C#'58
, and
422
868>#'58
for
&
are shown by bold lines. This is because these are conditions when the high-
423
order statistics deviate from the expected ranges in Fig. 6.
424
As shown in these figures, the PDFs at most heights are within the shaded areas even
425
below
(8#&
except for the PDFs depicted by bold lines (e.g., Fig. 8 (b, d),
# ' 868(#'58
).
426
This implies that Eq. (4) can describe the PDFs using GCS by incorporating
<-
and
=.
in the
427
distribution functions. Recalling the relationship in Fig. 6, these results indicate the
428
applicability of Eq. (4) based on the variation trends of higher-order statistics with order.
429
430
431
Fig. 7 Probability density functions at various heights for (a) streamwise velocity component
432
and (b) vertical velocity component.
433
434
15
435
Fig. 8 Probability density functions with respect to the standardized random variables for (a, b)
436
the streamwise velocity component and (c, d) vertical velocity components. (a, c) Positively
437
skewed condition, and (b, d) negatively skewed condition. The blue-shaded areas indicate the
438
predictable ranges by Eq. (4) using
<-
and
=.
. The grey solid lines indicate the Gaussian
439
distribution. The bold lines (conditions at
868(#'58
and
868>#'58
for
$
, and
868(#'58
,
440
868C#'58
, and
868>#'58
for
&
) show the conditions when the high-order statistics deviate
441
from the expected range in Fig. 6.
442
443
3.3 Percentile values of velocity components
444
Another noteworthy topic is the correlation of extreme wind events with other high-
445
order statistics. Therefore, in this section, we investigate the relationship between the percentile
446
values of the velocity components and the higher-order statistics.
447
Based on the PDF
/+$-
, the percentile,
$?
, is defined as the value when the
448
cumulative PDF adopts a certain value of
L
as follows:
449
L+$?-' * /+$-.$
96
#"
(16)
By definition, the small percentiles indicate the magnitudes of the rare weak wind events,
450
whereas the larger percentiles indicate the strengths of the rare strong wind events.
451
(a)
!"# !"#
!"!#
$ = (' ')/'!"# $ = (' ')/'!"#
!"!$
!"!%
!"!&
!"!'
!"!(
!"#$
!"#)
!"$*
!"&&
!"'$
+,+!"#
$") - .$- &"!
! - /%- !"0
(b)
$") - .$- &"!
1!"0 - /%- !
Gaussian
Gaussian
(c)
!"# !"#
$ = (+ +)/+!"# $ = (+ +)/+!"#
$") - .$- &"!
! - /%- !"0
(d)
$") - .$- &"!
1!"0 - /%- !
Gaussian
Gaussian
16
Fig. 9 shows the vertical profiles of several percentile values of
L ' 86(^
–99.9% for
452
the streamwise and vertical velocity components, as well as the mean values. Because three
453
independent measurements were conducted, the standard deviation,
Z
, of the three trials is also
454
shown in the figures to show the expected random errors. For the streamwise velocity, the
455
vertical profiles of the percentiles are highly similar to the mean velocity,
$
. The difference
456
between the percentiles and the mean value decreases gradually with height. For the vertical
457
velocity components, the low and high percentiles are negative and positive, respectively. In
458
addition, the magnitudes of the percentiles (which are considerably larger than the mean values)
459
increase with height. The mean value of the vertical velocity is almost zero owing to the mild
460
development of the boundary layer in the streamwise direction. Consequently, the percentiles
461
in the vertical velocity components tend to be akin to the mean streamwise velocity component
462
$
, but not to the mean vertical velocity component
&
. The decreasing tendencies of the low
463
percentiles with height are owing to the negative vertical velocity. This indicates that the
464
magnitude of the percentiles increases with height. In addition, it should be noted that the
465
profile shapes of the percentile magnitudes are similar to the integral length scales of the
466
vertical velocity component in Fig. 2(c). This is probably because the extreme values
467
represented by the percentiles are strongly influenced by the integral length scales, which
468
gradually becomes larger with the distance from the bottom wall (Fig. 2 (c)).
469
470
Fig. 9 Vertical profiles of the percentile values for (a) streamwise and (b) vertical velocity
471
components.
472
473
Although it appears that the percentiles of both
$
and
&
are affected strongly by the
474
mean streamwise velocity components, quantifying the influential statistics can help consider
475
how rare wind events can be determined. Therefore, we compare the percentiles of both
476
velocity components and three statistics mean, standard deviation, and skewness in Fig. 10. We
477
selected
$
,
$'()
, and
<-
for the comparisons because the higher-order statistics (
4 5 D
)
478
appear to display trends similar to those of the lower-order statistics (see Fig. 4). The
479
correlations of the percentiles with higher-order statistics are discussed subsequently. With
480
regard to the percentile values on the ordinate, we considered only the deviation from the mean
481
values (namely,
$?1$
or
&?1&
) to investigate the relationship between the extreme
482
values away from the mean and the statistics. The numbers in Fig. 10 indicate the correlation
483
coefficients,
_
, between the statistics on the abscissa and percentiles on the ordinate.
484
For the percentiles of the streamwise velocity component, the mean or skewness
485
evidently has a strong negative or positive correlation with the extreme values of occurrence
486
(a) (b)
!/!!"# !/!!"#
#/#!"# , #$/#!"#
$
%/#!"# , %$/#!"#
99.9%
99%
!
1%
0.1%
$
99.9%
99%
"
1%
0.1%
17
frequency of 99.9%, 99%, 1%, and 0.1%, although those of 90% and 10% show relatively weak
487
correlations with these statistics. The negative correlations between the percentiles and indicate
488
that the velocity variation ranges in the higher positions reduces probably because of the
489
reduction of the turbulence intensity. This is verified in Fig. 7(a). In contrast, the positive
490
correlations with skewness occur because the PDFs adopt long tail shapes in the positive ranges.
491
This indicates that the magnitudes of the strong wind speeds increase. The standard deviations
492
are less related to the percentile values. This may be owing to the limited ranges of the values
493
in the current approaching flow.
494
Fig. 10 (d–f) shows the same for the vertical velocity component. Although we
495
investigated the correlations with the statistics of the vertical velocity component and their
496
percentiles (not shown), the relationships were ambiguous. This could be anticipated from the
497
percentile profiles irrelevant to the mean vertical velocity component in Fig. 9 (b). Thus, we
498
show only the scatter charts of the percentiles and statistics of the streamwise velocity
499
component.
500
The vertical velocity component shows trends similar to those of the streamwise
501
component. That is, the mean and skewness are correlated strongly with the percentiles. Strong
502
wind events with occurrence frequencies of 99.9%–90% show positive correlations with
$
503
and negative correlations with
<-
. This is because the velocity ranges in the vertical velocity
504
component increase gradually with height (see Fig. 7) to cause larger fluctuations from the
505
mean with an increase in mean velocity
$
. In contrast, the skewness of the streamwise velocity
506
becomes negative and large with height, and that of the vertical velocity component exhibits
507
the converse trend with height (Fig. 5 (a)). This can also be understood from the fact that the
508
Reynolds stress is negative owing to the downward transport of momentum in the turbulent
509
boundary layers. The PDFs of the vertical velocity component become long-tailed in the
510
negative direction because of these relationships. This reduces the magnitudes of the strong
511
wind speeds (see Fig. 10 (f)).
512
For the weak wind events denoted by 10%–0.1%, the relationships between these
513
statistics and the percentiles are the converse of the above. Because the mean value of the
514
vertical velocity component is nearly zero, the weak wind events represented by 10%–0.1%
515
are negative or downward flows. Therefore, the negative correlation with the mean streamwise
516
velocity indicates that the magnitudes of the rare wind speeds increase because of the expansion
517
of the velocity ranges (see Fig. 7 (b)). Similar to the case of the streamwise velocity component,
518
the standard deviation is less related to the percentile values of the vertical velocity component.
519
As can be anticipated from Figs. 4 and 5, the vertical profiles of the higher-order
520
statistics are similar. However, the relationships are not monotonic with the order (see Fig. 6).
521
This implies that the correlations between percentile values and high-order statistics can vary
522
because of the dissimilar dependencies among the statistics and percentiles. To quantify this
523
aspect, Fig. 11 summarizes the correlation coefficients between the deviation of the percentile
524
values from the mean value and the statistics of the streamwise velocity component.
525
With regard to the percentiles of the streamwise velocity component, these
526
correlations show two important aspects for understanding the relationships among the
527
statistics and percentiles. First, the four statistics (mean, standard deviation, skewness, and
528
kurtosis) show different dependencies on the percentiles. Second, skewness or kurtosis
529
represents a similar correlation with high-order statistics with odd and even orders, respectively.
530
For the vertical velocity, the correlation coefficients vary between positive (for strong wind)
531
and negative (for weak wind) because of the negative percentile values. However, similar
532
dependencies of the percentiles on the statistics can be verified.
533
In the present approaching flow, the PDFs mostly follow the modified Gaussian
534
distributions described by Eq. (4). Thereby, higher-order statistics are predicted effectively by
535
Eqs. (5) and (6) (as verified in Fig. 6). This implies that the higher-order statistics (
4 5 P
) are
536
18
dependent on the lower-order statistics
<-
and
=.
. In addition, the applicability of Eqs. (5)
537
and (6) imply that the PDFs can be predicted using Eq. (4). This indicates that only four
538
independent parameters (namely, mean, standard deviation, skewness, and kurtosis) are
539
required to determine rare wind events. To validate this assumption, the dependency of the
540
percentile values is represented by only the first four statistics in Fig. 11, whereas
:12\:*/
541
simply show similar correlations as
<-
and
=.
with the percentiles.
542
Although we need to apply this analysis to the various types of turbulent flow to clarify
543
the relationships between the percentiles and high-order statistics, the present analysis has
544
shown that the dependency of the percentiles on the statistics is related to the number of
545
parameters incorporated in the PDF models. Therefore, the current study successfully
546
demonstrated the importance of scrutinizing high-order statistics as well as PDFs for estimating
547
rare wind events. The velocity fields and their PDFs around buildings are altered from the
548
original approaching flow owing to the presence of the buildings. This makes it important to
549
describe the probabilistic characteristics of the approaching flow to discuss PLWs, as identified
550
in this section.
551
552
553
Fig. 10 Correlations between the percentile values for (a–c) streamwise and (d–f) vertical
554
velocity components and the statistics of the streamwise velocity components. (a, d) Mean, (b,
555
e) standard deviation, and (c, f) skewness. The values
_
indicate the correlation coefficients
556
between the values on the two axes.
557
558
19
559
Fig. 11 Correlation coefficients between the statistics of streamwise velocity and percentile
560
values of (a) streamwise and (b) vertical velocity components.
561
562
4. Conclusions
563
In this study, wind tunnel experiments were conducted to generate an artificial
564
approaching flow following empirical equations for the mean streamwise velocity component,
565
standard deviation, integral length scale, and Karman spectrum. The objectives were to
566
understand how the probability density functions of the approaching flow can be expressed and
567
to clarify the correlation among high-order statistics, probability density functions (PDFs), and
568
extreme values.
569
The experimental data showed that the characteristics of the generated approaching
570
flow followed basic power-law vertical profiles over rough surfaces in category III. In addition,
571
the von Karman spectrum was reproduced effectively at various measurement heights.
572
To understand the theoretical relationship between high-order statistics and probability
573
density function (PDFs), we employed a modified Gaussian distribution function known as the
574
Gram–Charlier Series (GCS) by considering skewness and kurtosis. In addition, we clarified
575
how high-order statistics can be expressed by lower-order statistics.
576
Based on the time series data of the approaching flow, we determined ten statistics
577
from the mean to the 10th-order moments. Furthermore, the characteristics of the statistics were
578
examined. The relationship between the order and statistics was discussed for both odd- and
579
even-order statistics based on the PDFs modeled by the GCS. This indicated that high-order
580
statistics can be used as indices to validate whether the GCS model can be applied to the PDFs
581
of the approaching flow. The present approaching flow showed that the PDFs were modeled
582
effectively by the GCS by considering the mean, standard deviation, and third- and fourth-
583
order statistics (skewness and kurtosis).
584
In addition, the correlation between high-order statistics and extreme values
585
represented by percentiles based on PDFs was examined. This showed that three statistics
586
(mean, skewness, and kurtosis) are correlated strongly with the percentile velocity components.
587
Although a similar strong correlation between the percentiles and other high-order statistics
588
can be identified, the dependency of the statistics on each other shows that only fourth-order
589
statistics dominate in determining the PDFs of the approaching flow.
590
Although variations in PDFs is likely for various methods for generating the
591
approaching turbulent flow following empirical equations for the mean, turbulent intensity, and
592
power spectrum density, we showed the framework for displaying and considering the
593
stochastic information of the inflow when we consider the PLW based on stochastic analysis.
594
This study will expand the usage of the PDFs as a basic stochastic information when discussing
595
the effects inflow turbulence and buildings on the PLW because a prediction model of PDFs
596
! !!"# "$#%$&$'$($)$*$+,
%&!-' !( %&)-' )(
! !!"# "$#%$&$'$($)$*$+,
(a) (b)
0.1%
1%
99%
99.9%
20
based on GSC are introduced using the higher-order statistics. The most notable advantage of
597
the present model is the simple modification of the PDFs only by the high-order statistics,
598
which can be easily obtained in both WTE and CFDs. In addition, this study highlights the
599
importance of collecting the datasets of the high-order statistics which are not commonly
600
discussed in the previous studies.
601
We introduced the present model for describing the approaching turbulent flow;
602
However, the same method can also be applied for the modeling for PDFs at pedestrian levels
603
regardless of datasets from WTEs or CFDs. Applying the model for the various cases such as
604
various types of approaching flow, flows around buildings, and complex urban flow needs to
605
be conducted in a future study to prove the applicability.
606
607
Acknowledgments
608
This study was supported partially by a Grant-in-Aid for Scientific Research from JSPS, Japan,
609
KAKENHI (Grant no. JP 21K18770), FOREST program from JST, Japan (Grant No.
610
JPMJFR205O), and the Initiative for Realizing Diversity in the Research Environment. The
611
experiments were conducted in a wind tunnel facility at the Wind Engineering Research Center,
612
Tokyo Polytechnic University.
613
614
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615
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