Abstract
The wind pressure time history of high-rise building cladding is mostly non-Gaussian distribution, and there is a one-to-one correspondence between a specified guarantee rate and its corresponding peak factor. A stepwise search method for calculating the peak factor of non-Gaussian wind pressure and a gradual independent segmentation method for extracting independent peak values have been proposed to determine the relationship accurately in the previous study. Based on the given experiment and calculation results in the existing research results, more analysis can be given to enrich the study on this topic. In this paper, some characteristics of wind pressure coefficient time series in time and frequency domain are analysed. Based on the basic theory of fractal, the R/S analysis of wind pressure time series is made, and the fractal characteristics of wind pressure coefficient time series are explained. Based on the statistical theory, the relationship characteristics between high-order statistics and peak factors are studied. The correlation between the guarantee rate and the corresponding peak factor is analysed, and the guarantee rates calculated by the Davenport peak factor method are evaluated. The power spectrum characteristics of fluctuating wind pressure are analysed and the relationship between turbulence characteristic frequency and optimal observation time interval is discussed.
1. Introduction
In [1], a stepwise search method for calculating the peak factor of non-Gaussian wind pressure and a gradual independent segmentation method for extracting independent peak values have been proposed. Based on the pressure data of a high-rise building obtained from a rigid model wind tunnel test, the peak factors of non-Gaussian wind pressures on cladding are calculated and compared using several typical methods. The value of the peak factor and its error rate calculated by several methods is compared with the observed average peak value, and the conversion between the guaranteed rate and the peak factor is discussed. Based on the reliability theory, the true distribution of wind pressure time history is approached infinitely through an efficient numerical method in the process of stepwise search. In the discussion of calculation results, conversion between the guarantee rate and peak factor is given. In fact, more interesting conclusions can be obtained from the experiment and research in paper [1]. These conclusions not only help to illustrate the effectiveness of the algorithm but also are very meaningful for the study of wind pressure time series. In this paper, more analysis and conclusion are given to enrich the study in paper [1].
In the gradual independent segmentation method proposed in paper [1], the wind pressure time history is divided into several subintervals according to the optimal sectional capacity, and the maximum or minimum value (uniformly referred to as extreme value) is selected from each subinterval to form the maximum or minimum value time series (uniformly referred to as extreme value wind pressure time series). Because the location of the extremum of each segment is random, the data interval between adjacent subsamples may be less than the optimal segment capacity, which indicates that the statistical independence of the extracted values cannot be theoretically guaranteed [2,3]. Therefore, the gradual independent segmentation method is proposed to solve the problem of calculating the peak factor of non-Gaussian wind pressure. Complex wind field around high-rise buildings is one of the main reasons for non-Gaussian wind pressure [4]. There are hydrodynamic phenomena such as air separation, reattachment, and vortex. Scholars have explored the characteristics of fluid-induced vibration or flow field [5, 6], which is helpful for the research of wind load and wind pressure of high-rise buildings for wind-resistant design. With the progress of the technique for the measurement of unsteady aerodynamic force [7], wind pressure distributions and unsteady aerodynamic characteristics are investigated and analysed [8]. A series of wind tunnel experiments and numerical simulations have been carried out to determine aerodynamic forces and wind pressures acting on tall building models with various configurations including varying wind attack angles and wind velocities [9]. Through the experiments, the random characteristics of wind pressure time series are studied. The fractal theory is one of the basic theories used to study the random time series of wind pressure. In [1], the R/S sequence is only used to determine the optimal sectional capacity, and the R/S analysis of wind pressure coefficient time series is not given. In fact, the R/S series can also be used to study the fractal characteristics of time series [10, 11]. This paper further analyses the R/S characteristics of wind pressure series in the time domain based on the rescaled range analysis method, to explain the fractal judgment of wind pressure coefficient series. In addition, with more in-depth study of the algorithm in paper [1], the relationship among guarantee rate, peak factor, and non-Gaussian properties is discussed further, the characteristics of time series of wind pressure coefficients are analysed in the frequency domain, and the influence of turbulence characteristics on sectional capacity is also discussed.
2. Rescaled Range Segmentation Method and Hurst Number
“Rescaled range analysis method” is used to detect the long-term memory of wind pressure time history, determine the disappearance point of long-term memory behavior, and take this node as the segmentation point of segmented time length to ensure the independence of segmented time history. This method has been described in detail in [1], and it is briefly reviewed here.
In the study, the range R was removed by the standard deviation S of the observed value to obtain a dimensionless ratio called the R/S statistic. Suppose is the wind pressure time history, with a data length N and a corresponding time T. Then C is divided into segments , and the capacity of each segment , is . For the wind pressure time history C, when the segment length is n, the statistic of the kth segment is
The range of the kth segment is the standard deviation of the kth segment. When the segment length is n, the statistic of the wind pressure coefficient time history C is
The formula for the Q statistic is , which can be used to determine whether the time sequence has nonperiodic cycles and then determine the average cycle length; that is, the past trend affects the future time length. The critical point of the segment length (i.e., the sample capacity ) is then determined when the curve of changes markedly at a point such that its increasing trend stops, which indicates that the long-term memory of the stochastic process disappears [12, 13]. Therefore, the time history of wind pressure coefficient is divided into segments with length .
It is found that statistic is a power function of sample measurement capacity n (corresponding to measurement duration ): , where is a constant; H is Hurst index. Hurst index is an index based on rescaled range analysis. Taking logarithms on both sides of the listing equal sign, the following result can be got:
Research shows that the R/S analysis results of time series in natural phenomena generally are in the form of formula (4), which is consistent with the above formula about :where the H value is generally about 0.72, which is called the Horst phenomenon. The main method of time series analysis is based on the characteristics of time series data, through the establishment of an appropriate model for its approximate description. R/S analysis method provides a new way for time series analysis. According to the introduction of R/S analysis, the ratio R/S is related to the selected time range, so it can be defined as formula (5):
If we change the size of the time scale, there is (6). Formula (6) is the scaling characteristic of R/S analysis of time series:where is equivalent to the scaling factor of time scale transformation, that is, Horst index, which indicates that is self-similar. The formula indicates the relationship between scale transformation factor and fractal dimension. As the research of time series fractal theory is still developing and improving when the R/S analysis method is used to analyse the fractal characteristics of a time series, it is not necessary to assume that R/S measures the distribution characteristics of the time series; that is, it is relatively simple without considering the distribution that the time series follows. Therefore, this method is used to analyse the wind characteristics of the wind field based on R/S analysis results of wind pressure.
3. Research Methods Based on Wind Tunnel Test
3.1. Test Model
The rigid model for the wind tunnel pressure test is shown in Figure 1(a), together with the definitions of model orientation, wind direction angle, and coordinate axis of this rigid model. Each pressure signal was sampled with 6000 data points with a frequency of 312.5 Hz, which means the sampling time is 19.2 s. Because of the limitations of the channels of the electronic scan valve, the signals of measuring taps could not be collected simultaneously at a time and should be divided into 4 groups of A, B, C, and D. The way of grouping situation, the characteristic size of the model section, and the layout of surface pressure measurement taps are shown in Figure 2. It can be seen that pressure measurement taps were distributed on 22 levels, which were numbered from bottom to top of the model. There are 28 to 40 measurement taps in each level, changing with height. In this study, “L-i” is used to indicate the measurement taps, where L represents the number of levels and i represents the measurement point number at this level. For example, “5–17” represents the No. 17 measurement point of the 5th level.
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3.2. Test Conditions
Corresponding to the 1 : 350 length scales, the total height of this model is 1.4 m. In this test, the reference wind speed was 14 m/s, which was monitored by a Pitot tube at 1.2 m height in wind tunnel (above the gradient height, equivalent to the actual 420 m height), about 1.0 m away from the sidewall of the wind tunnel. Considering a recurrence period of 50 years, the actual wind speed corresponding to the reference point is 52.93 m/s according to the conversion of basic wind pressure of the C-type landform in Shanghai, and then the wind velocity ratio can be deduced to be . The time ratio is , which can be deduced by the dimensional analysis method. The definition of model azimuth, wind direction angle, and coordinate axis of synchronous pressure measurement wind tunnel test of this rigid model is shown in Figure 1(b).
3.3. Wind Field Simulation
This study takes the Shanghai World Financial Center as an engineering background, and the pressure tests were carried out on a rigid model in the TJ-2 Wind Tunnel at Tongji University. The TJ-2 Wind Tunnel is a boundary layer tunnel of closed-circuit type, and the working section of the tunnel is 3 m wide, 2.5 m high, and 15 m long. The achievable mean wind speed ranges from 0.5 m/s to 68.0 m/s, adjustable continuously. The mean wind speed profile together with the turbulence intensity for terrain category C was simulated mainly by using hybrid passive devices including roughness blocks, spirelets, and vertical-bar fences. The simulation result and the theoretical formula according to the load code for the design of building structures by the Ministry of Construction of the People’s Republic of China (GB50009–2001) are shown in Figure 3. The wind tunnel test design here fully considers meeting the similarity principle. When the experimental data are restored to the building prototype, the algorithm results are suitable for the actual building.
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4. Results and Parametric Studies
4.1. R/S Analysis of Wind Pressure Coefficient Series
Taking measurement points on the 5th floor as an example (see Figure 4), the relationship between log (R/s) and log (k) is drawn in the double logarithmic coordinate system. It can be seen from Figure 5 that the R/S analysis results of the wind pressure coefficient time history curve meet the law; that is, on the double logarithmic coordinates, log(R/S) to log(n) are straight lines, and the Hurst index is distributed in the range of 0.5∼1, which verifies that the wind pressure coefficient time history series has fractal characteristics. If the wind pressure coefficient time history sample analysed is a fractal signal, the Hurst value of it is the slope of the fitting line. Figure 5 is the wind pressure coefficient time history diagram of several typical measuring points and the corresponding R/S analysis results.
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4.2. The Relationship between Higher-Order Statistics, Peak Factor, and Guarantee Rate
4.2.1. Correlation between Wind Pressure Higher-Order Statistics and Peak Factors
The skewness and kurtosis of higher-order statistics are parameters closely related to non-Gaussian characteristics [14], so it is necessary to discuss their correlation. Figure 6 shows the relationship between skewness-peak factor and kurtosis-peak factor of 808 measurement taps at 0°, 45°, and 90° wind directions (the peak factor is obtained by different calculation models, which are the Davenport method [12], the stepwise search method proposed in this paper, the Kareem-Zhao method [13], and the Sadek-Simiu method [15]). It can be seen from Figure 6 that there is a negative linear correlation between skewness coefficient and peak factor in strong non-Gaussian region (Figures 6(a)–6(c) left side), while there is a nonlinear monotone positive correlation between kurtosis coefficient and peak factor in strong non-Gaussian region (Figures 6(a)–6(c) right side). With the increase of kurtosis (e.g., reaching 15–35, this value is related to wind direction angle), the correlation weakens. The results show that the skewness coefficient and kurtosis coefficient have a prominent correlation with peak factor, which indicates that it is reasonable to calculate peak factor in non-Gaussian region based on skewness coefficient and kurtosis coefficient. Different methods show different mathematical relationships between peak factor and skewness, peak factor, and kurtosis, which are influenced by six factors: building shape, wind direction angle, location of measurement taps, turbulence intensity, windward area, and calculation model. It is worth noting that (1) the results of the Davenport method do not coincide with the above general trend regularity (the area circled in each subfigure in Figure 6) because the peak factor method does not have a clear concept of reliability and cannot give the corresponding wind pressure extreme value with uniform guarantee rate with the increase of skewness and kurtosis. (2) There are exceptions to the results calculated by other methods except for the Davenport method for a few measurement taps; that is, the peak factor is very small when there are high skewness and large kurtosis, which indicates that the rule of high-order statistics to peak factor is not uniform under limited experimental conditions. From a statistical perspective, when most of the data in the sample are concentrated near the mean, the peak factor will be small even if there are high skewness and large kurtosis, whose mechanism needs further study.
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4.2.2. Correlation between Guarantee Rate and Corresponding Peak Factor
The variation of peak factor with the strength of non-Gaussian with the specified guarantee rate is discussed in this section. According to Section 4.2.1, the degree of non-Gaussian is measured by skewness and kurtosis. Therefore, it is needed to discuss the change of peak factor caused by the change of skewness and kurtosis. It is an analysis from the point of view of the original geometric concepts of skewness and kurtosis.
(1) Relationship between Skewness and Peak Factor. The skewness coefficient indicates the skew direction and skews degree of statistical data distribution, which is used to characterize the asymmetry degree of probability distribution density curve relative to the average value. The larger the absolute value of skewness coefficient is, the longer the tail of fluctuating wind pressure probability density curve is, that is, the greater the probability of extreme events is. The skewness coefficient of Gaussian distribution is 0. Since the skewness changes asymmetrically, for convenient analysis, the unilateral guarantee rate model is taken as the object to explain the relationship between the changes of skewness coefficient and the peak factor.
When the skewness satisfies , it is a positive (right) skewness (as shown in Figure 7(a)), and more data are located on the right side of the mean than on the left side.
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When the skewness continuously increases towards the right, the right tail of the probability density curve becomes longer, and the “right tailing” phenomenon will appear, which indicates that the probability of extreme events at the right tailing becomes greater; that is, the probability of maximum wind pressure coefficient becomes greater. The quantile value is selected near the maximum wind pressure coefficient (at the maximum value, the guarantee rate is calculated from the negative infinity of the horizontal axis to the quantile). Only the quantile moves from the original position to the right (equivalent to moving some samples to the left side of the quantile) can ensure achieving the same guarantee rate. In this case, the number of samples on the left side of the quantile remains unchanged (same for samples on the right side). Therefore, if the extremum (i.e., the quantile value) is larger, the corresponding peak factor will be larger.
When estimating the independent minimum wind pressure time series, it is needed to multiply the minimum wind pressure coefficient time history by −1 and then follow the steps of estimating the maximum wind pressure coefficient time history to calculate the expected value of the extreme value or the extreme value corresponding to other quantiles, which needs to be multiplied by −1 as the extreme value of the minimum wind pressure time history. From the process of estimating the extremum of minimum time series, it can be inferred that the analysis process of negative skewness is symmetrical with that of positive skewness.
When the skewness satisfies , it is a positive (right) skewness (as shown in Figure 7(b)), and more data are located on the left side of the mean than on the right side.
When the skewness continuously increases towards the left, the left tail of the probability density curve becomes longer, and the “left tailing” phenomenon will appear, which indicates that the probability of extreme events at the left tailing becomes greater; that is, the probability of minimum wind pressure coefficient becomes greater. The quantile value is selected near the minimum wind pressure coefficient (at the minimum value, the guarantee rate is calculated from the positive infinity of the horizontal axis to the quantile). The data as a whole tends to move to the left of the mean, and there are more samples on the left side of the quantile. Only the quantile moves to the left from the original position (equivalent to moving some samples to the right side of the quantile) can ensure achieving the same guarantee rate. In this case, the number of samples on the left side of the quantile remains unchanged, so the extreme value (i.e., the value of the quantile) decreases (the absolute value is increased), and the corresponding peak factor increases. With the increase of skewness, the quantile moves to which side the data moves to ensure the guarantee rate is unchanged.
According to the above analysis, under the condition of equal guarantee rate, when the skewness satisfies , the larger the skewness is, the larger the peak factor is; when the skewness satisfies , the larger the skewness is, the smaller the peak factor is. In brief, under the condition of equal guarantee rate, the larger the skewness is, the larger the peak factor is.
(2) Relationship between Kurtosis and Peak Factor. The kurtosis coefficient indicates the steepness in the middle of statistical data distribution curve and the length and width of the tail. The kurtosis coefficient of normal distribution is 3. The kurtosis coefficient greater than 3 indicates that the sample points are more concentrated near the mean value, the probability density curve is sharper, and there is a longer tail than the normal distribution, according to which a larger extreme value than the normal distribution can be obtained. The kurtosis coefficient less than 3 indicates that the sample points are more dispersed (farther from the mean value), the probability density graph is flat, the tail is shorter than that of the normal distribution, and the extremum that can be obtained is smaller than that of the normal distribution. As the kurtosis change is a symmetric process about the mean value, for convenient analysis, the bilateral guarantee rate model is taken as the object to explain the relationship between changes of the kurtosis coefficient change and the peak factor.
When the skewness satisfies (as shown in Figure 7(c)), the probability density curve is sharper than the normal distribution curve, which makes the area under the Gaussian distribution curve within the quantile according to the original guarantee rate smaller. Therefore, to keep the area unchanged (i.e., the guarantee rate unchanged), the quantiles should move symmetrically and synchronously to both sides away from the mean value, so the corresponding peak factor (extreme value) becomes larger.
When the skewness satisfies (as shown in Figure 7(d)), the probability density curve is smoother than the normal distribution curve, which makes the area under the Gaussian distribution curve within the quantile according to the original guarantee rate larger. Therefore, to keep the area unchanged (i.e., the guarantee rate remains unchanged), the quantiles should move symmetrically and synchronously close to the mean value, so the corresponding peak factor (extreme value) becomes smaller.
When the skewness satisfies (as shown in Figure 7(c), soft response state and softening process), the probability density curve is sharper, and the left and right tailing are longer than those of the Gaussian distribution curve. The probability of large positive and negative pressure pulse in the data is increased; that is, some maximum and minimum wind pressure coefficients move to both sides far away from the mean. In order to keep the samples between the quantiles unchanged, the quantiles should move symmetrically and synchronously to both sides away from the mean value, so the corresponding peak factor (extremum) becomes larger. The peak factor of the stepwise search method is larger than that of Davenport’s normal distribution calculation method (e.g., when the wind direction angle of 90o degrees in Table 1 is 5–17 point = 13.1109 > 3, it belongs to soft response state, the peak factor of stepwise search method is 6.9059, and the peak factor of Davenport’s method is 3.3035).
When the skewness satisfies (as shown in Figure 7(d), hard response state and hardening process), the probability density curve is smoother, and the left and right tailing are shorter than those of the Gaussian distribution curve. The probability of large positive and negative pressure pulse in the data is reduced; that is, some maximum and minimum wind pressure coefficients are moving towards the mean value. To keep the samples between the quantiles unchanged, the quantiles should move symmetrically and synchronously to both sides of the mean value, so the corresponding peak factor (extremum) becomes smaller. The peak factor of the stepwise search method is smaller than that of Davenport’s normal distribution calculation method (e.g., Kurtosis of 5–17 point = 2.8153 < 3 in Table 1 belongs to hard response state at 0° wind direction angle, the peak factor of stepwise search method is 3.4166, and that of Davenport’s method is 3.4192).
To sum up the above analysis, the conclusion is drawn: the higher the kurtosis coefficient is, the higher the peak factor is.
4.2.3. Evaluation of the Load Guarantee Rate Corresponding to the Davenport Peak Factor Method
The observed mean extremum may be used as a comparison object to evaluate the closeness degree between the wind pressure extreme value calculated by the peak factor method and the actual extremum probability model, through which the applicability of the Davenport peak factor to different measuring points can be evaluated. Based on the stepwise search method in this paper, a more accurate extremum probability model can be obtained, which is convenient for calculation and it is also suitable for comparison object consequently. Two points should be noted particularly in solving the problem: one is that the Davenport peak factor is calculated based on the extremum estimation method of sample parent distribution; the other is that the stepwise search method is based on the extremum estimation method of independent sample peak value. The two methods cannot be confused. The concrete operation is to calculate the wind pressure extreme value based on Davenport theory firstly and then substitute the extreme value results into the sample extreme value probability model established by numerical method, to obtain the corresponding guarantee rate.
The following measurement points of the 5th measurement level at 90o wind direction angle (refer to Figure 4) are taken as research objects. The load guarantee rates corresponding to the Davenport peak factor (i.e., the guarantee rates of extreme value estimates calculated by the peak factor method) are shown in Table 2 and the corresponding illustration is Figure 8, which are classified as Face D (left crosswind side), Face A (windward side), Face B (right crosswind side), Face C (leeward side), windward corner cut area (pocket area), and leeward corner cut area (wake area).
The wind load guarantee rate of measurement points in different regions is different. The guarantee rate of measurement points at the windward side is relatively high, up to about 80%. The guarantee rate on both crosswind and leeward sides is not high hovering at about 50%∼60%. The guarantee rate of measurement points at wake center with large negative pressure, leeward side, and windward side corner cut area with strong non-Gaussian is even lower. In Table 2, the guarantee rate of measurement points labeled with 5–7, 5–8, 5–17, and 5–18 in the area outlined by the red dotted line is the lowest, because these four points are located in the leading edge airflow separation zone on the left and right crosswind sides with prominent non-Gaussian effect. Based on the analysis of the above example, we can see that the stronger the non-Gaussian characteristics of the measurement points are, the smaller the wind load guarantee rate corresponding to the peak factor based on the Davenport theory is, and the error rate is more than 20%. Therefore, it is unreliable to use the Davenport method to estimate the wind pressure extreme value of the measuring points with strong non-Gaussian characteristics. The peak factor conceptually corresponds to mathematical expectations of the extremum, and the guarantee rate correspondingRelationship between Higher-Order Statistics, Peak Factor, and Guarantee Rate to mathematical expectations of extremum varies randomly with the specific distribution information (wind pressure time history information) of non-Gaussian distribution. It can be seen that the Davenport peak factor value of each measurement point is uncertain to obtain a certain guarantee rate.
The defect of the peak factor method lies in the lack of a clear concept of reliability and the inability to give the wind pressure extreme value with a uniform guarantee rate. Therefore, when the peak factor method is used in the wind-resistant design of structure cladding, the most unfavorable negative pressure is obviously underestimated for measurement points in the areas significantly affected by characteristic turbulence, such as the airflow separation area and wake area. The reason for this phenomenon is that the fluctuating wind pressures in the above-mentioned regions are controlled by organized vortex shedding, with strong spatial correlation and obvious non-Gaussian characteristics, and their corresponding extremum dispersion is also stronger. In this case, the basic assumption of the traditional peak factor method is no longer valid, and extreme value estimation methods applicable to the non-Gaussian statistical properties of samples should be adopted, such as the Sadek-Simiu method [15], the revised Hermite model method [16], and the stepwise search method.
4.3. Analysis of Wind Pressure Time Series in Frequency Domain
4.3.1. Power Spectrum Characteristics of Fluctuating Wind Pressure
Figure 9 shows cross section of the 41st layer of the experiment building model and deployment of measurement points at this measurement level. The following analysis is based on test on results of measure points at the 41st layer. Figure 10 shows the curve trend of three theoretical wind pressure spectra: Davenport spectrum, Kaimal spectrum, and Karman spectrum. Figure 11 shows the wind pressure spectrum of typical measurement points at 45° wind angle. This is the wind pressure spectrum on the windward side and the comparison with the downwind Davenport spectrum, Kaimal spectrum, and Karman spectrum [17].
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It can be seen from Figures 11 and 12 that the Davenport spectrum is very close to the actual wind pressure spectrum at both low and high frequencies. This is because the fluctuating wind pressure on the windward side of the building satisfies the quasisteady assumption for the incoming flow which is not disturbed by any building upstream. The ability to fluctuate wind pressure spectrum is concentrated in a wide frequency band, and there is no particularly prominent peak. The low-frequency energy is controlled, and this low-frequency energy mainly comes from the pulsation of the incoming flow; that is to say, the fluctuation of wind pressure on the windward side is mainly caused by the fluctuation of flow velocity. Therefore, the power spectrum of fluctuating wind pressure on the windward side is very close to that of fluctuating wind speed on the downwind side. On the nonwindward side, such as separation zone and wake zone, the fluctuating wind pressure spectrum is very different from that of incoming flow (see Figures 12(d), 12(g), and 12(h)). The comparison between the fluctuating wind pressure spectrum and Davenport spectrum in typical different areas is shown in Table 3.
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Figures 12(a)–12(e) and 12(g) show the time history of wind pressure coefficient and fluctuating wind pressure spectrum of measuring points 41–15, 41–20, and 41–25 in the top row of building facade under crosswind; it can be seen from the time history diagram that the wind pressure pulsation gradually transits from the large amplitude pulsation in the separation zone (measuring point 41–15) to the relatively stable pulsation in the attachment zone (measuring point 41–25). The corresponding wind pressure spectrum reflects the process of energy transfer from low frequency (reduction frequency is about 0.1) to high frequency (reduction frequency is about 1).
This shows the separation of the airflow at the leading edge and which generates large-scale and low-frequency vortexes. The vortexes roll downstream and reattach at the trailing edge of the side, resulting in a large number of small-scale and high-frequency vortexes, so that the high-frequency energy of the wind pressure spectrum at the measuring point is dominant. In the region between the separation point and reattachment point (measuring point 41–20), due to the combined action of large-scale and small-scale vortexes, the wind pressure spectrum curve reflects that the energy is mainly concentrated in two main frequencies, which form two peaks on the curve.
In this section, the above results are explained. The measuring point 41–17 is located in the separation zone, and there are two peaks on the curve of its fluctuating wind pressure spectrum, which indicates that there are two main frequencies, that is, one low frequency and one high frequency. Due to the separation of airflow, there are a lot of large-scale and low-frequency vortexes at the measuring point 41–17 in the upwind region of the crosswind surface. The energy of fluctuating wind pressure spectrum of large-scale vortexes is concentrated in the low-frequency band, and the reduction frequency is about 0.1. There is a 90° angle at the front of 41–17. The vortex generated at the separation point of 90° wind direction reattaches at the front of the crosswind surface, resulting in a large number of small-scale vortexes and falling off, which leads to a peak on the curve of the fluctuating wind pressure spectrum of measuring point 41–17 at high frequency, and its reduction frequency is about 1. Similarly, the measuring point 41–20 is located in a crosswind area, and there are two peaks on the curve of its fluctuating wind pressure spectrum. The low-frequency energy comes from the large-scale characteristic turbulence produced by the separation of the airflow, and the high-frequency energy comes from a large number of small vortexes produced by the reattachment of the large-scale characteristic turbulence [17]. The wind pressure spectrum of 41–20 in the middle vortex reattachment region shows obvious broadband characteristics. However, for measuring point 41–15, which is located at the front edge of the separation zone, there is only one wave peak on its curve of the fluctuating wind pressure spectra because there is no effect of vortex reattachment, and its reduction frequency is about 0.1.
Similarly, for the measuring points 41–25 and 41–24 located at the downwind area, the peaks of their curve of fluctuating wind pressure spectra at the low-frequency band are not obvious, especially for the measuring point 41–25, while the energy of the fluctuating wind pressure spectrum at the high-frequency band increases. A large number of large-scale and low-frequency turbulence conditions generated at the front edge of crosswind reattach and break down in the downwind region, resulting in a large number of small-scale and high-frequency vortexes. As a result, the energy of fluctuating wind pressure spectrum in the downwind region is concentrated in the high frequency, and its reduced frequency is about 1. Figures 12(f) and 12(h)–12(l) show the wind pressure coefficient time history and fluctuating wind pressure spectrum of measuring points 41–20, 39–23, and 28–39 in the middle vertical columns of building facade in the wake area. It can be seen from the wind pressure spectrum curve that the obvious broadband characteristics and the peak value in a very wide frequency band are not prominent. The energy of it is slightly raised in the reduced frequency range , and the distribution is relatively uniform in this range, which forms a horizontal line shape. This is because the regularity of the vortex is destroyed and the vortex structure distribution in the wake area becomes more uniform due to the influence of the wake. The fluctuating wind pressure spectrum accords with the quasisteady assumption. It can also be seen from the above analysis that energy is transferred from large-scale vortex to small-scale vortex. The frequency-domain characteristics of fluctuating wind pressure in typical areas are shown in Table 4.
To sum up, the power spectrum of fluctuating wind pressure on the windward side is very close to the Kaimal spectrum and Davenport spectrum describing the atmospheric flow, indicating that the conventional assumption can be adopted on the windward side. However, on the side wind surface parallel to the incoming flow, the power spectrum of the fluctuating wind pressure is quite different from that determined by the conventional assumption, which is due to the separation of the incoming flow, vortex shedding, and reattachment. Due to the influence of characteristic turbulence, the energy of fluctuating wind pressure spectrum of non-Gaussian measurement points is mostly concentrated in the high-frequency band, and the corresponding reduction frequency of its peak value is about 1, but the shape of fluctuating wind pressure spectrum is different. A few non-Gaussian measuring points are located near the edge of the building surface and the positive pressure area, due to the influence of the separated vortex and the airflow around the building edge, the wind pressure spectrum is more complex, and the difference is larger. Even on the same surface, the shapes of the wind pressure spectrum curve of non-Gaussian and Gaussian points will be different, so the curve characteristics of the wind pressure spectrum cannot be judged according to whether the wind pressure is Gaussian distribution or not.
4.3.2. Relationship between Turbulence Characteristic Frequency and Optimal Observation Time Interval
In paper [1], the gradual independent segmentation method to establish independent extremum sequence samples is introduced. The original intention of this method is as follows. Firstly, the optimal observation time interval is the shortest time delay of the autocorrelation coefficient falling from 1 to the value close to 0, which can only ensure that the segments (subintervals) are not correlated but cannot guarantee the independence of the segmented time histories. Secondly, for different wind pressure time histories, the optimal observation time interval is not unique. The common processing principle is to select the largest one among the multiple minimum time delays obtained by analysis as the optimal observation time interval. Thirdly, a certain wind pressure time history corresponds to an optimal segment capacity by default in the above description. However, the incoming flow in the actual wind field no longer meets the quasisteady assumption, the turbulence characteristic frequency (vortex shedding frequency) is a random variable, and the time interval between the nodal breakpoints of nonrandom walk of any random signal is not necessarily fixed. Therefore, the truly reasonable method is to obtain refined independent extremum sequence samples of each wind pressure time history by the “gradual independent segmentation method” similar to “frequency conversion technology” proposed in this paper.
When the data interval between adjacent subsamples is larger than , the adjacent subsamples are independent of each other. Otherwise, they are correlated. The data interval between the extremum values of the subsamples obtained by the “gradual independent segmentation method” is larger than , the numbers of these intervals are generally different, and the interval is changing with time. This is similar to the “frequency conversion” technology in electronic communications, where the “frequency conversion” means that period is variable; that is, the number of intervals is changing in real time to meet the independence of adjacent subsamples, which is determined by the internal information characteristics of the wind pressure time history.
From the analysis of the wind pressure power spectrum in the paper [17], it can be seen that the energy of fluctuating wind pressure mainly distributes within the reduction frequency range of 0.1 Hz to 1.0 Hz, the reduction frequency 0.1 corresponds to the frequency range of 1 Hz to 7.6 Hz in the wind tunnel test, and the reduction frequency 1.0 corresponds to the frequency range of 10 Hz to 76 Hz in the wind tunnel test. Therefore, the fluctuating period of wind pressure mainly falls in the range of 0.013 s to 1 s, which corresponds to the fluctuating period range of 0.44 s to 33.33 s of actual wind pressure. The designed time ratio of the wind tunnel test is 1 : 92.5803, and each pressure signal was sampled with 6000 data points at a frequency of 312.5 Hz, whose sampling time is 19.2 seconds (corresponding to the actual 30 minutes). Therefore, the relationship between the optimum sectional capacity and the optimal observation time interval is as follows: .
According to the statistical data of the calculation example, the average of the optimum sectional capacity of each measuring point working condition is 30.5725, the variation coefficient is between 0.05 and 0.2, and the values of under all measuring point working conditions are within the range of 7 to 56. The average of the optimal observation time interval for measurement points at actual building is about 9.1 seconds, which is in the same order of magnitude as 3 seconds in [18] and 6 seconds in [19], and the results of [18, 19] are all for low-rise buildings. With the increase of the height, the turbulence intensity decreases and the period of vortex motion becomes longer. The optimal observation time interval corresponding to high-rise buildings should be larger than that of low-rise buildings. Furthermore, to maintain the independence of each extremum, the interval between the extremum subsamples is larger than in the “gradual independent segmentation method,” and the shortest delay of the autocorrelation coefficient from 1 to 0 corresponds to the interval , which is also one of the reasons why the optimal observation time interval of high-rise buildings is longer than that of low-rise buildings in this paper.
In the atmospheric boundary layer, change of wind pressure on the structure cladding is caused by the change of the eddy on the cladding. The characteristic turbulence is caused by the separation, reattachment, and vortex shedding of the incoming flow on the building cladding. The characteristic turbulence makes the wind pressure acting on the local area have a strong spatial correlation because of the organized vortex structure above it. The period of such a complete correlation process is also the period of wind pressure fluctuation. That is to say, the fluctuation frequency of wind pressure is related to the frequency of vortex shedding, which is the reason why non-Gaussian wind pressure exhibits a distinct asymmetric distribution in time domain with intermittent large pulses. It is also the reason why the wind pressure time history has a phased memory, which determines the optimal segmentation capacity . The optimum segmentation capacity or the optimal observation time interval determines the independence between extremum samples. The above analysis can be summarized as follows. The independence of the extremum sample is determined by the turbulence characteristic frequency.
5. Conclusions
In this paper, some characteristics of wind pressure coefficient time series in time and frequency domain are analysed. Based on the experiments and data in paper [1], further analysis is given, and the following conclusions are obtained after detailed discussion:(1)The R/S analysis of wind pressure time series based on rescaled range method shows that the series has fractal characteristics.(2)(a) High-order statistics are closely related to the peak factor. In general, in the region with large absolute skewness (i.e., standard kurtosis coefficient minus 3), kurtosis beyond a certain value is also larger. (b) In general, for a given guarantee rate, the stronger the non-Gaussian region is, the larger the corresponding peak factor is.(3)The non-Gaussian characteristics of the wind pressure time series are mainly derived from the characteristic turbulence. The independence of subsample peaks is determined by the nodal point of nonrandom walking in wind pressure time history. The turbulence characteristic frequency (vortex shedding period) determines the best observation time interval . In the real wind field, the turbulence characteristic frequency is no longer a fixed value, but a random variable. The average length of the best section of the high-rise building is about 30.(4)The wind pressure extremum with a consistent assurance rate cannot be obtained by the peak factor method because the assurance rate corresponding to the expected value of the extremum changes randomly with the specific parameters of the distribution probability. The defect of the peak factor method is that it lacks a clear concept of reliability. The stronger the non-Gaussian is, the smaller the load guarantee rate of the peak factor method is. For each measuring point with the same load guarantee rate (equal guarantee rate), the Davenport peak factor of each measuring point is uncertain.
Data Availability
The data (float type, can be saved as in .mat file or .txt file) of pressure tests on a rigid model of Shanghai World Financial Center and data generated during research used to support the findings of this study were supplied by the State Key Laboratory of Disaster Reduction in Civil Engineering at Tongji University in China under license and so cannot be made freely available. Requests for access to these data should be made to the correspondence author.
Disclosure
Any opinions and concluding remarks presented here are entirely those of the authors.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work described in this study was supported by the Fostering Project of Innovation Team in Interdisciplinary Areas of Shanghai Science and Technology Commission (General Topic nso. 03DZ12039). The wind tunnel test data of this study come from this project, which provides a guarantee for the completion of this study.