Stochastic and statistical characteristics of artificially generated turbulent flow following
Karman spectrum in a wind-tunnel experiment
Muhd Azhar bin Zainol1, Naoki Ikegaya*1,2, Mohd Faizal Mohamad3
1 Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Japan
2 Faculty of Engineering Sciences, Kyushu University, Japan
3 Universiti Teknologi MARA, Malaysia
Kasuga-koen 6-1, Kasuga-shi, Fukuoka 816-8580, Japan
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.
Keywords: Turbulent inflow; high-order statistics; percentile wind speed; probability density;
Karman spectrum; Gram–Charlier Series, Wind tunnel
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,
To maintain safety and conformable environment in urban areas, pedestrian-level wind
(PLW) environments have been a main issue in the field of the outdoor environmental studies
(e.g. Blocken, 2014, 2015; Blocken et al. 2011, 2012, 2016; Moonen et al. 2012; Stathopoulos,
1997, 2002, 2006). This is because of the importance of understanding the aerodynamic effect
of urban geometry on PLW environments.
A significant concern in PLW studies in the past decades has been the clarification of
the effect of urban morphology on urban ventilation efficiency to dilute the excess heat and
scalar concentration (Ng, 2009) for better urban environments. The relationships between
urban geometry and PLWs in terms of mean wind speeds have been studied by wind tunnel
experiments (WTEs; e.g. Kubota et al. 2008) and computational fluid dynamics (CFDs: e.g.
Abd Razak et al. 2013; Hang et al. 2012; Hertwig et al. 2011, 2017a, 2017b; Yuan and Ng,
2012). According to these studies, it is well established that the mean PLW speeds decrease
monotonically with an increase in building packing densities. This results in low urban
ventilation (Ikeda et al. 2015; Ikegaya et al 2017a). In contrast, the height variation of buildings
in urban areas can enhance the introduction of flow at the pedestrian level. This would cause
an increase in PLW to alleviate ineffective ventilation situations in dense conditions (Abd
Razak et al. 2013; Ikegaya et al. 2017a; Kubota et al. 2008). These relationships between PLW
and urban geometries are also modeled by simple empirical equations to assess urban
ventilation efficiency (Ikegaya et al. 2017a; Kubota et al. 2008; Yuan and Ng, 2012).
In addition to these studies on urban ventilation, the turbulent characteristic of PLWs
in complex urban areas has been studied using statistical analysis. For example, Ikegaya et al.
(2017b) showed the probability density functions (PDFs) of three velocity components at the
pedestrian level to identify infrequent albeit influential wind phenomena (i.e. infrequent wind
events). This study was extended by Kawaminami et al. (2018) to identify the relationship
between the PDFs of the velocity components and scalar concentrations of a contaminant
released from the bottom surface of simplified urban areas. Their study showed that a positive
correlation exists between the mean and strong PLW speeds, whereas the occurrence frequency
of a low-wind-speed event is inversely proportional to the mean wind speed. In addition,
extreme scalar concentrations are correlated negatively with mean and strong PLWs. Although
their studies employed idealized and generic block arrays, it is noteworthy that they introduced
a statistical approach based on the PDFs of the PLWs to quantify the magnitude of wind events
with a low occurrence frequency in terms of urban ventilation.
Another main concern in PLW studies is the evaluation of strong wind-speed
phenomena in terms of pedestrian safety. With regard to the prediction and evaluation of strong
PLW speeds, stochastic evaluations are more important. This is because gust events occur as a
result of the random process of the airflow caused by the strong shear owing to buildings in
urban areas. For example, He and Song (1999) conducted a large-eddy simulation (LES) to
investigate gust events. Based on their observations, they proposed an effective peak gust using
the mean and standard deviation of the PLWs for simplified buildings in a city model. This
study showed the importance of considering the unsteadiness of urban airflow while discussing
the gust events of PLWs. In addition, Ahmad et al. (2017) recently reported an unsteady
simulation over a realistic district in Tokyo using lattice Boltzmann LES to determine the
maximum PLW speeds. The study showed that the maximum wind speed decreases
monotonically with urban building packing density, by directly analyzing the strong wind
events from instantaneous phenomena.
It is necessary to obtain the probability information to understand the stochastic
phenomena of shear-driven flow in a more representative manner. For example, Ikegaya et al.
(2020) employed LES to scrutinize the effect of rare wind speeds and their occurrence
frequency for an isolated building exposed to turbulent inflow. They verified that the PDFs of
the velocity components can be approximated by Gaussian distributions, except for regions
where the wind speeds become marginal. Tominaga and Shirzadi (2021a, 2021b) conducted
WTEs to identify how high-rise buildings modified the PDFs of the PLW. Furthermore, H’ng
et al. (2022) conducted WTEs to classify the PDFs around a realistic building model, to clarify
It is essential to investigate the stochastic characteristics of turbulent winds (Carta et
al. 2009; Efthimiou et al. 2017; He et al. 2010; Karthikeya et al. 2016; Masseran et al. 2013;
Morgan et al. 2011). Efthimiou et al. (2017) used turbulent flow datasets in the atmospheric
surface layer of a realistic district to perform a probabilistic analysis of PLWs. Their objective
was to demonstrate the superiority of the beta distribution over the commonly used Weibull
distribution. Accordingly, Wang and Okaze (2022) proposed a noteworthy procedure to predict
the peak factor by using mean, standard deviation, and skewness as representative statistics. It
is based on the assumption that PLWs are modeled by Weibull distributions. Although the
model still has limitations in describing the PDFs of a bi-modal shape, the study opened new
approaches to evaluating the rare wind speeds based on the modeling of PDFs and then,
quantifying the rare albeit strong PLWs.
To summarize, an understanding of the stochastic characteristics of PLWs is important
for assessing wind environments for both urban ventilation and strong wind predictions. In
most studies that addressed the external airflow around a building or buildings, the approaching
flow was verified to reproduce the turbulent characteristics regardless of CFD approaches or
WTEs in terms of turbulent statistics (Ikegaya et al 2019; Okaze et al. 2017b, 2021). This was
because the characteristics of an approaching flow dramatically alter the resultant PLWs
(Tominaga et al. 2008; Yoshie et al. 2007). The difficulty in reproducing approaching flow also
hinders the practical use of LES. This has resulted in the development of several methods for
artificially generating turbulent inflows (e.g., Lund et al. 1998; Okaze and Mochida 2017a; Xie
and Castro 2008).
Conventionally, inflow characteristics are discussed based on the mean and turbulence
intensity profiles, power spectral density, and turbulence length scale (AIJ, 2019). This is
because these are known to influence the resultant turbulent flow fields in terms of the
evaluation of turbulent statistics. However, the characteristics of the PDFs and rare wind speeds
of turbulent approaching flows have not been discussed thoroughly. It is essential to identify
such characteristics of turbulent approaching flows (which satisfy the fundamental statistical
criteria based on the conventional approach) to consider the current trends by applying a
stochastic approach for PLW evaluation, and to separately understand the original stochastic
characteristic of the approach flow and building aerodynamic effect. In addition, appropriate
modeling of the PDFs of approaching flows is required to understand the building effect on the
PDFs of PLWs.
To address these issues, this study aimed to 1) generate an artificial approaching flow
using empirical equations (namely, power-law profiles for the mean streamwise velocity
component, standard deviation, integral length scale, and Karman-type spectrum) to investigate
statistical quantities such as high-order statistics to predict and understand the characteristics
of the PDFs of the approaching flow and 2) consider the correlation between high-order
moments and rare wind speeds. In addition, we illustrated the framework for displaying the
stochastic information of the inflow when we consider PLW based on stochastic analysis. In
Section 2, theoretical and experimental methods for expressing the PDF of an approaching flow
are described. In Section 3, the results are discussed. Section 4 concludes this study.
2.1 Experimental description
The experiments were performed at Tokyo Polytechnic University in 2018 to measure
the vertical profile of the streamwise and vertical velocity components according to the
empirical equations for atmospheric velocity fields. The empirical equations were explained in
Section 2.3. The wind tunnel has a width, height, and length of 2.2 m, 1.8 m, and 19.1 m,
respectively (see Fig. 1). The inlet and outlet of the wind tunnel were exposed to a large
experimental room. Here, the flow from the outlet returned to the inlet. Although the inlet and
outlet were exposed to the room, these conditions can maintain a constant temperature within
the tunnel, which is suitable for velocity measurements using hot-wire anemometry (HWA).
The coordinates of
are defined as the streamwise, spanwise, and vertical
directions, respectively. In addition, the velocity components in the
and z coordinates
are denoted as
, respectively. Several meshes, honeycombs, and a contraction part
were installed at the upwind position of the test section to ensure uniformity of inflow. Spires,
a barrier, and roughness elements were installed in the upstream area of the test section as
shown in the Fig. 1 to generate the artificial approaching flow. The roughness element was
installed only in the test section, and no roughness element was installed in the measurement
An x-type (0249R-T5, KANOMAX) hot-wire probe was used to determine the two
. The hot-wire probe was calibrated with a probe calibrator
(KANOMAX, MODEL 1065). It can adjust both wind speed and angle of the flow approaching
the wire to determine the wire angle. The calibration coefficients are sensitive to temperature.
These were determined for each experiment in the wind tunnel. The mean wind speeds
determined by the i-type and x-type HWAs were compared to verify the validity of the x-type
HWA. The difference in streamwise velocity component between these probes was less than
0.2% at z = 0.1 [m], where the vertical velocity components were less than 1% of the
streamwise velocity component.
Measurements to obtain the vertical profile of the two velocity components were
conducted at the spanwise center 15.0 m downstream of the location where the spires and
barrier were installed. The measurement duration was 180 s at 1000 Hz. The measurements
were repeated three times at each measurement point. In a preliminary measurement, we
verified that the integral time scale under the present condition is approximately 0.05–0.06 s.
This indicates that the vortex turnover time in the present measurement period is sufficiently
large as 3000 times of the integral time scale in each measurement. This is notwithstanding our
repetition of the same experiment three times to verify the expected variations in the higher-
order statistics at each position.
Barrier & Spire
Fig. 1 Photograph of the interior of the wind tunnel and the arrangement of roughness, barrier,
and spires. The vertical profile of
was measured by an x-type hot wire anemometer
15 m downstream from the barrier and spires.
2.2 Probability density and high-order moment
To discuss the relationship between the PDFs and high-order moments, we define the
mean and deviation of the velocity component
$ ' (
!., ' * /+$-$
$$' $ 1$
is the averaging duration,
are the mean and the deviation of
mean, respectively; the overbar indicates the temporal averaging operation; and
PDF of the velocity component
. In a general expression, the nth-order statistic (
4 5 (
integer) is defined as
!., ' * /+$-+$ 1$-%
From Eq. (2),
$7 ' 8
are equivalent to the variance of
. In this study, the standard
is denoted by
). The higher-order statistics normalized by
are defined as normalized statistics
:%' $$% ;$'()
. By definition,
correspond to the skewness
Meanwhile, we can consider that the PDF at each height of the approaching flow
follows modified Gaussian distributions. This is because although
do not satisfy
the Gaussian values of
, they do not deviate significantly from these values
(we recall these values in Section 3.1). When these statistics do not deviate significantly from
the Gaussian distribution values, we can formulate the PDFs using the Gram–Charlier series
(Hald 2000). Herein, the distribution function is modified to satisfy the given higher-order
statistics. When the standardized random variable is defined as
? ' +$ 1 $-;$'()
, the PDF
incorporating the modification by
is expressed as
/+?-' @(A <-
Eq. (4) is the GCS considering the fourth-order normalized statistics of
Based on Eq. (4), the even- and odd-order normalized statistics are formulated
is an integer starting from two (
4 ' CI 1(
for odd moments and
4 ' CI
moments). This is because Eqs. (5) and (6) provide a higher-order moment than
respectively. These relationships also imply that nth-order normalized statistics can be
expressed by the general function
for odd statistics and
for even statistics as
:&3#* ' :&63#*7#* L4
:&3 ' :&
These relationships imply that statistics of higher than that of
by the recurrence formula with lower-order statistics. For example, for the Gaussian
:&3#* ' 8
demonstrates that the higher-order statistics of the random variable following a Gaussian
distribution are determined recurrently by the order of the statistics. Because these parameters
are normalized by
, this also indicates that
is the only statistical parameter that
determines the higher-order statistics in the Gaussian distribution. Similarly, when the PDF of
the velocity component can be expressed by Eq. (4), this assumption indicates that higher-order
statistics should satisfy the recurrence relationship in Eqs. (7) and (8). Therefore, higher-order
4 O P
) must be expressed by
These formulations are derived mathematically from the GCS. Consequently, Eq. (4)
reveals the relationships between the high-order statistics of the PDFs. However, it is
noteworthy that these formulations clearly demonstrate the importance of investigating high-
order statistics. Mean and standard deviation are important parameters in typical wind
engineering studies because these are considerably influential for turbulent flow fields.
Accordingly, skewness and kurtosis (or third- and fourth-order statistics) can be considered in
certain cases when the intermittency of velocity components is considered (Hagishima et al.
2009). In contrast, the significance of determining higher-order statistics (
4 O P
) appears to be
obscure. Hence, to our knowledge, no information on higher-order statistics has been presented
in previous studies. However, according to the discussion, higher-order statistics can be used
as indices to clarify whether the PDFs of velocity components follow a certain PDF distribution.
To summarize, we can examine whether the PDFs are bell-shaped or modified bell-shaped
functions by calculating these statistics. We recall this aspect in the following section by
scrutinizing the values of high-order statistics and PDFs derived from time-series data.
2.3 Generated artificial turbulent flow
In this section, we describe the fundamental characteristics of the turbulent inflow
generated in the present experimental setup. In general, the atmospheric boundary layer profiles
of the mean and turbulent intensities can be expressed using power law. Here, we reproduced
the profiles based on the 1 / 4th power law for the mean streamwise velocity and turbulent
is the reference wind speed of 11.6 m/s measured at the reference height
1.125 m at the experimental scale.
According to the standard surface geometry classification in Japan (AIJ, 2019),
m for full-scale velocity profile category III. The scale employed in the
experiment was 1 / 400. Therefore, all the length and velocity components were normalized by
, respectively, to clarify the relative length and strength of the entire boundary
layer depth and wind speed. In addition, the turbulent length scale
was formulated by the
following empirical equation above
#;#'58 O 868C
in category III (AIJ, 2019):
These vertical profiles were targeted a standard atmospheric flow in this study. Fig. 2
shows the vertical profiles of the velocity components
, turbulence intensities
, and integral length scales
. The integral length scales can
be calculated as
based on Taylor’s frozen hypothesis (Stull, 1988).
represent the integral time scales based on the autocorrelation functions of
, respectively. The blue- and red-shaded areas show the standard deviations of the
three trials. The gray dashed lines indicate the targeted empirical profiles in Eqs. (9)–(11).
The results indicate that the streamwise velocity and turbulence intensity are
consistent with those expressed by Eqs. (1) and (2), respectively, at the largest height. However,
there is a certain difference between the equations and measured values when
#;#'58 T 86(
That is, the streamwise velocity was larger than that in Eq. (1), whereas the turbulence intensity
was smaller than that in Eq. (2). This is probably because of the roughness arrangement in the
aligned array in the spanwise directions, resulting in insufficient velocity reduction and
turbulent generation especially in the lower ranges. For the vertical velocity components, the
mean values were less than 2% of
, and the turbulent intensity was comparable with
was larger than that in Eq. (11) at
#;#'58 T 86>
. However, it showed good
agreement with Eq. (11) for
#;#'58 O 86>
The integral length scale of the vertical velocity
was similar in order with
was always smaller than
Although we can admit differences in the targeted and generated profiles especially in the lower
ranges, they do not cause substantially effects in the following analyses.
In addition to the vertical profiles of the statistics, we verified the power spectral
densities. The power spectral densities of the streamwise and vertical components are denoted
, respectively. In typical atmospheric turbulent flows, the spectrum follows the
Karman spectrum (Von Karman, 1948, Zhang et al. 2015). They can be expressed as
. Note that
is also used for the normalized frequency in
dominant advection is in the streamwise direction.
Fig. 3 shows both components at
2#;#'58 ' 86(
with their standard deviations
represented by the blue and red shaded areas in the three trials. The experimental data were
smoothed by applying a Hanning filter 500 times to discuss the overall tendency in the spectra
of the peak location and energy cascade. The results show that the power spectral densities
agree well with the Karman-type spectrum at a reference height from
addition, the peaks of
indicates that most energy-containing eddies have time
in each velocity component, respectively. This is because the peaks
. The power spectral density appears to become larger than that of the Karman
. This is
notwithstanding the smaller relative magnitudes of the peak values of the spectrum.
Fig. 2 Vertical profiles of streamwise and vertical velocity components. (a) Mean velocity
, (b) turbulence intensities
, and (c) integral
. The integral length scales were determined from the integral time
by assuming Taylor’s frozen hypothesis.
The blue and red shaded areas show their standard deviations for the three trials. The grey
dashed lines indicate the targeted empirical profiles.
are the reference wind
speed and height, respectively.
Fig. 3 Power spectral densities for streamwise and vertical velocity components,
respectively. These values are scaled by their standard deviation,
frequency f in the horizontal axis is normalized by
. The blue and
red shaded areas show their standard deviations for the three trials. The grey dashed line
indicates the Karman-type spectrum by Eqs. (12) and (13).
(% = 0.2)
(% = 0.2)
#!$%/#, $!$%/# *&/!!"#, *'/!!"#
3.1 Profiles of high-order statistics
In this section, we scrutinize the vertical profiles of high-order statistics of the
streamwise and vertical velocity components. Similar to Eqs. (5) and (6), the normalized
statistics tend to increase with the order of statistics. To consider only the vertical distributions
of the statistics, we introduce the following scaling based on the order indices
of the n-th
order statistics (
4 ' CI 1(
for odd moments and
4 ' CI
for even moments):
Y&3#* ' :&3#* >K C3#*+I 1 C-J
Y&3 ' :&3 C3IJ
Figs. 4 and 5 show the vertical profiles of
4 ' >
to n =
streamwise and vertical velocity components, respectively. To confirm random errors in the
measurements, an independent measurement was repeated three times, and the standard
were calculated for each measurement position. The standard deviations of
are indicated by the blue shaded areas in Figs. 4 and 5. In addition, the red dashed lines are
added to the graphs for the even-order moments. These show the values of
random variable follows a Gaussian distribution (e.g.,
Y&3 ' (
In Figs. 4 and 5, the standard deviations of
increase gradually with the increase in
the order of the statistics because high-order statistics are determined by the power of the
deviation part of the velocity components. In addition, the statistics of the vertical velocity
component in Fig. 5 are larger than the streamwise velocity component. This is probably
because of the smaller velocity magnitude in the vertical direction compared with the
streamwise component. However, these deviations of
are smaller than the
. Therefore, we can discuss the general tendency of the high-order statistics in
Figs. 4 and 5 by considering the expected error ranges.
The odd-order statistics of the streamwise velocity components
are shown in
Fig. 4 (a)–(d). The vertical profile of
, or skewness
, implies that the PDFs of the
streamwise velocity component below
are skewed positively and those above
are skewed negatively.
is approximately zero in the middle range of
# T 86>#'58
. These tendencies are commonly observed while developing turbulent boundary
layers. This is because sweep-dominant events (high-momentum downflow,
$3 O 8
) contribute mainly to the momentum transfer within the lower part of the boundary layers,
whereas ejection-dominant events (low-momentum upwind,
$3 T 8
&3 O 8
) occur in the
upper part of the boundary layers. For example, Raupach (1981) compared the skewness of the
streamwise and vertical velocity components over various types of rough and smooth surfaces.
This revealed that the values of skewness are positive and large near the surface, are
approximately zero in the middle range of the boundary layers, and decrease with height and
peak near the boundary layer depth. In addition, Hagishima et al. (2009) showed vertical
profiles of skewness over block arrays. They explained that the contribution of ejection events
prevails within the outer layers. The similar tendencies observed in
indicate that the current
approaching flow generated artificially using large spires and barriers (whose sizes can cover
the entire boundary layer depth (i.e.,
)) also follows the typical turbulent characteristics
within the turbulent boundary layers.
The higher-order statistics of
in Fig. 4 (b–d) show a noteworthy characteristic:
the gradient of the statistics decreases gradually when the order of the statistics increases. In
are nearly zero in the range between
# ' 86(#'58
is wider than the range wherein
is approximately zero. In contrast, these statistics remain
although the slopes of the profiles reduce with increase in order. These
tendencies of the odd-order statistics indicate that the PDFs at the middle height follow non-
skewed, symmetric distributions, whereas those in the range of the lower and upper parts of
the boundary layer are skewed because of the interaction with the floor surface or upper free-
The even-order statistics from
are shown in Fig. 4 (e)–(h), respectively.
increases with the increase in order because even-order statistics normalized using Eq. (15)
still contain the order index
. Note that the abscissa in Figs. 4 (e, f) and Fig. 4 (g, h) shows
becomes 1.0 when
, as shown in the figures. The vertical profile
is less sensitive to the height than that of
. However, a marginal increase could be
. Above this height,
is approximately constant and marginally
. This tendency implies that the PDFs of the streamwise velocity component
tend to be flattened marginally compared with the Gaussian distributions. The higher-order
are also constant above
, whereas these increase near the surface.
Fig. 5 shows the same profiles for the vertical velocity components. The Reynolds
stress within the boundary layer is generally negative because of the downward momentum
transport. Consequently, the odd order statistics of
generally has a sign opposite to that of
are smaller than those for
, these values are nearly zero below
0.3zref and increase gradually with height. In addition, the higher-order statistics show less
sensitivity to height, similar to those of
. With regard to the even-order statistics,
w are larger than those of
. This indicates that the PDFs of the vertical velocity component
are sharpened compared with the Gaussian distributions. This trend is the converse of that of
the PDFs of
The higher-order statistics of the velocity components have not been discussed
adequately (except for skewness and kurtosis). However, their investigation can help
understand whether the PDFs can be expressed using parameters represented by the skewness
and kurtosis. As shown in Figs. 4 and 5, the vertical profiles of the higher-order statistics should
be consistent with the lower-order statistics if these represent the probability distributions.
Fig. 4 Vertical profiles of the coefficient determined by high-order moments for streamwise
velocity component. (a)–(d) odd-order moments from
, and (e)–(h) even-order
. The blue shaded area shows their standard deviations for the three
trials. The red dashed lines in (e)–(h) indicate the values for
Fig. 5 Identical to Fig. 4 except for the vertical velocity component.
3.2 Relationship between statistics and PDF
To quantify the relationship between the high-order statistics shown in the vertical
profiles, Fig. 6 shows the variations in the statistics with the order indices
4 ' CI
4 ' CI 1(
for odd-order statistics) for the streamwise and vertical velocity
components. According to Eqs. (5) and (6),
can be expressed by
when the velocity components follow the modified Gaussian distribution by the GCS. The
shaded areas in Fig. 6 show the ranges of
expressed by Eqs. (5) and (6) assuming
for the even-order statistics and
186P T <-T 86P
for the odd-order statistics. These
ranges of the statistics can vary depending on the random variables to be considered. However,
here, we determined these based on the vertical profiles of the velocity components in Figs. 4
and 5 to investigate whether the high-order statistics can follow the prediction by Eqs. (5) and
(6), respectively. When the PDFs follow a Gaussian distribution, the even-order statistics
indicated by the gray line in Fig. 6 (a, c), and the odd-order statistics in
Fig. 6 (b, d) become zero (for example,
in Eqs. (5) and (6), respectively).
The symbols in Fig. 6 are denoted by open squares when any of the statistics deviates from the
shared areas (which indicates that the PDFs cannot be expressed by Eq. (4)).
As is evident from the figure, the statistics at most heights fall within the shaded areas,
whereas the values at
# T 868>#'58
depart from the areas owing to the influence of the strong
shear caused by the bottom surface. These results imply that the PDFs of the velocity
components are probably expressed by modified Gaussian distributions based on the GCS in
Eq. (4) at the most heights. As shown in Section 3.1, we can qualitatively discuss the PDFs
based on the skewness and kurtosis values. However, the applicability of Eq. (4) is uncertain
when these values are considered. Therefore, it is appropriate to scrutinize the trends of higher-
order statistics with respect to the order indices (as shown in Fig. 6) to assess whether the
distributions follow Eq. (4) or not.
Fig. 6 Relationships between the high-order statistics and the orders for (a) even-order and (b)
odd-order statistics for
and (c, d) for
. The yellow lines in (a, c) indicate the statistics of
the variable following the Gaussian distribution. The blue-shaded areas denote the expected
ranges by Eq. (6) when
C6] T =.T D68
, and by Eq. (5) when
186P T <-T 86P
To verify the relationships between the PDFs and statistics in Fig. 6, Figs. 7 shows the
PDFs at the corresponding heights of the statistics in Fig. 6 for the streamwise and vertical
velocity components. Fig. 7 shows the original PDFs with respect to
respectively. Fig. 7 (a) shows that as the streamwise velocity component increases in the
vertical direction, the PDFs of
shift gradually in the positive direction with the increase in
. To summarize, the PDFs are skewed positively at lower heights, whereas these are skewed
negatively at higher positions. For the PDFs of
, the distribution ranges broaden when the
positions become higher because of the increase in the turbulence scales (in Fig. 2).
To demonstrate whether these skewed PDFs can be represented by Eq. (4), Fig. 8
shows the PDFs with respect to the standardized random variable
, respectively. The PDFs are categorized into two groups based
(Fig.8 (a) and (c)] and
(Fig. 8 (b) and (d)]. In addition, two distribution
functions are plotted: a Gaussian distribution (grey solid line), and a modified Gaussian
distribution in Eq. (4) with the expected ranges of
. The ranges of
consistent with those shown in Fig. 6. The ordinate is displayed on the logarithmic axis to
emphasize the prediction accuracy at a low probability. In addition, the PDFs at the
are shown by bold lines. This is because these are conditions when the high-
order statistics deviate from the expected ranges in Fig. 6.
As shown in these figures, the PDFs at most heights are within the shaded areas even
except for the PDFs depicted by bold lines (e.g., Fig. 8 (b, d),
# ' 868(#'58
This implies that Eq. (4) can describe the PDFs using GCS by incorporating
distribution functions. Recalling the relationship in Fig. 6, these results indicate the
applicability of Eq. (4) based on the variation trends of higher-order statistics with order.
Fig. 7 Probability density functions at various heights for (a) streamwise velocity component
and (b) vertical velocity component.
Fig. 8 Probability density functions with respect to the standardized random variables for (a, b)
the streamwise velocity component and (c, d) vertical velocity components. (a, c) Positively
skewed condition, and (b, d) negatively skewed condition. The blue-shaded areas indicate the
predictable ranges by Eq. (4) using
. The grey solid lines indicate the Gaussian
distribution. The bold lines (conditions at
) show the conditions when the high-order statistics deviate
from the expected range in Fig. 6.
3.3 Percentile values of velocity components
Another noteworthy topic is the correlation of extreme wind events with other high-
order statistics. Therefore, in this section, we investigate the relationship between the percentile
values of the velocity components and the higher-order statistics.
Based on the PDF
, the percentile,
, is defined as the value when the
cumulative PDF adopts a certain value of
L+$?-' * /+$-.$
By definition, the small percentiles indicate the magnitudes of the rare weak wind events,
whereas the larger percentiles indicate the strengths of the rare strong wind events.
$ = (' − ')/'!"# $ = (' − ')/'!"#
$") - .$- &"!
! - /%- !"0
$") - .$- &"!
1!"0 - /%- !
$ = (+ − +)/+!"# $ = (+ − +)/+!"#
$") - .$- &"!
! - /%- !"0
$") - .$- &"!
1!"0 - /%- !
Fig. 9 shows the vertical profiles of several percentile values of
L ' 86(^
the streamwise and vertical velocity components, as well as the mean values. Because three
independent measurements were conducted, the standard deviation,
, of the three trials is also
shown in the figures to show the expected random errors. For the streamwise velocity, the
vertical profiles of the percentiles are highly similar to the mean velocity,
. The difference
between the percentiles and the mean value decreases gradually with height. For the vertical
velocity components, the low and high percentiles are negative and positive, respectively. In
addition, the magnitudes of the percentiles (which are considerably larger than the mean values)
increase with height. The mean value of the vertical velocity is almost zero owing to the mild
development of the boundary layer in the streamwise direction. Consequently, the percentiles
in the vertical velocity components tend to be akin to the mean streamwise velocity component
, but not to the mean vertical velocity component
. The decreasing tendencies of the low
percentiles with height are owing to the negative vertical velocity. This indicates that the
magnitude of the percentiles increases with height. In addition, it should be noted that the
profile shapes of the percentile magnitudes are similar to the integral length scales of the
vertical velocity component in Fig. 2(c). This is probably because the extreme values
represented by the percentiles are strongly influenced by the integral length scales, which
gradually becomes larger with the distance from the bottom wall (Fig. 2 (c)).
Fig. 9 Vertical profiles of the percentile values for (a) streamwise and (b) vertical velocity
Although it appears that the percentiles of both
are affected strongly by the
mean streamwise velocity components, quantifying the influential statistics can help consider
how rare wind events can be determined. Therefore, we compare the percentiles of both
velocity components and three statistics mean, standard deviation, and skewness in Fig. 10. We
for the comparisons because the higher-order statistics (
4 5 D
appear to display trends similar to those of the lower-order statistics (see Fig. 4). The
correlations of the percentiles with higher-order statistics are discussed subsequently. With
regard to the percentile values on the ordinate, we considered only the deviation from the mean
) to investigate the relationship between the extreme
values away from the mean and the statistics. The numbers in Fig. 10 indicate the correlation
, between the statistics on the abscissa and percentiles on the ordinate.
For the percentiles of the streamwise velocity component, the mean or skewness
evidently has a strong negative or positive correlation with the extreme values of occurrence
#/#!"# , #$/#!"#
%/#!"# , %$/#!"#
frequency of 99.9%, 99%, 1%, and 0.1%, although those of 90% and 10% show relatively weak
correlations with these statistics. The negative correlations between the percentiles and indicate
that the velocity variation ranges in the higher positions reduces probably because of the
reduction of the turbulence intensity. This is verified in Fig. 7(a). In contrast, the positive
correlations with skewness occur because the PDFs adopt long tail shapes in the positive ranges.
This indicates that the magnitudes of the strong wind speeds increase. The standard deviations
are less related to the percentile values. This may be owing to the limited ranges of the values
in the current approaching flow.
Fig. 10 (d–f) shows the same for the vertical velocity component. Although we
investigated the correlations with the statistics of the vertical velocity component and their
percentiles (not shown), the relationships were ambiguous. This could be anticipated from the
percentile profiles irrelevant to the mean vertical velocity component in Fig. 9 (b). Thus, we
show only the scatter charts of the percentiles and statistics of the streamwise velocity
The vertical velocity component shows trends similar to those of the streamwise
component. That is, the mean and skewness are correlated strongly with the percentiles. Strong
wind events with occurrence frequencies of 99.9%–90% show positive correlations with
and negative correlations with
. This is because the velocity ranges in the vertical velocity
component increase gradually with height (see Fig. 7) to cause larger fluctuations from the
mean with an increase in mean velocity
. In contrast, the skewness of the streamwise velocity
becomes negative and large with height, and that of the vertical velocity component exhibits
the converse trend with height (Fig. 5 (a)). This can also be understood from the fact that the
Reynolds stress is negative owing to the downward transport of momentum in the turbulent
boundary layers. The PDFs of the vertical velocity component become long-tailed in the
negative direction because of these relationships. This reduces the magnitudes of the strong
wind speeds (see Fig. 10 (f)).
For the weak wind events denoted by 10%–0.1%, the relationships between these
statistics and the percentiles are the converse of the above. Because the mean value of the
vertical velocity component is nearly zero, the weak wind events represented by 10%–0.1%
are negative or downward flows. Therefore, the negative correlation with the mean streamwise
velocity indicates that the magnitudes of the rare wind speeds increase because of the expansion
of the velocity ranges (see Fig. 7 (b)). Similar to the case of the streamwise velocity component,
the standard deviation is less related to the percentile values of the vertical velocity component.
As can be anticipated from Figs. 4 and 5, the vertical profiles of the higher-order
statistics are similar. However, the relationships are not monotonic with the order (see Fig. 6).
This implies that the correlations between percentile values and high-order statistics can vary
because of the dissimilar dependencies among the statistics and percentiles. To quantify this
aspect, Fig. 11 summarizes the correlation coefficients between the deviation of the percentile
values from the mean value and the statistics of the streamwise velocity component.
With regard to the percentiles of the streamwise velocity component, these
correlations show two important aspects for understanding the relationships among the
statistics and percentiles. First, the four statistics (mean, standard deviation, skewness, and
kurtosis) show different dependencies on the percentiles. Second, skewness or kurtosis
represents a similar correlation with high-order statistics with odd and even orders, respectively.
For the vertical velocity, the correlation coefficients vary between positive (for strong wind)
and negative (for weak wind) because of the negative percentile values. However, similar
dependencies of the percentiles on the statistics can be verified.
In the present approaching flow, the PDFs mostly follow the modified Gaussian
distributions described by Eq. (4). Thereby, higher-order statistics are predicted effectively by
Eqs. (5) and (6) (as verified in Fig. 6). This implies that the higher-order statistics (
4 5 P
dependent on the lower-order statistics
. In addition, the applicability of Eqs. (5)
and (6) imply that the PDFs can be predicted using Eq. (4). This indicates that only four
independent parameters (namely, mean, standard deviation, skewness, and kurtosis) are
required to determine rare wind events. To validate this assumption, the dependency of the
percentile values is represented by only the first four statistics in Fig. 11, whereas
simply show similar correlations as
with the percentiles.
Although we need to apply this analysis to the various types of turbulent flow to clarify
the relationships between the percentiles and high-order statistics, the present analysis has
shown that the dependency of the percentiles on the statistics is related to the number of
parameters incorporated in the PDF models. Therefore, the current study successfully
demonstrated the importance of scrutinizing high-order statistics as well as PDFs for estimating
rare wind events. The velocity fields and their PDFs around buildings are altered from the
original approaching flow owing to the presence of the buildings. This makes it important to
describe the probabilistic characteristics of the approaching flow to discuss PLWs, as identified
in this section.
Fig. 10 Correlations between the percentile values for (a–c) streamwise and (d–f) vertical
velocity components and the statistics of the streamwise velocity components. (a, d) Mean, (b,
e) standard deviation, and (c, f) skewness. The values
indicate the correlation coefficients
between the values on the two axes.
Fig. 11 Correlation coefficients between the statistics of streamwise velocity and percentile
values of (a) streamwise and (b) vertical velocity components.
In this study, wind tunnel experiments were conducted to generate an artificial
approaching flow following empirical equations for the mean streamwise velocity component,
standard deviation, integral length scale, and Karman spectrum. The objectives were to
understand how the probability density functions of the approaching flow can be expressed and
to clarify the correlation among high-order statistics, probability density functions (PDFs), and
The experimental data showed that the characteristics of the generated approaching
flow followed basic power-law vertical profiles over rough surfaces in category III. In addition,
the von Karman spectrum was reproduced effectively at various measurement heights.
To understand the theoretical relationship between high-order statistics and probability
density function (PDFs), we employed a modified Gaussian distribution function known as the
Gram–Charlier Series (GCS) by considering skewness and kurtosis. In addition, we clarified
how high-order statistics can be expressed by lower-order statistics.
Based on the time series data of the approaching flow, we determined ten statistics
from the mean to the 10th-order moments. Furthermore, the characteristics of the statistics were
examined. The relationship between the order and statistics was discussed for both odd- and
even-order statistics based on the PDFs modeled by the GCS. This indicated that high-order
statistics can be used as indices to validate whether the GCS model can be applied to the PDFs
of the approaching flow. The present approaching flow showed that the PDFs were modeled
effectively by the GCS by considering the mean, standard deviation, and third- and fourth-
order statistics (skewness and kurtosis).
In addition, the correlation between high-order statistics and extreme values
represented by percentiles based on PDFs was examined. This showed that three statistics
(mean, skewness, and kurtosis) are correlated strongly with the percentile velocity components.
Although a similar strong correlation between the percentiles and other high-order statistics
can be identified, the dependency of the statistics on each other shows that only fourth-order
statistics dominate in determining the PDFs of the approaching flow.
Although variations in PDFs is likely for various methods for generating the
approaching turbulent flow following empirical equations for the mean, turbulent intensity, and
power spectrum density, we showed the framework for displaying and considering the
stochastic information of the inflow when we consider the PLW based on stochastic analysis.
This study will expand the usage of the PDFs as a basic stochastic information when discussing
the effects inflow turbulence and buildings on the PLW because a prediction model of PDFs
! !!"# "$#%$&$'$($)$*$+,
%&!-' !( %&)-' )(
! !!"# "$#%$&$'$($)$*$+,
based on GSC are introduced using the higher-order statistics. The most notable advantage of
the present model is the simple modification of the PDFs only by the high-order statistics,
which can be easily obtained in both WTE and CFDs. In addition, this study highlights the
importance of collecting the datasets of the high-order statistics which are not commonly
discussed in the previous studies.
We introduced the present model for describing the approaching turbulent flow;
However, the same method can also be applied for the modeling for PDFs at pedestrian levels
regardless of datasets from WTEs or CFDs. Applying the model for the various cases such as
various types of approaching flow, flows around buildings, and complex urban flow needs to
be conducted in a future study to prove the applicability.
This study was supported partially by a Grant-in-Aid for Scientific Research from JSPS, Japan,
KAKENHI (Grant no. JP 21K18770), FOREST program from JST, Japan (Grant No.
JPMJFR205O), and the Initiative for Realizing Diversity in the Research Environment. The
experiments were conducted in a wind tunnel facility at the Wind Engineering Research Center,
Tokyo Polytechnic University.
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