Nonlinear dynamic analysis of wind pressure variability on standard tall building

Abstract

Understanding the variability of wind pressure on tall building surfaces is of essential importance regarding the control of wind-induced response. In this study, advanced time series analysis techniques, e.g., phase space reconstruction and recurrence analysis, were applied to diagnose the wind pressure variability from a nonlinear dynamic perspective. It is shown that, the wind pressure acting on tall building surfaces exhibit distinct chaotic nature. Due to the complex flow patterns around the building, the underlying dynamics of wind pressure is subject to pronounced region-to-region variability. For windward surface, the wind pressure dynamics at the stagnation region is more deterministic than those of downstream. For the side surfaces, the distribution of recurrence quantification analysis (RQA) indicators follows a clear pattern, where the values are larger at the near-front edge, and decreases towards the rear. The distribution of RQA indicators on the leeward surface is opposite to that of windward surface, in which larger values occur more often at near-ground level. Further upstream, the values are found to decrease. The outcomes provide valuable insights to the wind-structural interaction, thus could help wind-resistant design.

Keywords

Commonwealth Advisory Aeronautical Research Council standard tall building surface wind pressure nonlinear dynamic analysis phase space reconstruction recurrence analysis

Introduction

It is apparent that, with the rapid developments in construction technologies and in the context of substantial urban growth, tall buildings have been increasingly constructed worldwide (Sanyal and Dalui, 2020; Chen et al., 2022a, 2022b). These buildings are generally light-weighted and flexible, which are vulnerable to wind loads. On this account, estimation of wind-induced loads on tall buildings is imperative regarding its safety and serviceability.

Oftentimes, the aerodynamic behavior of tall buildings subjected to wind is investigated by evaluating forces and pressure coefficients, as well as obtaining a physical interpretation of the observed phenomenon (Carassale and Brunenghi, 2011, 2012). Multi-channel surface pressure measurements, in particular, have been frequently carried out for these activities since they can provide accurate characterization of both local and global wind-induced actions. Meanwhile, the analysis of the obtained wind pressure time series may also reveal valuable information on the qualitative behavior of the flow, detecting the signature of the vortexes that develop in the neighborhood of the building (Carassale, 2012; Carassale and Brunenghi, 2012).

As have been well documented, the wind pressures acting on building surfaces have a random nature, which tend to fluctuate spatiotemporally in a complicated manner (Kikuchi et al., 1997). This is due primarily to the presence of turbulence in the approaching flow together with signature (body-induced turbulence) (Qiu et al., 2014). For this reason, the wind pressures are often identified as realizations of multi-variate random process (Carassale, 2012), and the characteristics and information involved in random fluctuations of building surface wind pressures have usually been investigated via their probability and statistical parameters (Tamura et al., 1999). Note that qualitative analysis of wind pressures cannot be easily performed via common statistical methods (e.g., correlation and coherence functions). Alternatively, various modal representation techniques are more preferred. For instance, reduced-order models (ROMs) have been applied as a common diagnostic tool for complicated flow field (Beltrán et al., 2019). The core of ROMs is to find a simple low-dimension system to describe a chaotic, high-dimension system via a process of decomposition, truncation and error estimation (Zhou et al., 2021a, 2021c). From a practical standpoint, proper orthogonal decomposition (POD) and Independent component analysis (ICA) are two most common ROMs models used for extracting dominant patterns and eliminating noise. In the POD, the original data is projected into several spatial orthogonal modes based on optimal standard orthogonal bases, and the obtained modes are sorted based on the order of energy. Thus, the main features of the original data can be extracted using the dominant POD modes. In wind engineering community, POD analysis has been successfully applied by several scholars to diagnose the random wind pressure field on structures (Baker, 2000; Chen and Kareem, 2005; Carassale and Brunenghi, 2011). However, Zhou et al. (2021a) reported that, while POD is effective to extract the prevailing pressure pattern on building surfaces, high-order statistics and nonlinear transformation feature cannot be properly reflected by POD. Moreover, it is to be noted that the orthogonality of POD is not identified from a physical perspective. On this account, the ICA was proposed to determine the pressure patterns using non-orthogonal decomposition. While the ICA can identify the pressure pattern more clearly as compared to POD, it also has its limitation (Kim et al., 2021).

The identification of a nonlinear dynamical model that relates directly to the underlying dynamics of the system being modeled remains a challenging and active research topic. A problem frequently encountered in nonlinear time-series analysis is appropriate characterization of changes in the system dynamics (Zhou et al., 2021). More recently, the concept of nonlinear dynamic analysis has gained increasing attention across various disciplines. There exits several nonlinear dynamic analysis techniques, notably among which includes correlation dimension, Lyapunov exponent, phase space reconstruction, approximate entropy and recurrence analysis, etc. These techniques are powerful in terms of exploring the underlying dynamics of time-dependent variables. The authors have carried out a series of studies, where the usability of different nonlinear dynamic analysis techniques for wind engineering researches has been discussed (Shu et al., 2020, 2021a, 2021b, 2021c). In particular, Shu and Li (2022) used recurrence plot and recurrence quantification analysis to identify and describe the chaotic nature of wind pressure fluctuation underneath the separated shear layer, and compared their dynamic characteristics as a function of inflow turbulence and leading edge shapes.

The practical importance of understanding the dynamics of wind pressure fluctuation has been emphasized by several scholars. Zhou et al. (2021) once highlighted that the random fluctuation of wind pressures on building surfaces can be considered as a high-dimensional dynamic system, and proper assessment of the dominant features and dynamic characteristics can further advance the understanding of fluid-structural interaction, and therewith facilitate the control of wind-induced responses. Likewise, Barahona and Poon (1996) reported that proper identification of deterministic dynamics in time series can lead to new insights regarding the underlying physical processes, and enable better prediction. Therefore, the main goal of this study is to extend the concept of recurrence analysis for diagnosing the dynamics associated with wind pressure fluctuation on tall buildings, and thus facilitate the assessment of wind loads and the control of wind-induced responses.

Methodology

Experimental set-up

The natural wind in the atmosphere is known to be chaotic (Yan et al., 2020, 2021; Shu et al., 2020, 2021a, 2021b, 2021c), which often leads to random-like behavior in time domain. Hence, in order to properly diagnose the wind effects on buildings and structures, several different methods have been proposed, in which physical simulation in wind tunnel is one of the most popular and well used methods. In this study, a series of wind tunnel tests were carried out in the high-speed test section in the boundary layer wind tunnel (BLWT) at Changsha University of Science and Technology (see Figure 1), concerning the dynamic characteristics of surface wind pressure on the Commonwealth Advisory Aeronautical Research Council (CAARC) building. The CAARC building is the industry standard model for diagnosing the wind-induced effects on tall building. Particularly, the surface wind pressure characteristics on CAARC building have been examined in numerous studies by means of wind tunnel and numerical simulations. Hu et al. (2019) gave a detailed assessment on the surface pressures of the CAARC building fitted with a modified double-skin façade with vertical openings in the external skin mounted in front of the windward surface. Meng et al. (2018) examined the sensitivity of CAARC wind pressure on various geometric and computational parameters, e.g., turbulence model, inflow wind speed and grid type, using CFD simulation. Yang et al. (2022) focused on the effects of the turbulence integral scale on the non-Gaussian properties and extreme wind loads of surface wind pressure, in which two CAARC scaled models were measured in three turbulent flow fields with different turbulence integral scales. Chen et al. (2022c) used a 1:300 scaled CAARC building model to experimentally evaluate the effects of various façade appurtenances on surface wind pressures. Likewise, Li et al. (2020) concentrated on the effects of corner chamfers on the aerodynamic performance of CAARC building, in which five building models with corner modification rates ranging from 0% to 20% were tested for pressure measurement in wind tunnel. It is worth noting that, the CAARC building has been extensively adopted as the first-generation benchmark tall building. More recently, the concept of second-generation benchmark tall building was proposed, which was designed with an asymmetrical structural configuration and natural eccentricities. In comparison with the CAARC standard tall building, the second-generation benchmark tall building is able to generate more complex lateral-torsion vibration models (Zhou et al., 2021b, 2022; Tse et al., 2005).

Figure 1.

The test CAARC building model in BLWT.

The configuration of the test section is about 4.0 m (width) * 3.0 m (height)* 21.0 m (length). The attainable wind speed ranges from 1 m/s to 45 m/s. For the reproduction of a realistic boundary layer flow, spires and various-sized ground roughness elements were positioned in the upstream of the test model. The consequent wind profiles are shown in Figure 2, which characterizes a sub-urban area flow with a power-law exponent of 0.15. The length scale of the simulated boundary layer was 1:300.

Figure 2.

The approaching wind field simulated in this study. The red lines indicate the equations specified in Chinese design code (GB50009-2012, 2012).

On the other hand, the CAARC standard tall building was likewise scaled down at 1:300, as shown in Figure 3. To ensure its rigidity, the model was constructed using ABS (acrylonitrile-butadiene-styrene) plates. The overall blockage ratio was less than 2%, thus no further correction was applied. To better characterize the wind loads on tall building, the pressure field on building surfaces is usually examined since it can reflect the wind load characteristics of building when it is embedded in the turbulent boundary layer (Hu et al., 2017). In this study, multi-channel simultaneous pressure measurements were conducted for such purpose. The pressure measurement taps, with a total number of 260, were uniformly divided into 13 layers, with 20 taps per layer (see Figure 3). These taps were connected to the electronic scanning pressure measuring system (Scanivalve, Inc) via vinyl tubes, with 1.1 mm in diameter and 900 mm in length. To eliminate aliasing effects, a low-pass filter was installed in each data acquisition channel. Moreover, the tubing effects were numerically compensated using the gain and phase shift characteristics of the pressure measurement system (Chen, Wang, et al., 2021a, 2021b). The sampling rate was set as 350 Hz, and the sampling duration was 60 s. This is to ensure that the recorded wind pressure data is statistically stable.

Figure 3.

Definition of wind directions and location of pressure taps.

Data processing

To better describe the wind load characteristics, a non-dimensional time varying pressure coefficient was defined as:

C_{p} (t) = \frac{p (t) - p_{0}}{0.5 ρ {V_{H}}^{2}}

(1)

where

C_{p} (t)

is the instantaneous pressure coefficient pressure.

p (t)

is the time-varying wind pressure,

p_{0}

is the static pressure of the wind tunnel,

V_{H}

is the reference wind speed measured by Cobra probe at near-top height.

ρ

is the air mass density (taken as

ρ = 1.25 kg / m^{3}

). Both the reference wind speed and static pressure were measured at location that it much less affected by the existence of the building model. In the following analysis, the wind pressure characteristics are presented in two different forms:

(a) The mean pressure coefficient, $C_{p, m e a n}$

C_{p, m e a n} = \frac{\bar{p}}{\bar{p_{H}}}

(2)

(b) The root-mean-square (RMS) pressure coefficient, $C_{p, r m s}$

C_{p, r m s} = \frac{\sqrt{\bar{{(p (t) - \bar{p})}^{2}}}}{\bar{p_{H}}}

(3)

where

\bar{p}

is the time-averaged wind pressure,

p (t)

is the instantaneous pressure recorded at time t,

\bar{p_{H}}

is the reference dynamic wind pressure calculated at the rooftop height,

0.5 ρ {V_{H}}^{2}

(Meddage et al., 2022).

Nonlinear dynamic analysis

For predicting the time-dependent behavior of variables, it is important to model the underlying physical mechanism responsible of their generation. Note that most real-world dynamic system signals are nonlinear and nonstationary, which can lead to random-like behavior in time-domain (Coulibaly and Baldwin, 2005). In the context of chaos theory, such random behavior can arise in deterministic nonlinear system with a few degrees of freedom (Henry et al., 2001). Essentially, nonlinear dynamic analysis provides a universal tool for mathematically describing the deterministic part of the dynamic systems generating the time series. A flowchart of the nonlinear dynamic analysis in this study is shown in Figure 4.

Figure 4.

Flowchart of the nonlinear dynamic analysis in this study.

To characterize the dynamic of a given time series, the first step is to reconstruct the embedded phase space. Phase space is a mathematical space spanned by the dynamic variables of the system. The state of the dynamic system at any given time is represented by a point in this phase space. Hence, as the dynamic variables change in time, the representative point traces out a path in the phase space (Henry et al., 2001). Such path is usually referred to as the phase trajectory, which is indicative of how the dynamic system evolves from its initial state. The region of attraction (i.e., the area where these trajectories overlap in the phase space) provides qualitative information regarding the nature of the system dynamics, such as the degree of complexity or variability (Sivakumar et al., 2007).

From a practical point of view, the most popular method for reconstructing the phase space is the time delay method proposed by Takens (Takens, 1981), in which the dynamics of a single variable time series $x (t_{i}), i = 1,2 \dots N$ , can be reconstructed in a m-dimensional phase space as: (Lekscha and Donner, 2018)

x (t_{i}) \to Y (t_{i}) = [x (t_{i}), x (t_{i} - τ), \dots x (t_{i} + (m - 1) τ)],

(4)

where

Y (t_{i})

is the reconstructed state vectors,

τ

is the time delay, and m is the embedding dimension that reflects the minimum number of state variables needed to describe the system. Figure 5 illustrates an instance of the reconstruction of an attractor trajectory in the state space, i.e., phase space reconstruction.

Figure 5.

Examples of phase space reconstruction in the state space using time delay method (Ohtaki et al., 2005).

Recurrence analysis is a well-established method for investigating the deterministic dependence in time series, which has been widely applied in dynamic analysis. The recurrence of a state is a fundamental property of deterministic dynamic systems, which reflects a time that the trajectory in the phase space returns to a location where it has visited before (Marwan et al., 2002). The recurrences occur in a system’s phase space, and in order to measure the recurrence of a trajectory, the recurrence plot (RP) is frequently used. A RP consists of a two-dimensional binary diagram, which indicates how likely it is that the behavior observed in a m-dimensional phase space is recurrent. The time-dependent behavior of the dynamic system can be pictured as a trajectory $\vec{x_{i}} \in R^{m} (i = 1, . . N)$ in the m-dimensional phase space:

R_{i j} = Θ (ε_{i} - ‖ \vec{x_{i}} - \vec{x_{j}} ‖), i, j = 1, \dots, N

(5)

in which

R_{i j}

is the recurrence matrix, Θ(x) is the Heaviside function,

ε_{i}

is the predefined cut-off distance between two neighbouring points, ‖.‖ is a norm (e.g., the Euclidean norm), and

\vec{x_{i}}

and

\vec{x_{j}}

define the point in phase space at which the system is situated at times i and j, respectively. In this case, a recurrence of a state at time i at a different time j is represented by a two-dimensional squared matrix with ones and zeros in which axes of the recurrence matrix are time axes (Marwan et al., 2002). The ones in the recurrence matrix,

R_{i j}

, is marked by a black dot in the position (i, j) in the graphic representation. It is to be noted that a RP often consists of different typical patterns, whose structure is reflective of the underlying dynamics of the time series (Shu et al., 2021a; Marwan, 2003; Mocenni et al., 2011). These patterns can be broadly divided into two categories: typology and textures. In specific, the typology describes the global appearance of the RP, e.g., points with relatively homogeneous distribution is usually linked to a stationary stochastic process, whereas long diagonal lines parallel to the Line of Identity (LOI) represent periodic behaviors. In addition, white bands in the RP are indicative of the non-stationarity and abrupt changes in the underling dynamics. By comparison, the texture mainly catches the local structures existed in the patterns in the RP. For instance, isolated single points denote that the state does not persist for a long time, while the vertical and horizontal lines represent that the state of the dynamic system changes in relatively slow manner in the time domain.

It was well noted that recurrence analysis is more preferrable when describing the complex patterns in system dynamics. Specifically, recurrence analysis are adequate for analyzing relatively small data set. It is also applicable for both linear or nonlinear data sets. More importantly, it is not stymied by nonstationary data drifts, and enables definitive identification of dynamical transitions between different states (Webber and Zbilut, 1996; Belaire-Franch et al., 2002). The structure in RP is visually appealing and allows the investigation of high-dimensional dynamics in a simpler manner (Yan et al., 2021). Given its broad applicability, recurrence analysis has been adopted across various fields, e.g., quantifying the temporal dynamics of fixation sequences that helps understanding both fine and global aspects of the temporal structure of eye movements (Anderson et al., 2013); diagnosing the deterministic dependence in international stock market (Bastos and Caiado, 2011) and facilitating the characterization of the molecular dynamics simulation (Manetti et al., 1999).

Note that the RP is simply a visualization tool, which can only provide qualitative information of the system dynamics. One of the limitations of RPs is that the interpretation of different structures in RPs can be sometimes subjective, which depends heavily on the experience of the observer. Furthermore, once the capacity of data exceeds certain limit, the RP can be more or less misleading since the visualization of RP can be affected by screen resolution (Mocenni et al., 2011; Webber and Marwan, 2015). To cope with these limitations, the recurrence quantification analysis (RQA) has been introduced, in which several indicators of complexity that quantify the small-scale structures in RPs have been proposed. These indicators are mainly determined based on the recurrence point density, as well as the diagonal and vertical line structures in RPs, which is useful for identifying bifurcation points, particularly for chaos-order transitions. Table 1 lists some of the common indicators in RQA. For example, the recurrence rate (RR) is the simplest measure in the RQA, which is a measure of the density of recurrence point in the RP. The determinism defines the ratio of recurrence points that form diagonal line structures to all recurrence points. The Lmax is the length of the longest diagonal line in the RP, and ENT reflects the complexity of the RP in respect to the diagonal lines. Given their definitions, it is understandable that the larger the magnitude of these indicators, the higher the likelihood of the recurrence of states, and thus the lower degree of complexity the system dynamics. All the nonlinear dynamic analysis involved in this study were performed using the CRP toolbox in MATLAB (Marwan, 2010).

Table 1.

Summary of common recurrence quantification indicators.

RQA indicators	Equation
Recurrence rate	$R R = \frac{1}{N^{2}} \sum_{i, i = 1}^{N} R_{i j}$ where N is the number of states
Maximal length of diagonal structures	$L_{MAX} = \max ({l_{i}; i = 1, . ., N})$
Determinism	$DET = \frac{\sum_{l = l_{\min}}^{N} l P (l)}{\sum_{i j}^{N} R_{i j}}$
Entropy	$E N T = - \sum_{l = l_{\min}}^{N} P (l) \ln P (l)$

From equation (5), it is obvious that the recurrence matrix is strongly tied with the threshold distance. Several different criteria have been applied in previous studies to determine a suitable threshold distance, such as the 10% of mean space diameter, 5%–6% of the maximum space diameter, 25% of standard deviation, and a fixed percentage of RR. Among these criteria, a fixed percentage of RR, with the value spanning from 1.5% to 15%, is most frequently adopted. Hence, in this study, the recurrence analysis was carried out with a fixed threshold RR of 10%, unless otherwise specified, following the suggestion of Marwan (2010).

Results and discussion

In wind engineering community, the wind-building interaction has long been a major concern, and the wind flow patterns around tall building, in particular, have been frequently investigated, as shown in Figure 6.

Figure 6.

Illustration of wind flow around a tall building (Moonen et al., 2012).

As the wind approaches the building, part of the flow is guided over the building top, and part is directed around the vertical edges. On the windward side, a flow downwash can be observed where the flow is deviated to the ground-level, forming a standing-vortex that subsequently wraps around the corners and joins the overall flow around the building. Note that the standing-vortex and corner streams are the regions where high wind speed are more likely to occur. Further upstream, there exits a stagnation region on the windward surface where relatively low wind speeds can be found. Downstream of the building, the wind flow patterns are usually complex and strongly transient.

Understandably, such complex wind flow patterns can lead to different wind load characteristics at different regions on the building surfaces. To illustrate, Figure 7 depicts the time series of wind pressure coefficients obtained at different pressure taps on the same level. Clearly, the fluctuation of WPC between these taps is somewhat different.

Figure 7.

Time series of wind pressure coefficients obtained at selected positions at level J.

To further diagnose and compare the variability of WPC at different regions on building surfaces, the time series of WPC were analyzed from a nonlinear dynamic perspective. As discussed hereinabove, the first step of dynamic analysis usually consists reconstructing the phase space, in which a suitable pair of time delay $τ$ and embedding dimension m is of paramount importance (Li et al., 2010). Specifically, a too-large time delay can lead to vectors whose components are much less correlated and there would be no (or little) dynamic correlation evidence. However, if the time delay is too small, the consequent phase space coordinates would not be independent, which may lead to the loss of information regarding the characteristics of the attractor structure (Camplani and Cannas, 2009; Dhanya and Kumar, 2010). For the time delay estimation (TDE), there exists several different methods (Bjorklund and Ljung, 2003). Traditionally, the most popular technique for TDE was based on generalized cross-correlation (GCC), in which the relative delay can be obtained by maximizing the cross-correlation between filtered versions of the received signals. However, it is noteworthy that the performance of GCC-based TDE techniques degrades dramatically when the noise in signal is high. From a fundamental perspective, GCC is a second-order-statistics measure of the dependence among two random variables. Therefore, it is most suitable Gaussian source signals (Wen and Wan, 2009). Comparatively, the mutual information technique is more ideal in nonlinear dynamic analysis (Strozzi et al., 2002), because it takes into account the nonlinear correlation. This is important for diagnosing the dynamics of time series with inherent nonlinearity. Figure 8(a) shows the results of mutual information function, where the position of the first local minimum is considered as an appropriate criterion for optimal time delay (Fraser and Swinney, 1986).

Figure 8.

Determination of (a) optimal time delay and (b) minimum embedding dimension for nonlinear dynamic analysis. The arrow indicates the first zero value of FNN.

On the other hand, the selection of embedding dimension for dynamic analysis is also important, not only for the phase space reconstruction but also for nonlinear prediction (Takens, 1981). According to the delay embedding theorem, when the embedding dimension is larger than the minimum embedding dimension, the original phase space of a system can be well embedded in the reconstructed phase space (Qing-Fang et al., 2007). However, the selection of the embedding dimension is not entirely objective. From a practical standpoint, when the embedding dimension is too small, the phase space cannot be fully unfolded, thus the reconstructive system cannot accurately reveal the true system (Chun-Hua, Xin-Bao, 2004; Han et al., 2015). By contrast, if a too-large embedding dimension is used, the process of phase space reconstruction can be much more time-consuming. It was also reported that a greater embedding dimension cannot significantly improve the results (Chen et al., 2014). The most popular method for determining the embedding dimension is the false nearest neighbor (FNN) method (Kennel et al.,1992). Figure 8(b) illustrates the results of FNN, in which the arrow indicates the first percentage of FNN at which it approaches zero. It is worth mentioning that, given the inherent variability of wind pressure data, the optimal time delay and embedding dimension may vary depending on the time series to be considered. For ease of comparison, a consistent time delay of 20 and an embedding dimension of six were used for all wind pressure time series in this study, unless otherwise specified.

Once the optimal time delay and embedding dimension were determined, the phase space of the wind pressure time series can be reconstructed following equation (4), and the results at selected measurement positions are shown in Figure 9. It is clear that the projections of two-dimensional attractors at different measurement positions are somewhat different, which reflects the different degree of complexity in terms of underlying dynamics in wind pressure. Overall, the trajectories are neither randomly scattered nor constrained in a well-defined region, rather exhibiting some signature of attractor-like shape. Such phase space trajectories confirm the existence of chaos in the hidden structure of wind pressure time series.

Figure 9.

Phase space reconstruction based on wind pressure time series at selected positions.

Figures 10 and 11 illustrate respectively the distribution of mean and rms wind pressure coefficients at different building surfaces when the test flow approaches from 0°. It is clear that, the Cpmean on the windward surface remains mostly positive. The largest Cpmean on the windward surface is found near the stagnation region. The magnitude of Cpmean gradually decreases towards the ground-level. Due to the occurrence of flow separation, the Cpmean on the side surfaces are mostly negative (i.e., suction pressure), in which larger suction pressures are found at the near-rear edges at about 2/3 building height, while relatively smaller suction pressures are found at the near-rear edges at ground-level. For the leeward surface, the Cpmean is found to vary between −0.42 and −0.28, and the smaller suction pressure occurs predominantly at the central region.

Figure 10.

Distribution of mean WPC at different surfaces of CAARC building at 0° wind direction.

Figure 11.

Distribution of rms WPC at different surfaces of CAARC building at 0° wind direction.

As for the Cprms, the magnitudes on the side surfaces are generally larger than those on the windward and leeward surfaces. The largest Cprms on the windward surface, with a magnitude of about 0.12, can be found at the stagnation region. By contrast, the smallest Cprms on the leeward surface also occurs at the 2/3 building height, and the largest Cprms are found at 1/3-1/2 building height on both vertical edges. On the side surfaces, the largest Cprms, with a magnitude of about 0.25, consistently occur at the near-front edge, which is partly due to the corner streams.

To describe the nonlinear dynamics of wind pressure on the CAARC tall building, the time series of Cp recorded at each measurement position was analyzed by means of recurrence analysis. To add statistical reliability of the analysis, recurrence plot and recurrence quantification analysis were performed on the time series segment with a fixed window-length of 1000, and a half-length sliding window technique was likewise applied. Figure 12 reveals the example of recurrence plots based on recorded time series at different measurement positions. As can be seen, the recurrence plots consists of different small-scale structures, which is indicative of different dynamic properties. Diagonal line structures can be consistently observed, which implies the existence of deterministic component in the time series. However, white bands are also common in these plots. This means that the underlying dynamics of time series is subjected to transition or abrupt change. Overall, the results of recurrence plots agree with those of phase space reconstruction, which clearly indicates the chaotic nature of wind pressure time series.

Figure 12.

Illustration of recurrence plots using wind pressure coefficients at different positions.

Furthermore, Figures 13 –15 show the distribution of various RQA indicators at each building surface. As can be seen, the distribution of these RQA indicators are overall consistent, which, to some extent, evidences the reliability of RQA. On the windward surface, the distribution of RQA indicators is somewhat similar to that of Cpmean, where relatively larger values are observed at stagnation region. Further downstream, the values of RQA indicators are found to decrease. This implies that the underlying dynamics of wind pressure fluctuation at the stagnation region is more well-organized than those of downstream, which can lead to less randomness in the time-dependent behavior. The dynamics associated with horseshoe vortex formed in the bottom of the windward surface is noted to be complex, which leads to much smaller values of RQA. On the side surfaces, the variation of RQA indicators follows a well-defined pattern, where the values are larger at the near-front edge, and decreases towards the rear. This is in good agreements with the results by Shu and Li (2022), who reported that within the separation bubble, the dynamics of wind pressure becomes less deterministic as the positions moves more and more downstream. This can be partly related to the evolution of vortex within the bubble. On the leeward surface, the distribution of RQA indicators appears to be opposite to that of windward surface, in which larger values are observed mostly at near-ground level, while lower values are observed at 2/3 building height. The larger RQAs are mostly lined to the cavity zone where flow fluctuation is attenuated. However, the elevated vortex on the leeward surface is shown to exhibit more complex behavior, reflected by the lower RQAs.

Figure 13.

Distribution of Lmax at different surfaces of CAARC building.

Figure 14.

Distribution of DET at different surfaces of CAARC building.

Figure 15.

Distribution of ENT at different surfaces of CAARC building.

Concluding remarks

Proper identification and characterization of the underlying dynamics of wind pressure on building surfaces provides a new avenue to diagnose the wind load characteristics, and therewith facilitate the assessment and control of wind-induced responses. In this study, advanced nonlinear dynamic analysis techniques, namely phase space reconstruction, recurrence plot and recurrence quantification analysis, were applied for such purpose. The main outcomes of this study are summarized as follows:

• The wind pressures acting on tall building surfaces exhibit distinct chaotic nature, which has been well identified via phase space reconstruction and recurrence plot.

• The chaotic properties of wind pressure are subject to region-to-region variability, which can be attributed to the flow patterns resulted from wind-building interaction.

• For windward surface, the wind pressure dynamics at the stagnation region is more deterministic than those of downstream (i.e., where standing vortex prevails).

• For the side surfaces, the distribution of RQA indicators follows a clear pattern, where the values are larger at the near-front edge, and decreases towards the rear. Such phenomenon is closely tied to the flow separation.

• The distribution of RQA indicators on the leeward surface is opposite to that of windward surface, in which larger values occur more often at near-ground level. Further upstream, the values are found to decrease.

The outcomes in this study are expected to provide implication for better understanding of the wind-induced effect on tall buildings. However, the wind pressure data analyzed in this study were measured at the standard test condition (i.e., 0 approaching wind, standard upstream terrain). For more comprehensive assessment of the wind pressure fluctuation on tall buildings, further studies with emphasis on the effects of inflow wind direction and upstream terrain conditions are also recommended.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the grants from National Natural Science Foundation of China (Project no.: 52208515).

ORCID iD

ZR Shu

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