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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Wind Tunnel Experiment

#### 2.1. Wind Tunnel Flow

_{z}/U

_{600}and turbulence intensity I

_{u}above the centre of the turntable in the wind tunnel are shown in Figure 2, where U

_{z}is the wind speed at a height of z mm. Figure 3a shows the dimensionless power spectral density (PSD), fS

_{u}(f)/σ

^{2}, of the gradient flow at the reference height H (z = 375 mm), where f is the frequency, S

_{u}(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 L

_{z}= 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 L

_{z}with the recommended value in the AIJ-RLB for suburban exposure (Terrain Category III), L

_{z}seems to be smaller but is within the range of variability of the observations.

#### 2.2. Experiments Using a Rigid Model

_{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 ${\overline{C}}_{pe}$ and the fluctuating wind pressure coefficient ${\tilde{C}}_{pe}$ for smooth and gradient flows for V

_{r}= 9.7 are shown in Figure 7 and Figure 8. The wind pressure coefficients ${\overline{C}}_{pe}$ and ${\tilde{C}}_{pe}$ are defined as follows:

_{r}on the wind pressure coefficient distribution is almost negligible.

#### 2.3. Experiments Using an Elastic Model

^{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-freedom 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.

_{1}, calculated by Equation (3) using the ratio d of amplitudes per period, and the dimensionless displacements δ/H is shown in Figure 12, Figure 13, Figure 14 and Figure 15. 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.

_{r}= 8.8, V

_{r}= 9.7 are shown in Figure 16, Figure 17, Figure 18 and Figure 19. For V

_{r}= 9.7, ${\overline{C}}_{pe}$ at the upper part of both side surfaces was smaller than that in the case of V

_{r}= 8.8. ${\tilde{C}}_{pe}$ 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 ${\overline{C}}_{pe}$ and ${\tilde{C}}_{pe}$ for h = 1%, h = 2%, V

_{r}= 9.7 in the gradient flow are shown in Figure 20 and Figure 21. In the gradient flow, the wind pressure distributions are almost the same regardless of h or V

_{r}.

_{me}: displacement of the elastic model by measurement.

_{ae}: displacement by time-history response analysis using the wind force on the elastic model.

_{ar}: displacement by time-history response analysis using the wind force on the rigid model.

_{me}/H and δ

_{ae}/H values are almost the same in the along- and across-wind directions. Meanwhile, the RMS and maximum values of δ

_{me}/H and δ

_{ae}/H are larger than those of δ

_{ar}/H in across-wind direction around V

_{r}= 9.7 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.

_{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.

## 3. Complex Proper Orthogonal Decomposition

#### 3.1. Evaluation Method

^{th}eigenvector as j-column; $\Lambda $ 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:

_{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 ${\tilde{p}}_{j}\left(t\right)$ shown in Equation (11), and the eigenmodes and eigenvalues are obtained using a complex eigenvalue analysis for

**R**. Mode ${\varphi}_{j}$ shall have orthogonality in the broad sense using $A$, and matrix $\Phi $ is normalized as follows:

#### 3.2. Contribution Ratio

^{th}contribution ratio C

_{k}was obtained using Equation (13).

_{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 Figure 32 and Figure 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.

#### 3.3. Eigenmodes

^{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 Figure 34 and Figure 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 Figure 36 and Figure 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.

## 4. Evaluation of Phase Characteristics

#### 4.1. Symmetricity of Fluctuating Wind Pressure Mode

#### 4.1.1. Inner Product

#### 4.1.2. Element Exchange Vector

#### 4.1.3. Symmetry Index

#### 4.1.4. Evaluation Results

#### 4.2. Similarity Rate of Fluctuating Wind Pressure Fields

## 5. Recomposition of Fluctuating Wind Pressure Fields

#### 5.1. Principal Coordinate

^{th}-row element is expressed as the product of CPOD mode matrix

**Φ**and the principal coordinate vector $\tilde{\mathit{a}}={\left\{{\tilde{a}}_{1}\left(t\right),\cdots ,{\tilde{a}}_{N}\left(t\right)\right\}}^{T}$ as follows:

^{th}-principal coordinate, ${\tilde{a}}_{j}\left(t\right)$, is:

_{H}, and variance of the principal coordinates σ

^{2}.

_{H}= 0.1 because the symmetry index of the symmetry mode of Type 1 has a relatively small value of 0.88.

_{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.

#### 5.2. Correlation of Wind Pressure Field

_{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.

#### 5.3. Wind Pressure Fields of Symmetric Mode

^{th}recomposed fluctuating wind pressure field ${\mathit{p}}_{j}={\left\{{p}_{j1}\left(t\right),\cdots ,{p}_{jN}\left(t\right)\right\}}^{T}$, represented by mode ${\mathit{\varphi}}_{j}$ can be expressed using the j

^{th}principal coordinate ${\tilde{a}}_{j}\left(t\right)$ as follows, where ${p}_{jk}\left(t\right)$ is the j

^{th}recomposed fluctuating wind pressure at point k:

- 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%

_{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.

#### 5.4. Wind Pressure Fields of Anti-Symmetric Mode

_{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.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Characteristics of gradient flow: (

**a**) PSD at the height of z = 375 mm; (

**b**) turbulence scale observation in AIJ-RLB.

**Figure 7.**Wind pressure coefficient distributions (rigid model in smooth flow, V

_{r}= 9.7): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 8.**Wind pressure coefficient distributions (rigid model in gradient flow, V

_{r}= 9.7): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 12.**Relationship between the displacement ratios and damping ratios of the free vibration experiment (elastic model in smooth flow, h = 1%): (

**a**) along wind; (

**b**) across wind.

**Figure 13.**Relationship between the displacement ratios and damping ratios of the free vibration experiment (elastic model in smooth flow, h = 2%): (

**a**) along wind; (

**b**) across wind.

**Figure 14.**Relationship between the displacement ratios and damping ratios of the free vibration experiment (elastic model in gradient flow, h = 1%): (

**a**) along wind; (

**b**) across wind.

**Figure 15.**Relationship between the displacement ratios and damping ratios of the free vibration experiment (elastic model in gradient flow, h = 2%): (

**a**) along wind; (

**b**) across wind.

**Figure 16.**Wind pressure coefficient distributions (elastic model in smooth flow, V

_{r}= 8.8, h = 1%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 17.**Wind pressure coefficient distributions (elastic model in smooth flow, V

_{r}= 9.7, h = 1%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 18.**Wind pressure coefficient distributions (elastic model in smooth flow, V

_{r}= 8.8, h = 2%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 19.**Wind pressure coefficient distributions (elastic model in smooth flow, V

_{r}= 9.7, h = 2%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 20.**Wind pressure coefficient distributions (elastic model in gradient flow, V

_{r}= 9.6, h = 1%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 21.**Wind pressure coefficient distributions (elastic model in gradient flow, V

_{r}= 9.6, h = 2%): (

**a**) ${\overline{C}}_{pe}$; (

**b**) ${\tilde{C}}_{pe}$.

**Figure 23.**Vibration–displacement ratio (smooth flow, h = 1%, along wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 24.**Vibration–displacement ratio (smooth flow, h = 1%, across wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 25.**Vibration–displacement ratio (smooth flow, h = 2%, along wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 26.**Vibration–displacement ratio (smooth flow, h = 2%, across wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 27.**Vibration–displacement ratio (gradient flow, h = 1%, along wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 28.**Vibration–displacement ratio (gradient flow, h = 1%, across wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 29.**Vibration–displacement ratio (gradient flow, h = 2%, along wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 30.**Vibration–displacement ratio (gradient flow, h = 2%, across wind): (

**a**) mean; (

**b**) RMS; (

**c**) max.

**Figure 34.**Eigenmodes (rigid model in smooth flow, V

_{r}= 9.7): (

**a**) 1st-mode, absolute value; (

**b**) 1st-mode, phase; (

**c**) 2nd-mode, absolute value; (

**d**) 2nd-mode, phase.

**Figure 35.**Eigenmodes (elastic model in smooth flow, V

_{r}= 9.7, h = 1%): (

**a**) 1st-mode, absolute value; (

**b**) 1st-mode, phase; (

**c**) 2nd-mode, absolute value; (

**d**) 2nd-mode, phase.

**Figure 36.**Eigenmodes (rigid model in gradient flow, V

_{r}= 9.6): (

**a**) 1st-mode, absolute value; (

**b**) 1st-mode, phase; (

**c**) 2nd-mode, absolute value; (

**d**) 2nd-mode, phase.

**Figure 37.**Eigenmodes (elastic model in gradient flow, V

_{r}= 9.6, h = 1%): (

**a**) 1st-mode, absolute value; (

**b**) 1st-mode, phase; (

**c**) 2nd-mode, absolute value; (

**d**) 2nd-mode, phase.

**Figure 39.**Symmetry index of eigenmodes (elastic model in smooth flow, h = 1%): (

**a**) 1st-mode; (

**b**) 2nd-mode.

**Figure 40.**Symmetry index of eigenmodes (elastic model in smooth flow, h = 2%): (

**a**) 1st-mode; (

**b**) 2nd-mode.

**Figure 42.**Symmetry index of eigenmodes (elastic model in gradient flow, h = 1%): (

**a**) 1st-mode; (

**b**) 2nd-mode.

**Figure 43.**Symmetry index of eigenmodes (elastic model in gradient flow, h = 2%): (

**a**) 1st-mode; (

**b**) 2nd-mode.

**Figure 44.**Similarity of fluctuating wind pressure fields consisting of two principal eigenmodes: (

**a**) similarity to the rigid model case of V

_{r}= 9.7 in smooth flow; (

**b**) similarity to the rigid model case of V

_{r}= 9.5 in gradient flow.

**Figure 45.**Power spectral densities (PSDs) of the principal coordinate: (

**a**) symmetric mode; (

**b**) anti-symmetric mode.

**Figure 46.**Recomposition of fluctuating wind pressure fields (symmetric mode): (

**a**) Type 1; (

**b**) Type 2; (

**c**) Type 3.

**Figure 47.**Recomposition of fluctuating wind pressure fields (anti-symmetric mode): (

**a**) Type 1; (

**b**) Type 2; (

**c**) Type 3.

**Table 1.**Corresponding structural specifications between the target building and the experimental model.

Structural Specification | Target Building | Experimental Model | ||
---|---|---|---|---|

Natural frequency (Hz) | 0.25 | 8.34 | ||

Wind speed at reference height (m/s) | 31.0–126.9 | 2.6–10.6 | ||

Reduced wind velocity V_{r} | 4.1–16.9 | 4.1–16.9 | ||

Building density (kg/m^{3}) | 300 | 307 | ||

Generalized mass (kg) | 13,500,000 | 0.216 | ||

Damping ratio h (%) | Smooth flow | Along wind | 1 | 1.08 |

Across wind | 1.02 | |||

Gradient flow | Along wind | 1.04 | ||

Across wind | 0.96 | |||

Smooth flow | Along wind | 2 | 2.03 | |

Across wind | 1.97 | |||

Gradient flow | Along wind | 1.97 | ||

Across wind | 2.00 |

Type of Fluctuating Wind Pressure Field | Model | Flow | h | V_{r} |
---|---|---|---|---|

Type 1 (Smooth flow without resonance) | Rigid | Smooth | - | All measured V_{r} |

Elastic | 1% | Except 9.7 and 10.7 | ||

2% | All measured V_{r} | |||

Type 2 (Smooth flow with resonance) | Elastic | Smooth | 1% | 9.7, 10.7 |

Type 3 (Gradient flow) | Rigid | Gradient | - | All measured V_{r} |

Elastic | 1% | |||

2% |

Type of Fluctuating Wind Pressure Field | 1 | 2 | 3 | ||
---|---|---|---|---|---|

1 | Smooth flow without resonance | - | 0.701 | 0.632 | $\mathrm{Symmetric}\phantom{\rule{0ex}{0ex}}\mathrm{mode}\text{}\left({r}_{sym}\right)$ |

2 | Smooth flow with resonance | 0.907 | - | 0.352 | |

3 | Gradient flow | 0.547 | 0.670 | - | |

$\mathrm{Anti-symmetric}\text{}\mathrm{mode}\text{}({r}_{anti}$) |

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## Share and Cite

**MDPI and ACS Style**

Murakami, T.; Nishida, Y.; Taniguchi, T.
Study on Phase Characteristics of Wind Pressure Fields around a Prism Using Complex Proper Orthogonal Decomposition. *Wind* **2023**, *3*, 35-63.
https://doi.org/10.3390/wind3010004

**AMA Style**

Murakami T, Nishida Y, Taniguchi T.
Study on Phase Characteristics of Wind Pressure Fields around a Prism Using Complex Proper Orthogonal Decomposition. *Wind*. 2023; 3(1):35-63.
https://doi.org/10.3390/wind3010004

**Chicago/Turabian Style**

Murakami, Tomoyuki, Yuichiro Nishida, and Tetsuro Taniguchi.
2023. "Study on Phase Characteristics of Wind Pressure Fields around a Prism Using Complex Proper Orthogonal Decomposition" *Wind* 3, no. 1: 35-63.
https://doi.org/10.3390/wind3010004