Study on Phase Characteristics of Wind Pressure Fields around a Prism Using Complex Proper Orthogonal Decomposition

: Wind loads for the design of wind-resistant high-rise buildings are generally evaluated based on spectral modal analysis or time-history response analysis using wind pressure data obtained from wind tunnel experiments with rigid models. The characteristics of the ﬂuctuating wind pressures around vibrating buildings must be evaluated for relevant wind-resistant designs because the wind pressures around buildings are affected by their vibrations. One of the methods to investigate ﬂuctuating ﬁelds is complex proper orthogonal decomposition (CPOD), which can express complicated pressure ﬁelds, including advection phenomena, as coherent structures. This paper presents the phase characteristics of ﬂuctuating wind pressures around rigid and elastic models of a square-sectioned prism evaluated via CPOD analysis using the results of wind tunnel experiments. The evaluation procedure for the symmetricity of the ﬂuctuating wind pressure modes obtained via CPOD is presented. The similarity of ﬂuctuating wind pressure ﬁelds is evaluated as the congruency of the planes formed by the 1st-and 2nd-eigenmodes. With symmetricity and similarity, the ﬂuctuat-ing wind pressure ﬁelds are classiﬁed into three types: resonant and non-resonant states in smooth ﬂow, and in gradient ﬂow. The characteristics of the three types of wind pressure ﬁelds are shown, respectively, in the symmetric and anti-symmetric modes.


Introduction
For a rational design of wind-resistant high-rise buildings, it is crucial to correctly estimate the wind pressures around them. Accordingly, several researchers have estimated the wind pressures around high-rise buildings [1][2][3][4][5][6][7][8]. In general, the wind pressures are evaluated through wind tunnel experiments with rigid models. The characteristics of wind pressures must be studied to consider building vibrations because they can be affected by the vibration of buildings. Additionally, proper orthogonal decomposition (POD) has been frequently used as a method for detecting the coherent structures of fluctuating wind pressure fields when studying the wind pressures around buildings.
Tamura [9] validated POD by evaluating the fluctuating wind pressure fields around low-and high-rise building models using it. Li [10] also evaluated the eigenvalues and contribution ratio of POD for the wind pressures measured on the sloped roof surfaces of a low-rise building in a wind tunnel experiment to demonstrate the validity and usefulness of the method. Kim [11] conducted a wind tunnel experiment on two linked super-high-rise buildings and used POD analysis to evaluate the fluctuating wind pressure fields around the two building models. Jeong [12] used POD to analyse the fluctuating wind pressure fields on the roof surfaces of a low-rise building model in occurrences where the wind pressure measurement holes were both uniformly and non-uniformly placed. Cao [13] used POD to evaluate the peak pressure of a high-rise building at two localized regions with

Wind Tunnel Flow
Wind tunnel experiments were conducted in an Eiffel-type wind tunnel at the General Building Research Corporation of Japan. The wind tunnel is shown in Figure 1. This study used two types of wind tunnel flows: a smooth flow and a gradient flow. The smooth flow had a uniform vertical wind speed profile and little turbulence, and the gradient flow was in a turbulent boundary layer generated on the wind tunnel floor by a spire and roughness blocks. The power-law exponent of the gradient flow, α, for the mean wind-speed profile was approximately 0.20. The profiles of the mean wind speed ratio Uz/U600 and turbulence intensity Iu above the centre of the turntable in the wind tunnel are shown in Figure 2, where Uz is the wind speed at a height of z mm. Figure 3a shows the dimensionless power spectral density (PSD), fSu(f)/σ 2 , of the gradient flow at the reference height H (z = 375 mm), where f is the frequency, Su(f) is the power spectrum of the fluctuating wind speed, and σ 2 is the variance of the wind speed. The general shape of the PSD is consistent with the Karman-type spectrum with Lz = 0.41 m, which corresponds to 164 m when multiplied by the scale factor of the model (=1/400, as described later in this section). Figure 3b shows a comparison of the turbulence scale (approximately 224 m) between the wind tunnel gradient flow and the value in the AIJ Recommendations for Loads on Buildings [37] (hereafter referred to as the AIJ-RLB). Comparing the value of Lz with the recommended value in the AIJ-RLB for suburban exposure (Terrain Category III), Lz seems to be smaller but is within the range of variability of the observations.

Experiments Using a Rigid Model
The target of this study was a high-rise building that was 30 m wide × 30 m deep × 150 m high. The model scale was set at 1/400, and the experimental rigid model was a square-sectioned prism 75 mm wide × 75 mm deep × 375 mm high. An overview of the rigid model is presented in Figure 4. The arrangement of the wind-pressure measurement points is shown in Figure 5. In total, 120 wind pressure measurement points were established with 30 points on each side of the model. A brass tube (40 mm in length, 0.8 mm in inner diameter) and a vinyl tube (1000 mm in length, 1.0 mm in inner diameter) were attached to the wind pressure measurement hole (0.8 mm in diameter). The rigid model was made of acrylic material, and vinyl tubing was led from the inside to the bottom of the model.

Experiments Using a Rigid Model
The target of this study was a high-rise building that was 30 m wide × 30 m deep × 150 m high. The model scale was set at 1/400, and the experimental rigid model was a square-sectioned prism 75 mm wide × 75 mm deep × 375 mm high. An overview of the rigid model is presented in Figure 4. The arrangement of the wind-pressure measurement points is shown in Figure 5. In total, 120 wind pressure measurement points were established with 30 points on each side of the model. A brass tube (40 mm in length, 0.8 mm in inner diameter) and a vinyl tube (1000 mm in length, 1.0 mm in inner diameter) were attached to the wind pressure measurement hole (0.8 mm in diameter). The rigid model was made of acrylic material, and vinyl tubing was led from the inside to the bottom of the model.

Experiments Using a Rigid Model
The target of this study was a high-rise building that was 30 m wide × 30 m deep × 150 m high. The model scale was set at 1/400, and the experimental rigid model was a square-sectioned prism 75 mm wide × 75 mm deep × 375 mm high. An overview of the rigid model is presented in Figure 4. The arrangement of the wind-pressure measurement points is shown in Figure 5. In total, 120 wind pressure measurement points were established with 30 points on each side of the model. A brass tube (40 mm in length, 0.8 mm in inner diameter) and a vinyl tube (1000 mm in length, 1.0 mm in inner diameter) were attached to the wind pressure measurement hole (0.8 mm in diameter). The rigid model was made of acrylic material, and vinyl tubing was led from the inside to the bottom of the model.    A schematic of the wind tunnel experiment using the rigid model is shown in Figure  6. The reference velocity pressure was measured with a pitot-static tube A installed on the windward side of the model. The wind pressure at each measurement point was led from a vinyl tube to the pressure gauge. The pressure on the model surface was measured as the differential pressure from the static pressure obtained by a pitot-static tube B installed at the top of the model. The amplitude and phase distortions of the wind pressure time series data owing to the vinyl tubing were corrected using an appropriate transfer function. The experimental wind speed comprised 15 wind speed steps from approximately 3 to 11 m/s at the reference height of the model, H = 375 mm. Thereafter, the wind speed was expressed as the reduced wind velocity Vr = UH/f0B, which is the experimental wind speed at the reference height, UH, divided by the natural frequency of the elastic model, f0 = 8.34 Hz, and the model width, B = 0.075 m. The experimental wind direction was the only direction in which the wind was normal to the front side of the model. The datasampling frequency for the measurements was 800 Hz. The number of measured data A schematic of the wind tunnel experiment using the rigid model is shown in Figure 6. The reference velocity pressure was measured with a pitot-static tube A installed on the windward side of the model. The wind pressure at each measurement point was led from a vinyl tube to the pressure gauge. The pressure on the model surface was measured as the differential pressure from the static pressure obtained by a pitot-static tube B installed at the top of the model. The amplitude and phase distortions of the wind pressure time series data owing to the vinyl tubing were corrected using an appropriate transfer function. The experimental wind speed comprised 15 wind speed steps from approximately 3 to 11 m/s at the reference height of the model, H = 375 mm. Thereafter, the wind speed was expressed as the reduced wind velocity V r = U H /f 0 B, which is the experimental wind speed at the reference height, U H , divided by the natural frequency of the elastic model, f 0 = 8.34 Hz, and the model width, B = 0.075 m. The experimental wind direction was the only direction in which the wind was normal to the front side of the model. The data-sampling frequency for the measurements was 800 Hz. The number of measured data points was 129,465, which corresponds to 90 min in real time, considering the experimental similarity rule for elastic models described later. The distributions of the mean wind pressure coefficients C pe and the fluctuating wind pressure coefficient C pe for smooth and gradient flows for V r = 9.7 are shown in Figures 7 and 8. The wind pressure coefficients C pe and C pe are defined as follows: where t is the time, P and P denote the mean and root mean square (RMS) of the wind pressure time series at a measurement point P(t), respectively, q H = ρU H 2 /2 is the velocity pressure, and ρ is the air density.
The distributions of C pe and C pe exhibited different trends depending on the experimental flow. The effect of V r on the wind pressure coefficient distribution is almost negligible.

Experiments Using an Elastic Model
An overview of the elastic model is shown in Figure 9. The geometry of the model and the wind pressure measurement point locations were similar to those of the rigid model. The elastic model was made of balsa wood, and vinyl tubes were fixed inside the model to prevent vinyl tubes from vibrating. A schematic of the wind tunnel experiments using the elastic model is shown in Figure 10. The elastic model was attached to the vibration test apparatus at the bottom of the turntable and elastically supported by springs. The displacements of the model were measured with laser displacement meters at a position 207.5 mm downward from the centre of rotation. The measured displacement data were low-pass filtered at 20 Hz to remove the noise components at high frequencies. The displacements were measured in two directions: along-wind and across-wind. The definitions of the axes are shown in Figure 11. smooth and gradient flows for Vr = 9.7 are shown in Figures 7 and 8. The wind pressure coefficients ̅ and are defined as follows: where t is the time, and denote the mean and root mean square (RMS) of the wind pressure time series at a measurement point ( ), respectively, = /2 is the velocity pressure, and is the air density. The distributions of ̅ and exhibited different trends depending on the experimental flow. The effect of Vr on the wind pressure coefficient distribution is almost negligible.
where t is the time, and denote the mean and root mean square (RMS) of the wind pressure time series at a measurement point ( ), respectively, = /2 is the velocity pressure, and is the air density. The distributions of ̅ and exhibited different trends depending on the experimental flow. The effect of Vr on the wind pressure coefficient distribution is almost negligible.

Experiments Using an Elastic Model
An overview of the elastic model is shown in Figure 9. The geometry of the model and the wind pressure measurement point locations were similar to those of the rigid model. The elastic model was made of balsa wood, and vinyl tubes were fixed inside the model to prevent vinyl tubes from vibrating. A schematic of the wind tunnel experiments tion test apparatus at the bottom of the turntable and elastically supported by spri displacements of the model were measured with laser displacement meters at a 207.5 mm downward from the centre of rotation. The measured displacement d low-pass filtered at 20 Hz to remove the noise components at high frequencies. placements were measured in two directions: along-wind and across-wind. Th tions of the axes are shown in Figure 11.   The target of this study was a high-rise RC building that was 30 m wide × 30 3 Figure 10. Schematic of wind tunnel experiments using the elastic model.  The target of this study was a high-rise RC building that was 30 m wide × 30 m deep × 150 m high, with a building density of 300 kg/m 3 and a natural frequency of 0.25 Hz. The damping ratios h were 1% and 2%, respectively. In this experiment, the two-degree-of-  The target of this study was a high-rise RC building that was 30 m wide × 30 m deep × 150 m high, with a building density of 300 kg/m 3 and a natural frequency of 0.25 Hz. The damping ratios h were 1% and 2%, respectively. In this experiment, the two-degree-offreedom rocking vibration was reproduced by matching the reduced wind velocity, mass ratio, and damping ratio between the target building and experimental model according to the similarity rule of the vibration experiment. Table 1 presents the correspondence between the structural properties of the target building and the experimental model. The damping ratios and natural frequencies were evaluated based on the results of the free vibration experiments under no-wind conditions. Free vibration experiments were conducted before and after each experiment in each case. The damping ratios shown in Table 1 were calculated using the following equations by using the average value of the ratio d, which is the amplitude of the free vibration waveforms obtained from the free vibration experiments for each adjacent period for approximately 4 s.
(3) The relationship between the damping ratios h 1 , calculated by Equation (3) using the ratio d of amplitudes per period, and the dimensionless displacements δ/H is shown in  Here δ is the model top displacement. In each experiment case, there is almost no amplitude dependency of the damping ratios, and generally stable values are obtained.       The distributions of C pe and C pe for the elastic model in the smooth flow for h = 1%, h = 2%, V r = 8.8, V r = 9.7 are shown in Figures 16-19. For V r = 9.7, C pe at the upper part of both side surfaces was smaller than that in the case of V r = 8.8. C pe shows large values in the leeward region of the model side surfaces compared to those in the case of V r = 8.8 regardless of damping ratio. The different distribution compared to those for the rigid model could be attributed to the large vibration of the elastic model caused by the resonance phenomenon because the reduced wind velocity V r = 9.7 is near the resonance wind speed. The same pattern was observed for the case of h = 2% in the smooth flow. However, the change in wind pressure distribution was smaller than that for the case of h = 1%. Furthermore, the wind pressure distribution in the two cases of V r = 9.7 and V r = 10.7 was similar to each other for h = 1%, and whereas those for other V r were almost the same as those at V r = 8.8. For h = 2%, the wind pressure distribution is almost the same for all other wind speed ranges except for V r = 9.7. The distributions of C pe and C pe for h = 1%, h = 2%, V r = 9.7 in the gradient flow are shown in Figures 20 and 21. In the gradient flow, the wind pressure distributions are almost the same regardless of h or V r . nance phenomenon because the reduced wind velocity Vr = 9.7 is near the resonance wind speed. The same pattern was observed for the case of h = 2% in the smooth flow. However, the change in wind pressure distribution was smaller than that for the case of h = 1%. Furthermore, the wind pressure distribution in the two cases of Vr = 9.7 and Vr = 10.7 was similar to each other for h = 1%, and whereas those for other Vr were almost the same as those at Vr = 8.8. For h = 2%, the wind pressure distribution is almost the same for all other wind speed ranges except for Vr = 9.7. The distributions of ̅ and for h = 1%, h = 2%, Vr = 9.7 in the gradient flow are shown in Figures      The generalized wind force F(t) was calculated using P(z) obtained by linear interpolation of the wind force (P1-P6) between each layer, assuming a linear mode μ(z), as shown in Figure 22, using the following equation: The generalized wind force F(t) was calculated using P(z) obtained by linear interpolation of the wind force (P1-P6) between each layer, assuming a linear mode μ(z), as shown in Figure 22, using the following equation: The generalized wind force F(t) was calculated using P(z) obtained by linear interpolation of the wind force (P1-P6) between each layer, assuming a linear mode µ(z), as shown in Figure 22, using the following equation: The generalized wind force F(t) was calculated using P(z) obtaine lation of the wind force (P1-P6) between each layer, assuming a li shown in Figure 22, using the following equation: The time-history response analysis was performed by solving the of motion, using as input the generalized wind force time series F(t) wind pressure data for the rigid and elastic models. Newmark's beta celeration method) was used for numerical integration. The time-history response analysis was performed by solving the following equation of motion, using as input the generalized wind force time series F(t) calculated from the wind pressure data for the rigid and elastic models. Newmark's beta method (linear acceleration method) was used for numerical integration.
where m is the generalized mass of 0.216 kg, and k is the generalized stiffness of 592 N/m. The damping coefficient, c kg/s, was obtained from the damping ratio h for each case presented in Table 1 using the following equation: The number of data points for the generalized wind force was set to 143,850, and the first 14,385 data points corresponding to the first 10 min of real time were envelopeprocessed. In Figures 23-30, the following three displacements are shown for comparison.
δ me : displacement of the elastic model by measurement. δ ae : displacement by time-history response analysis using the wind force on the elastic model. δme/H and δae/H are larger than those of δar/H in across-wind direction around Vr = 9.7 to 10.7 at h = 1% and Vr = 9.7 at h = 2%. This is attributed to the effect of unsteady wind forces generated by the vibration of the elastic model. δme/H was larger than δar/H and δae/H in the same wind speed range along the wind direction. This is because when the model is oscillating significantly across-wind, it oscillates in an elliptical orbit in the XY plane, which also affects the along-wind vibration displacement.  10.7 at h = 1% and Vr = 9.7 at h = 2%. This is attributed to the effect of unsteady wind forces generated by the vibration of the elastic model. δme/H was larger than δar/H and δae/H in the same wind speed range along the wind direction. This is because when the model is oscillating significantly across-wind, it oscillates in an elliptical orbit in the XY plane, which also affects the along-wind vibration displacement.    The vibration-displacement ratios of the along-and across-wind directions in the gradient flow are shown in  There was no significant difference in the alongwind vibration displacement among δme/H, δae/H, and δar/H for the experimental results for the gradient flow. Meanwhile, there was a difference in the RMS and maximum values of the across-wind vibration displacement for δar/H, δme/H, and δae/H, although the difference was smaller than that in the case of smooth flow.  The vibration-displacement ratios of the along-and across-wind directions in the gradient flow are shown in Figures 27-30. There was no significant difference in the alongwind vibration displacement among δme/H, δae/H, and δar/H for the experimental results for the gradient flow. Meanwhile, there was a difference in the RMS and maximum values of the across-wind vibration displacement for δar/H, δme/H, and δae/H, although the difference was smaller than that in the case of smooth flow.

Evaluation Method
The CPOD analysis was performed on the wind pressure data obtained from wind tunnel experiments using rigid and elastic models by solving the following eigenvalue problem:

Evaluation Method
The CPOD analysis was performed on the wind pressure data obtained from wind tunnel experiments using rigid and elastic models by solving the following eigenvalue problem: to 10.7 at h = 1% and V r = 9.7 at h = 2%. This is attributed to the effect of unsteady wind forces generated by the vibration of the elastic model. δ me /H was larger than δ ar /H and δ ae /H in the same wind speed range along the wind direction. This is because when the model is oscillating significantly across-wind, it oscillates in an elliptical orbit in the XY plane, which also affects the along-wind vibration displacement.
The vibration-displacement ratios of the along-and across-wind directions in the gradient flow are shown in Figures 27-30. There was no significant difference in the along-wind vibration displacement among δ me /H, δ ae /H, and δ ar /H for the experimental results for the gradient flow. Meanwhile, there was a difference in the RMS and maximum values of the across-wind vibration displacement for δ ar /H, δ me /H, and δ ae /H, although the difference was smaller than that in the case of smooth flow.

Evaluation Method
The CPOD analysis was performed on the wind pressure data obtained from wind tunnel experiments using rigid and elastic models by solving the following eigenvalue problem: where Φ is a mode matrix with the j th eigenvector as j-column; Λ is an eigenvalue matrix with the j th eigenvalues λ j as j-row and j-column diagonal elements; and A is a diagonal matrix with a j-row and j-column element, A j , which is the ratio of the burden area at point j to the total side area. R is a complex covariance matrix, and assuming that the wind pressure time series at point j, p j (t), is a stationary process with period T, the j-row and k-column elements R jk are obtained as follows: p j (t) = ∑ ω n ≥0 2P j (ω n )·e iω n t .
where i denotes an imaginary unit, * is the complex conjugate, and P j (ω n ) is the Fourier coefficient of the wind pressure at point j, p j (t), with respect to frequency ω n . In this method, the covariance of the fluctuating wind pressure is obtained using the analytical signal p j (t) shown in Equation (11), and the eigenmodes and eigenvalues are obtained using a complex eigenvalue analysis for R. Mode φ j shall have orthogonality in the broad sense using A, and matrix Φ is normalized as follows: where, † is the Hermitian conjugate, and I is the unit matrix.

Contribution Ratio
From the eigenvalues obtained by the CPOD analysis, the k th contribution ratio C k was obtained using Equation (13).
where N is the maximum mode order. The contribution ratios of the 1st-to 5th-mode, C 1 to C 5 , in smooth and gradient flows for the rigid model are shown in Figure 31. C 1 and C 2 are remarkably higher than those of the other modes for both smooth and gradient flows. C k for the elastic model is shown in Figures 32 and 33. In the smooth flow of V r = 9.7 and V r = 10.7, C 1 is remarkably large for h = 1%. In the case of h = 2%, C 1 is much larger than the others when V r = 9.7 in the smooth flow. However, the contribution ratios, C k , in the gradient flow are almost the same regardless of the damping ratio, h, reduced velocity, V r , and whether the model is rigid or elastic.
where N is the maximum mode order. The contribution ratios of the 1st-to 5th-mode, C1 to C5, in smooth and gradient flows for the rigid model are shown in Figure 31. C1 and C2 are remarkably higher than those of the other modes for both smooth and gradient flows. Ck for the elastic model is shown in Figures 32 and 33. In the smooth flow of Vr = 9.7 and Vr = 10.7, C1 is remarkably large for h = 1%. In the case of h = 2%, C1 is much larger than the others when Vr = 9.7 in the smooth flow. However, the contribution ratios, Ck, in the gradient flow are almost the same regardless of the damping ratio, h, reduced velocity, Vr, and whether the model is rigid or elastic.

Eigenmodes
The eigenmodes obtained by the CPOD analysis are complex and have real and imaginary parts. The absolute value and phase of the eigenmodes, and , are = ∑ . (13) where N is the maximum mode order. The contribution ratios of the 1st-to 5th-mode, C1 to C5, in smooth and gradient flows for the rigid model are shown in Figure 31. C1 and C2 are remarkably higher than those of the other modes for both smooth and gradient flows. Ck for the elastic model is shown in Figures 32 and 33. In the smooth flow of Vr = 9.7 and Vr = 10.7, C1 is remarkably large for h = 1%. In the case of h = 2%, C1 is much larger than the others when Vr = 9.7 in the smooth flow. However, the contribution ratios, Ck, in the gradient flow are almost the same regardless of the damping ratio, h, reduced velocity, Vr, and whether the model is rigid or elastic.

Eigenmodes
The eigenmodes obtained by the CPOD analysis are complex and have real and imaginary parts. The absolute value and phase of the eigenmodes, and , are where N is the maximum mode order. The contribution ratios of the 1st-to 5th-mode, C1 to C5, in smooth and gradient flows for the rigid model are shown in Figure 31. C1 and C2 are remarkably higher than those of the other modes for both smooth and gradient flows. Ck for the elastic model is shown in Figures 32 and 33. In the smooth flow of Vr = 9.7 and Vr = 10.7, C1 is remarkably large for h = 1%. In the case of h = 2%, C1 is much larger than the others when Vr = 9.7 in the smooth flow. However, the contribution ratios, Ck, in the gradient flow are almost the same regardless of the damping ratio, h, reduced velocity, Vr, and whether the model is rigid or elastic.

Eigenmodes
The eigenmodes obtained by the CPOD analysis are complex and have real and imaginary parts. The absolute value and phase of the eigenmodes, and , are

Eigenmodes
The eigenmodes obtained by the CPOD analysis are complex and have real and imaginary parts. The absolute value and phase of the eigenmodes, φ kj and arg φ kj , are important for considering the characteristics of phenomena such as fluctuating wind pressure fields around buildings. φ kj represents the j-row and k-column elements of Φ and the element at point k of the j th eigenmode. The 1st-and 2nd-eigenmodes for the rigid model and the elastic model in the smooth flow of reduced velocity V r = 9.7 are shown in Figures 34 and 35, respectively. Regarding the phases on the right and left sides of the rigid model, the 1st-mode is the symmetric mode, whereas the 2nd-mode is the anti-symmetric mode. Meanwhile, the eigenmodes of the elastic model shown in Figure 35 are opposite to those of the rigid model, with the 1st-mode being the anti-symmetric mode and the 2nd-mode being the symmetric mode. The eigenmodes for the rigid model and the elastic model in the gradient flow of reduced velocity V r = 9.6 are shown in Figures 36 and 37, respectively. The 1st-mode is the symmetric mode and the 2nd-mode is the anti-symmetric mode, with no significant difference in absolute value and phase for both the rigid and elastic models. However, the absolute value distributions and phase characteristics are different from those in the smooth flow.
important for considering the characteristics of phenomena such as fluctuating wind pressure fields around buildings. represents the j-row and k-column elements of and the element at point k of the j-th eigenmode. The 1st-and 2nd-eigenmodes for the rigid model and the elastic model in the smooth flow of reduced velocity Vr = 9.7 are shown in Figures 34 and 35, respectively. Regarding the phases on the right and left sides of the rigid model, the 1st-mode is the symmetric mode, whereas the 2nd-mode is the anti-symmetric mode. Meanwhile, the eigenmodes of the elastic model shown in Figure 35 are opposite to those of the rigid model, with the 1st-mode being the anti-symmetric mode and the 2nd-mode being the symmetric mode. The eigenmodes for the rigid model and the elastic model in the gradient flow of reduced velocity Vr = 9.6 are shown in Figures 36  and 37, respectively. The 1st-mode is the symmetric mode and the 2nd-mode is the antisymmetric mode, with no significant difference in absolute value and phase for both the rigid and elastic models. However, the absolute value distributions and phase characteristics are different from those in the smooth flow. sure fields around buildings. represents the j-row and k-column elements of and the element at point k of the j-th eigenmode. The 1st-and 2nd-eigenmodes for the rigid model and the elastic model in the smooth flow of reduced velocity Vr = 9.7 are shown in Figures 34 and 35, respectively. Regarding the phases on the right and left sides of the rigid model, the 1st-mode is the symmetric mode, whereas the 2nd-mode is the anti-symmetric mode. Meanwhile, the eigenmodes of the elastic model shown in Figure 35 are opposite to those of the rigid model, with the 1st-mode being the anti-symmetric mode and the 2nd-mode being the symmetric mode. The eigenmodes for the rigid model and the elastic model in the gradient flow of reduced velocity Vr = 9.6 are shown in Figures 36  and 37, respectively. The 1st-mode is the symmetric mode and the 2nd-mode is the antisymmetric mode, with no significant difference in absolute value and phase for both the rigid and elastic models. However, the absolute value distributions and phase characteristics are different from those in the smooth flow.  (c) (d)

Symmetricity of Fluctuating Wind Pressure Mode
This section proposes a method for quantitatively evaluating the symmetricity of the fluctuating wind pressure modes of CPOD.

Symmetricity of Fluctuating Wind Pressure Mode
This section proposes a method for quantitatively evaluating the symmetricity of the fluctuating wind pressure modes of CPOD.

Inner Product
The vectors a and b are assumed in Equations (14) and (15). a = a 1 , · · · , a j , · · · , a N T , The inner product a·b of vectors a and b is defined as follows: where A j is the barden area of point j.
The magnitudes of a, |a|, whose elements are complex, are defined as follows:

Element Exchange Vector
The element exchange vector of a is defined as Equation (18): a e = a 1 , · · · , a j , · · · , a N T (18) where j' denotes the number of points symmetrical to point j with respect to the central section (X = 0 in Figure 11). Because the pressure measuring points are located symmetrically with respect to the central section, the weights A j and A j are equal, A j = A j . In the case where point j is on the centerline of the front side, j = j. According to Equations (14)- (18), the inner product a·a e and |a e | can be expressed as follows: Because the complex conjugate of a * j A j a j a * j A j a j , are always included in Equation (19), a·a e is always a real number. Considering Equations (17) and (20), all of a * j A j a j are included in Equation (17) and vice versa, so the following equation holds:

Symmetry Index
The symmetry index of vector a, I s , is defined in Equation (22): Wind 2023, 3

54
Here, it is an example of a symmetry index I s using eigenmodes consisting of three elements with the same burden area. If points 1 and 3 are symmetrically located and 2 = 2, the element exchanged vector of the vector a = {a 1 , a 2 , a 3 } T is the vector a e = {a 1 , a 2 , a 3 } T = {a 3 , a 2 , a 1 } T . When a 1 = a 3 , the symmetry index I s (a) = 1, and when a 1 = −a 3 and a 2 = 0, I s (a) = −1. Thus, the symmetry index I s (a) expresses the symmetry of vector a.

Evaluation Results
The relationship between I s and V r for a rigid model in the smooth flow is shown in Figure 38. I s for the rigid model varies significantly between −1 and 1 based on the changes in V r . Meanwhile, the symmetry indices for the elastic model for h = 1%, shown in Figure 39, show a clear pattern where I s (φ 1 ) = −1 and I s (φ 2 ) = 1 in the resonant region of V r = 9.7 or 10.7. Therefore, the 1st-mode is anti-symmetric, the 2nd-mode is symmetric, and the anti-symmetric mode becomes dominant, considering the contribution ratio in Figure 32. In the case of h = 2% in Figure 40, this pattern is limited to the reduced velocity V r = 9.7. However, the opposite trend is observed in the other velocity regions.
Here, it is an example of a symmetry index using eigenmodes consisting of three elements with the same burden area. If points 1 and 3 are symmetrically located and 2′ = 2, the element exchanged vector of the vector = { , , } is the vector = { , , } = { , , } . When = , the symmetry index ( ) = 1, and when = and = 0, ( ) = 1. Thus, the symmetry index ( ) expresses the symmetry of vector .

Evaluation Results
The relationship between and for a rigid model in the smooth flow is shown in Figure 38. for the rigid model varies significantly between −1 and 1 based on the changes in . Meanwhile, the symmetry indices for the elastic model for h = 1%, shown in Figure 39, show a clear pattern where ( ) = 1 and ( ) = 1 in the resonant region of = 9.7 or 10.7. Therefore, the 1st-mode is anti-symmetric, the 2nd-mode is symmetric, and the anti-symmetric mode becomes dominant, considering the contribution ratio in Figure 32. In the case of h = 2% in Figure 40, this pattern is limited to the reduced velocity = 9.7. However, the opposite trend is observed in the other velocity regions. Figure 41 shows the relationship between and for the rigid model in the gradient flow. The symmetry indexes of the 1st-mode are relatively large, despite the pattern being less explicit compared to the cases of smooth flow. The anti-symmetricities of the 2nd-mode are weaker than those in the case of smooth flow. The trends in the case of the elastic model shown in Figures 42 and 43 are almost identical to those in the case of the rigid model.  Here, it is an example of a symmetry index using eigenmodes consisting of three elements with the same burden area. If points 1 and 3 are symmetrically located and 2′ = 2, the element exchanged vector of the vector = { , , } is the vector = { , , } = { , , } . When = , the symmetry index ( ) = 1, and when = and = 0, ( ) = 1. Thus, the symmetry index ( ) expresses the symmetry of vector .

Evaluation Results
The relationship between and for a rigid model in the smooth flow is shown in Figure 38. for the rigid model varies significantly between −1 and 1 based on the changes in . Meanwhile, the symmetry indices for the elastic model for h = 1%, shown in Figure 39, show a clear pattern where ( ) = 1 and ( ) = 1 in the resonant region of = 9.7 or 10.7. Therefore, the 1st-mode is anti-symmetric, the 2nd-mode is symmetric, and the anti-symmetric mode becomes dominant, considering the contribution ratio in Figure 32. In the case of h = 2% in Figure 40, this pattern is limited to the reduced velocity = 9.7. However, the opposite trend is observed in the other velocity regions. Figure 41 shows the relationship between and for the rigid model in the gradient flow. The symmetry indexes of the 1st-mode are relatively large, despite the pattern being less explicit compared to the cases of smooth flow. The anti-symmetricities of the 2nd-mode are weaker than those in the case of smooth flow. The trends in the case of the elastic model shown in Figures 42 and 43 are almost identical to those in the case of the rigid model.

Similarity Rate of Fluctuating Wind Pressure Fields
As mentioned in the previous section, the symmetry index of the 1st-and 2nd-modes of the fluctuating wind pressure varied with the experimental parameters. This section proposes a method to evaluate the similarity of fluctuating wind pressure fields as the projection ratio of the planes formed by the 1st-and 2nd-eigenmodes, and , to that

Similarity Rate of Fluctuating Wind Pressure Fields
As mentioned in the previous section, the symmetry index of the 1st-and 2nd-modes of the fluctuating wind pressure varied with the experimental parameters. This section proposes a method to evaluate the similarity of fluctuating wind pressure fields as the projection ratio of the planes formed by the 1st-and 2nd-eigenmodes, and , to that

Similarity Rate of Fluctuating Wind Pressure Fields
As mentioned in the previous section, the symmetry index of the 1st-and 2nd-modes of the fluctuating wind pressure varied with the experimental parameters. This section proposes a method to evaluate the similarity of fluctuating wind pressure fields as the projection ratio of the planes formed by the 1st-and 2nd-eigenmodes, and , to that

Similarity Rate of Fluctuating Wind Pressure Fields
As mentioned in the previous section, the symmetry index of the 1st-and 2nd-modes of the fluctuating wind pressure varied with the experimental parameters. This section proposes a method to evaluate the similarity of fluctuating wind pressure fields as the projection ratio of the planes formed by the 1st-and 2nd-eigenmodes, a 1 and a 2 , to that of the other two eigenmodes, e 1 and e 2 . Where vectors a 1 and a 2 , e 1 and e 2 are normalized, the projection of mode a 1 onto the plane formed by modes e 1 and e 2 , is expressed as proj(a 1 ) = (e 1 ·a 1 )e 1 + (e 2 ·a 1 )e 2 . (23) Wind 2023, 3

56
The projection of mode a 2 onto that plane, proj(a 2 ), can also be obtained, and the following relationship is obtained: proj(a 1 ) proj(a 2 ) = e 1 ·a 1 e 2 ·a 1 e 1 ·a 2 e 2 ·a 2 a 1 a 2 .
The 2 × 2 matrix on the right side of Equation (24) is considered the Jacobi matrix, whose determinant represents the projection ratio between the plane formed by a 1 and a 2 and that formed by e 1 and e 2 . Subsequently, the Jacobian, which represents the magnification rate of the transformation, can be expressed by the following equation: det (e 1 ·a 1 ) (e 2 ·a 1 ) (e 1 ·a 2 ) (e 2 ·a 2 ) = |(e 1 ·a 1 )(e 2 ·a 2 ) − (e 2 ·a 1 )(e 1 ·a 2 )|.
The projection rate expresses the similarity of the fluctuating pressure fields consisting of the 1st-and 2nd-eigenmodes. The results of the quantitative evaluation of the similarity between the fluctuating wind pressure modes are shown in Figure 44. The principal axes of the 1st-and 2nd-eigenvectors, e 1 and e 2 , are the eigenmodes of the measurement cases of (a) and (b) shown in Figure 44. In the smooth flow for h = 1% and h = 2%, the congruency of the plane formed by the two principal eigenmodes by CPOD was different in the resonant wind speed range, indicating different fluctuating wind pressure fields compared to those in other cases. Notably, the damping ratios may affect the fluctuating wind pressure fields by altering the vibration amplitude. Furthermore, an insignificant difference was noted in the characteristics of fluctuating wind pressure fields in the gradient flow. The previous section showed that the symmetricity of fluctuating wind pressure modes tends to vary with the measurement conditions and wind speed. However, by evaluating the similarities of the pressure fields consisting of the 1st-and 2nd-eigenvectors, it appears that the fluctuating wind pressure fields can be classified into the three categories presented in Table 2.
following relationship is obtained: The 2 2 matrix on the right side of Equation (24) is considered the Jacobi matrix, whose determinant represents the projection ratio between the plane formed by and and that formed by and . Subsequently, the Jacobian, which represents the magnification rate of the transformation, can be expressed by the following equation: The projection rate expresses the similarity of the fluctuating pressure fields consisting of the 1st-and 2nd-eigenmodes. The results of the quantitative evaluation of the similarity between the fluctuating wind pressure modes are shown in Figure 44. The principal axes of the 1st-and 2nd-eigenvectors, and , are the eigenmodes of the measurement cases of (a) and (b) shown in Figure 44. In the smooth flow for h = 1% and h = 2%, the congruency of the plane formed by the two principal eigenmodes by CPOD was different in the resonant wind speed range, indicating different fluctuating wind pressure fields compared to those in other cases. Notably, the damping ratios may affect the fluctuating wind pressure fields by altering the vibration amplitude. Furthermore, an insignificant difference was noted in the characteristics of fluctuating wind pressure fields in the gradient flow. The previous section showed that the symmetricity of fluctuating wind pressure modes tends to vary with the measurement conditions and wind speed. However, by evaluating the similarities of the pressure fields consisting of the 1st-and 2nd-eigenvectors, it appears that the fluctuating wind pressure fields can be classified into the three categories presented in Table 2.

Recomposition of Fluctuating Wind Pressure Fields
In this section, the characteristics of the fluctuating wind pressure fields for the three types in Table 2 are discussed using the recomposition of the principal modes.

Principal Coordinate
A wind pressure vector p = { p 1 (t), · · · , p N (t)} T with the analytical signal, p j (t), of the wind pressure time series at measurement point j, p j (t), as the j th -row element is expressed as the product of CPOD mode matrix Φ and the principal coordinate vector a = { a 1 (t), · · · , a N (t)} T as follows: Assuming the normality of eigenmodes in Equation (12), the following relationship is obtained from Equation (26): Therefore, the j th -principal coordinate, a j (t), is: The PSDs of the principal coordinates are shown in Figure 45. The power spectrum s(f ) is nondimensionalized by the model width B, mean wind speed at the reference height U H , and variance of the principal coordinates σ 2 .

Recomposition of Fluctuating Wind Pressure Fields
In this section, the characteristics of the fluctuating wind pressure fields for the three types in Table 2 are discussed using the recomposition of the principal modes.

Principal Coordinate
A wind pressure vector = { ( ), ⋯ , ( )} with the analytical signal, ( ), of the wind pressure time series at measurement point j, ( ), as the j th -row element is expressed as the product of CPOD mode matrix Φ and the principal coordinate vector = { ( ), ⋯ , ( )} as follows: Assuming the normality of eigenmodes in Equation (12), the following relationship is obtained from Equation (26): Therefore, the j th -principal coordinate, ( ), is: The PSDs of the principal coordinates are shown in Figure 45. The power spectrum s(f) is nondimensionalized by the model width B, mean wind speed at the reference height UH, and variance of the principal coordinates σ 2 . The PSDs have relatively high power in the low-frequency region for all types in the symmetric mode. The PSD of Type 1 has a sharp peak around fB/UH = 0.1 because the symmetry index of the symmetry mode of Type 1 has a relatively small value of 0.88.  The PSDs have relatively high power in the low-frequency region for all types in the symmetric mode. The PSD of Type 1 has a sharp peak around fB/U H = 0.1 because the symmetry index of the symmetry mode of Type 1 has a relatively small value of 0.88.
In the case of the anti-symmetric mode, the PSDs of Type 1 and Type 2 have a sharp peak around fB/U H = 0.1 because of the Kalman vortex shedding. In the case of Type 2, the natural frequencies of the elastic model and vortex shedding frequencies are similar; therefore, the resonance phenomenon affects the wind pressure fields. In the case of Type 3, there is a gradual peak at a frequency of approximately 0.1. The peak is not as sharp as that of Type 1 and Type 2 because the vortices are not cleanly generated owing to the high turbulence intensity of the flow.

Correlation of Wind Pressure Field
The correlation coefficient of the eigenmodes is expressed by the following equation: where Φ sym,Typej and Φ anti,Typej are the eigenvectors of the symmetric and anti-symmetric modes in the case of Type j, respectively. The correlation coefficients between Types j and k, r sym , jk , r anti , jk , are listed in Table 3. In the case of symmetric modes, the fluctuating wind pressure field in the smooth flow without resonant phenomena, Type 1, is relatively high correlated to that of Types 2 and 3. In the case of anti-symmetric modes, the fluctuating wind pressure fields in the case of smooth flow tend to be close, regardless of resonant phenomena, although the wind pressure field in the gradient flow is different from that in the smooth flow.

Wind Pressure Fields of Symmetric Mode
The j th recomposed fluctuating wind pressure field p j = p j1 (t), · · · , p jN (t) T , represented by mode φ j can be expressed using the j th principal coordinate a j (t) as follows, where p jk (t) is the j th recomposed fluctuating wind pressure at point k: The recomposed fluctuating wind pressure fields were investigated for the 1st-and 2nd-modes using p jk (t). Hereafter, the characteristics of the fluctuating wind pressure fields for each type in Table 2 are shown by comparing the following three measurement cases: • Typical measurement case of Type 1: Rigid model, smooth flow, V r = 9.7. • Typical measurement case of Type 2: Elastic model, smooth flow, V r = 9.7, h = 1% • Typical measurement case of Type 3: Elastic model, gradient flow, V r = 9.6, h = 1% The recomposed fluctuating wind pressure fields in the symmetric mode for each type are shown in Figure 46. Furthermore, the pressure rise, or recovery region, is shown in red, the pressure drop region in blue, and the intensity of the fluctuating wind pressure is indicated by the shade of the colour. The wind pressure was divided by the velocity pressure. case of Type 3, shown in Figure 46c, the pressure on the front surface increases simultaneously with the pressure drop on the side surface. For Types 2 and 3, the fluctuating wind pressure fields on the side surface have very different trends, and for this reason the correlation coefficient, rsym,23, is likely to be low. However, for Types 1 and 3, the fluctuating wind pressure fields appear to be different on both the front and side surfaces, although rsym,31 is relatively high. The reason for this is unknown.

Wind Pressure Fields of Anti-Symmetric Mode
The recomposed fluctuating wind pressure fields of the anti-symmetric mode for each type are shown in Figure 47.  For Type 1 and Type 2, on the side surface, the pressure drop region slowly moved from the area near the bottom to the top. The relatively high correlation of r sym , 12 = 0.701 may be because of the similar trends in the fluctuating wind pressure fields on the side surface. For Type 3, on the side surface, a pressure drop region appeared near the height of 2H/3 on the windward side and moved toward the bottom of the leeward side. In the case of Type 3, shown in Figure 46c, the pressure on the front surface increases simultaneously with the pressure drop on the side surface. For Types 2 and 3, the fluctuating wind pressure fields on the side surface have very different trends, and for this reason the correlation coefficient, r sym , 23 , is likely to be low. However, for Types 1 and 3, the fluctuating wind pressure fields appear to be different on both the front and side surfaces, although r sym , 31 is relatively high. The reason for this is unknown.

Wind Pressure Fields of Anti-Symmetric Mode
The recomposed fluctuating wind pressure fields of the anti-symmetric mode for each type are shown in Figure 47. For Type 1, the pressure drop region moves toward the leeward edge near the bottom of the model on the side surface. For Type 2, the pressure drop region near the windward edge at a height of approximately H/4 moves toward the leeward edge of the side at a height of approximately 3H/4 on the side surface. The reason for the very high correlation of r anti , 12 = 0.907 is that the pressure drop region exhibits a similar tendency. For Type 3, the pressure drop region near the windward edge of the side at a height of 3H/4 moves to the leeward edge near the bottom of the model on the side surface. Additionally, the pressure rise region appearing at the right edge of the front extends over the entire front surface, as shown from 62.055 to 62.119 s. This is because the relatively low correlation of r anti , 31 = 0.547 and r anti , 23 = 0.670 may be the differences in the fluctuating wind pressure fields on the front surface.

Concluding Remarks
The aim of this study was to evaluate the phase characteristics of fluctuating wind pressure fields around buildings and to provide valuable knowledge for the design of wind-resistant high-rise buildings. For these purposes, wind tunnel experiments were wind [13.186sec] [13.194sec

Concluding Remarks
The aim of this study was to evaluate the phase characteristics of fluctuating wind pressure fields around buildings and to provide valuable knowledge for the design of wind-resistant high-rise buildings. For these purposes, wind tunnel experiments were conducted in smooth and gradient flows, measuring the pressures around a rigid and elastic model and the displacement of the elastic model. The characteristics of the wind pressure distribution around the rigid and elastic models were clarified based on the experimental results. In addition, the characteristics of the measured displacement of the elastic model are explained.
A CPOD analysis was performed to investigate the fluctuating wind pressure fields around the prism. The contribution ratio of the anti-symmetric mode for the elastic model with a damping ratio of 1% in the smooth flow of the resonant velocity is much higher than that of the rigid model. Therefore, the fluctuating wind pressure fields around the prism are affected by the vibration of the model. Furthermore, there were no significant differences in the contribution ratios and eigenmodes between the rigid and elastic models in the gradient flow.
A method for evaluating the symmetricity of the eigenmodes was proposed to clarify the phase characteristics of the wind pressure fields around buildings. In the case of the elastic model in the smooth flow, symmetric or anti-symmetric modes are clearly affected by whether the wind speed is in the range of the resonance wind speed or not. In contrast, the symmetricity of the eigenmodes for the rigid model in the smooth flow and for the rigid and elastic models in the gradient flow showed different trends with wind speed. This may be because of the close proximity of the 1st-and 2nd-eigenvalues.
In addition, a method for evaluating the similarity of fluctuating wind pressure fields composed of the 1st-and 2nd-modes as the projection ratio of the plane formed by the two principal eigenmodes was proposed. The results show that fluctuating wind pressure fields can be classified into three types: near the resonant and non-resonant states in smooth flow, and in gradient flow.
The three types of fluctuating wind pressure fields were recomposed to show the phase characteristics of the fluctuating wind pressure fields of the symmetric and anti-symmetric modes. In the case of symmetric modes, the fluctuating wind pressure fields in the smooth flow without the resonance phenomenon tend to be similar to those in the case of smooth flow with resonance and gradient flow. However, the fluctuating wind pressure fields in the smooth flow without resonance and the gradient flow show different patterns. In the case of anti-symmetric modes, the fluctuating wind pressure fields in the case of smooth flow without resonance and with resonance tend to be very close. However, the case of gradient flow shows different patterns. Despite the contribution ratios being different for the rigid and elastic models, the phase characteristics of the fluctuating wind pressure fields for both models are almost the same.