Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition

Abstract

Investigating flow separation regions on the surfaces of three-dimensional bluff bodies in turbulent flows is important because these regions can induce significant aerodynamic loads. Separation bubbles can generate extreme pressures, making the roof components of low-rise buildings vulnerable. In this study, proper orthogonal decomposition (POD) was applied to wind-induced roof pressures to elucidate the physical significance of the dominant modes. Based on the interpretation of the first mode from the POD, the mean reattachment length of the roof separation bubbles on a low-rise building model in turbulent flow was determined. The mean reattachment length derived from the POD was then compared with the length obtained from an aerodynamic database. For the centerline of the roof, the mean reattachment length based on the POD aligned well with that from the aerodynamic database, showing a difference of less than 5%. This study highlights the efficacy of POD as a powerful tool for estimating the reattachment length of separation bubbles on bluff bodies.

1. Introduction

Flow separation and reattachment on sharp-edged, elongated bluff bodies profoundly influence aerodynamic loads. The associated pressure fluctuations on these bodies’ surfaces have been a focal point in numerous research endeavors [1,2,3,4,5]. Such fluctuations can impose significant loads on low-rise building roofs [6,7] and lead to failures in cladding and structural members, as illustrated in Figure 1. Therefore, flow separation and reattachment phenomena on building roofs exert a significant influence on various aspects, from structural safety [8,9] to architectural design [10,11].

Figure 2 provides a detailed depiction of flow separation and reattachment over sharp-edged bluff bodies. In the figure, the separation bubble in a turbulent flow over sharp-edged, elongated bluff bodies refers to a region where the flow detaches from the body’s surface due to an adverse pressure gradient, resulting in reversed or stagnant flow. This bubble is characterized by an initial separation point where the flow detaches, a subsequent region of reversed or recirculating flow, and finally a reattachment point where the turbulent flow, having regained sufficient momentum, reattaches to the surface of the body [12,13,14,15]. The mean reattachment length, denoted as $X_r$, represents the distance from the point where the flow separates from the body surface to the point where it reattaches. It serves as a measure of the representative length of the separation bubble [16]. This length encompasses the region from the flow separation point to the reattachment point, and within this region, significant pressure variations can occur.

Various methods have been used to assess the reattachment lengths of separation bubbles in wind tunnel tests [3,12,17,18,19]. Among these, particle image velocimetry (PIV) stands out as an effective tool for rapidly visualizing velocity fields within flow. It is especially proficient at accurately determining the mean reattachment length ($X_r$), as evidenced by its proven high accuracy [12,18]. However, this technique has its drawbacks, such as the need for costly equipment and software, as well as a complex setup and calibration processes to ensure accurate measurements [20].

Only utilizing the measured pressure data to decipher complex flow dynamics and pinpoint the mean reattachment length ($X_r$) poses inherent challenges. The prevailing method widely cited in the literature involves a scaling-reduced pressure coefficient (${C^{*}}_p$) based on the mean pressure coefficient ($C_{p,mean}$) and the minimum value of the mean pressure coefficient ($C_{p,min}$) on the surface under the separation bubble [17,21,22,23,24,25]. The attempt to apply this technique to evaluate the mean reattachment length ($X_r$) for incompressible flow can be traced back to the research of Roshko and Lau [23]. Subsequent research by Westphal et al. [25] affirmed that under conditions of low turbulence intensity, the distribution of the reduced pressure coefficient (${C^{*}}_p$) remains consistent at the reattachment point. Advancing this methodology, Akon [17] curated a database by drawing from previous measurements in the literature [3,21,24,26]. This database establishes the relationship between the reduced pressure coefficient (${C^{*}}_p$) and the mean reattachment length ($X_r$), which depends on the level of turbulence intensity, laying a foundation for an innovative estimation method. However, the database primarily applies to the roof centerline of bluff body building models with generic shapes and encompasses a limited range of model configurations and flow conditions. The method’s adaptability may diminish when applied to building models or flow dynamics that fall outside the database’s defined scope [22].

To overcome the limitations of evaluating mean reattachment length ($X_r$) using the reduced pressure coefficient (${C^{*}}_p$), it is necessary to numerically discern the patterns in pressure distribution to assess the mean reattachment length ($X_r$) accurately. The field of fluid mechanics and wind engineering extensively employs a variety of reduced-order models for analyzing complex flow dynamics, among which proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and higher-order dynamic mode decomposition (HODMD) are prominent methods.

Chen et al. [27,28] employed DMD to capture the dynamic behavior of fluid flows. In their studies, DMD was particularly adept at revealing the temporal evolution of coherent structures in fluid dynamics and was applied effectively in the analysis of prism wake dynamics. Zhou et al. [29] demonstrated that HODMD extends the capabilities of DMD by incorporating higher-order nonlinearities. This method excels in capturing complex, multi-scale dynamics in fluid flows and offers enhanced insights into the spatial–temporal evolution of wind pressures on square buildings.

In contrast, POD [30] offers a distinct advantage in terms of simplicity and robustness, especially when dealing with the estimation of the mean reattachment length ($X_r$) based on general pressure patterns on roofs. While DMD and HODMD provide a more detailed temporal analysis, they require more complex data processing and interpretation compared to POD. POD is advantageous for effectively understanding the characteristics of wind pressure, as it concisely and intuitively extracts the most energetically significant pressure characteristics.

POD, first introduced by Lumley [31] as the Karhunen–Loève expansion, offers a comprehensive framework for understanding and characterizing structures by decomposing physical fields based on the inherent variables they represent. In recent years, POD has emerged as a valuable tool in wind engineering for analyzing complex flow and pressure dynamics surrounding bluff bodies [32,33,34,35,36,37,38,39,40,41,42], expanding our understanding of wind pressure fields on bluff bodies.

Bienkiewicz et al. [33] applied POD to analyze the dynamic behavior of pressure on low-rise building roofs. They demonstrated that the first POD mode effectively captures the distribution of the pressure mean square, while the second mode aligns closely with the mean pressure derivative relative to the wind direction, underlining the method’s capability to discern directional pressure variations due to wind. Tamura et al. [35] utilized POD to dissect random wind pressure fields on building surfaces, offering crucial insights into identifying systematic structures amid random fluctuations. This study laid the groundwork for subsequent research focusing on more complex architectural forms. Expanding on these initial applications, Kim et al. [39,40] innovatively extended the POD’s utility to probe the aerodynamic characteristics and inter-building correlations in tall, linked buildings. Their research underscored the efficacy of POD in elucidating complex aerodynamic interactions. In a similar vein, Zhou et al. [41] employed POD to interpret the aerodynamic interference effects between tall buildings in various configurations, advancing our comprehension of building interactions. Additionally, Murakami et al. [42] applied complex proper orthogonal decomposition (CPOD) to examine the phase characteristics of fluctuating wind pressure fields around prisms, further broadening the application spectrum of POD in wind engineering.

Despite these advancements and the broad application of POD in aerodynamic force and pressure field analyses, as evidenced in the literature [32,33,34,35,36,37,38,39,40,41,42], a technical gap remains in employing POD for predicting the mean reattachment lengths ($X_r$) in flow separation and reattachment phenomena of low-rise building roofs. This research sought to bridge this gap by providing a novel application of POD, specifically tailored to estimate the mean reattachment lengths ($X_r$) on low-rise building roofs. This is a crucial aspect for understanding aerodynamic loads, particularly the effects on components and cladding.

The application of POD for identifying mean reattachment lengths ($X_r$) marks a significant advancement over the traditional method that utilizes reduced pressure coefficients (${C^{*}}_p$). The POD method not only provides a more comprehensive understanding of the physics underlying flow separation and reattachment phenomena but also delivers a more thorough analysis of mean reattachment lengths ($X_r$) on the envelopes of low-rise buildings. Its effectiveness is not constrained by variations in building shapes, such as aspect ratios and roof configurations, or by diverse flow conditions, including turbulence intensities and integral scales—areas where previous studies have noted limitations. Furthermore, the POD technique can be especially adept at accurately identifying the reattachment line across the entire roof surface. This capability stands in contrast to traditional methods, which typically focus on pinpointing a single mean reattachment point along the building’s roof centerline.

This study explores the potential of POD with the objective of assessing its effectiveness in quantifying features of roof separation bubbles. Specifically, the POD modes of the roof pressure field are determined and interpreted for their physical meanings. This information is then employed to calculate the mean reattachment length ($X_r$) of roof separation bubbles. This innovative approach offers a novel perspective on understanding flow behavior and pressure distribution on the roof surface under varying turbulence intensities and wind directions. Moreover, the POD results are validated against the established methodology [17] for the centerline of the roof. The mean reattachment length ($X_r$) derived from the POD analysis is compared to values from an aerodynamic database along the centerline of the roof to ensure the consistency and precision of the outcomes.

2. Theoretical Background of POD

The objective of POD is to identify a deterministic function, $\varphi$, that correlates the best with the entire ensemble of a physical field [32]. Given a random pressure, $p\left(x,y,t\right)$, the maximum of the projection of pressure $p\left(x,y,t\right)$ on the function $\varphi \left(x,y\right)$ can be sought by considering the following:

$$\int \frac{p\left(x,y,t\right)dxdy}{\sqrt{{\varphi }^2\left(x,y\right)dxdy}}=max$$

(1)

where $x$, $y$, and $t$ are the coordinates and time of the pressure.

The maximization condition, given by Equation (1), is typically implemented in the mean square sense. This can be summarized as the following integral equation [43]:

$$\int R_p\left(x,{y,x}^{\prime },y^{\prime }\right)\varphi \left(x^{\prime },y^{\prime }\right)d{x}^{\prime }{dy}^{\prime }=\lambda \varphi \left(x,y\right)$$

(2)

where $R_p\left(x,{y,x}^{\prime },y^{\prime }\right)=〈p\left(x,y,t\right)p\left(x^{\prime },y^{\prime },t\right)〉$ is the space covariance of pressure and 〈 〉 denotes the expectation operator. The solution of Equation (2) is a set of eigenvalues ${\lambda }_i$ and eigenvectors ${\varphi }_i\left(x,y\right)$, denoted hereafter as eigenmodes.

The POD eigenmodes $\varphi \left(x,y\right)$ satisfy the following orthogonality condition:

$$\int {\varphi }_i(x,y){\varphi }_j(x,y)dxdy=\lambda {\delta }_{ij}$$

(3)

where ${\delta }_{ij}$ is the Kronecker delta with the property of ${\delta }_{ij}$ = 1 or 0 if $i$ = $j$ or $i$ ≠ $j$.

The eigenmodes can be used as the base functions in a series expansion to reconstruct the pressure:

$$p\left(x,t\right)={\displaystyle {\sum }_i}{q}_i\left(t\right){\varphi }_i\left(x,y\right)$$

(4)

where the principal coordinates, $q_i\left(t\right)$, are as follows:

$$q_i\left(t\right)=\int p\left(x,y,t\right){\varphi }_i\left(x,y\right)dxdy.$$

(5)

The principal coordinates, $q_i\left(t\right)$, are statistically uncorrelated. They adhere to an orthogonality condition expressed as:

$$〈q_i\left(t\right)q_j\left(t\right)〉={\delta }_{ij}{\lambda }_i.$$

(6)

3. Methodology

3.1. Data Collection for Roof Pressure on Low-Rise Building

Wind-induced roof pressure data for this study were sourced from turbulent boundary layer simulations conducted in a wind tunnel at MS Engineering Corp. in Haman-gun, South Korea. The wind tunnel has external dimensions of 6.4 m (W) × 4.9 m (H) × 34.0 m (L), and the size of the test section is 3.0 m (W) × 2.0 m (H) × 15.0 m (L), as shown in Figure 3. It can generate wind speeds exceeding 15 m/s, with a turbulence homogeneity of ± 0.8% at 10 m/s.

In this study, hot-wire anemometry was utilized to measure wind velocity and turbulent intensity ($I_u$) profiles, while an electronically scanned pressure measurement system (Scanivalve MPS4264 miniature pressure scanner) was used to measure the roof pressure. The MPS4264 offers accuracy within ± 0.2% for ± 4 inches of water (in H₂O) of the full scale.

Prior to the experiments, the hot-wire anemometer was calibrated using a series of known airspeeds in a controlled environment. This calibration involved correlating the anemometer’s readings with the known airspeeds to establish a reliable relationship between the electrical signal and the wind velocity. The pressure measurement system was calibrated by applying a known standard pressure, adjusting its settings for alignment, and confirming the accuracy across its entire operating range.

The experimental setup is depicted in Figure 4. In this study, the building model was a rectangular prism, measuring 800 mm in width, 800 mm in length, and 400 mm in height with a geometry length scale of 1:40. Within the wind tunnel, this model produced a blockage ratio of 5.3%.

The experiment primarily focused on assessing the impact of turbulence changes on the reattachment lines that bound roof separation bubbles. Rather than simulating the boundary layer for a specific wind exposure category, the experiment emphasized variations in turbulence intensity ($I_u$) at the roof height of 400 mm, with wind velocities scaled down to 1:4.

Figure 5 shows the wind velocity and turbulence intensity ($I_u$) profiles for two turbulent flow conditions: intermediate and low turbulence. For the intermediate turbulence condition, the boundary layer was defined by a power law index of 0.20. At the roof height, the wind velocity and turbulence intensity ($I_u$) were approximately 6.8 m/s and 12.7%, respectively. On the other hand, under the low turbulence condition, the boundary layer had a power law index of 0.16, and at the roof height, the wind velocity and turbulence intensity ($I_u$) were approximately 8.0 m/s and 0.6%, respectively.

To examine the influence of wind direction on the mean reattachment length ($X_r$) of roof separation bubbles, wind pressures on the model’s roof were measured under intermediate turbulence intensity ($I_u$ = 12.7%) for two wind angles ($\alpha$) of 0° and 5°, as shown in Figure 6. Additionally, to assess the effect of different turbulence intensities on the mean reattachment length ($X_r$), wind pressures were measured under low turbulence intensity ($I_u$ = 0.6%) with a wind angle ($\alpha$) of 0°.

The roof pressure was measured at 486 taps using the MPS4264, with a sampling rate of 350 Hz. Each pressure record consisted of 16,384 data points per channel, ensuring the stationarity of signals.

3.2. Analysis of the Pressure Eigenmode Using POD

In this study, the pressure fluctuation component was subjected to POD to identify the dominant modes of pressure variability. For POD analysis, out of the 494 pressure taps shown in Figure 4, only 169 taps, evenly spaced at 60 mm intervals, were used. This selection was made to eliminate potential errors in the maximization of pressure projection, shown in Equation (2), caused by non-uniform pressure distributions.

The pressure data were organized into a matrix, and a covariance matrix was computed to capture the relationships between pressures at different locations. The covariance matrix then underwent eigenvalue decomposition to obtain the eigenvalues and corresponding eigenmodes. The application of a covariance matrix in the POD analysis was essential for accurately representing the spatial interrelationships of fluctuating wind pressures on the roof. By utilizing the covariance matrix, correlations between pressure measurements at various locations were quantitatively analyzed, revealing how individual fluctuations contributed to the overall pressure pattern on the roof. This methodology played a key role in identifying the spatial dynamics of wind pressures, which are crucial for understanding roof separation bubbles and their reattachment.

The POD eigenmodes identified distinctive pressure patterns associated with roof separation bubbles. The first three eigenmodes of the pressure under conditions of intermediate turbulence intensity ($I_u$ = 12.7%) with two wind angles ($\alpha$) of 0° and 5° were given special attention, as they provided a comprehensive understanding of flow features over the roof, including roof surface separation and reattachment phenomena, vortex inside the separation bubble, and the roof corner’s effects on flow.

3.3. Mean Reattachment Length ($X_r$) Evaluation Using POD and Validation

Based on the evaluation of the first three POD eigenmodes and their eigenvalues, information from the first POD eigenmodes, obtained under a wind angle ($\alpha$) of 0° for both low and intermediate turbulence intensities ($I_u$ = 0.6% and 12.7%, respectively), was utilized to calculate the mean reattachment lengths ($X_r$) of the roof separation bubbles. To validate the mean reattachment lengths ($X_r$) derived from the POD analysis, they were compared with values evaluated for the roof centerline using Akon’s [17] method, which utilizes reduced pressure coefficients (${C^{*}}_p$) from the aerodynamic database.

4. Results

4.1. Pressure Fluctuation Characteristics

In this study, the POD method was employed to assess the mean reattachment length ($X_r$) of roof separation bubbles based on the wind pressure coefficient fluctuation components. The characteristics of the pressure distribution were analyzed based on the standard deviations of the surface pressure coefficients, as depicted in Figure 7.

At an intermediate turbulence intensity ($I_u$ = 12.7%), when the wind angle ($\alpha$) is 0°, as depicted in Figure 7a, the standard deviation of the roof surface pressure coefficients shows marked fluctuations, especially near the left and right front roof corners and in an area approximately 200–300 mm from the front edge. Fluctuations at the corners are influenced by the roof corner’s impact on flow dynamics. Additionally, the high fluctuation components occurring inside the roof are believed to be associated with roof separation bubbles and the vortices within them. When the flow has a wind angle ($\alpha$) of 5° (as seen in Figure 7b), the center of the prominent fluctuation inside the roof shifts slightly leftward from the centerline, influenced by the wind direction.

Under conditions of low turbulence intensity ($I_u$ = 0.6%), as illustrated in Figure 7c, the area within the roof that exhibited high standard deviation values under intermediate turbulence intensity ($I_u$ = 12.7%) appears to have reduced standard deviations and moved further inward on the roof surface. This may be due to the center of the recirculation region beneath the separated and reattaching streamline moving away from the leading edge as the turbulence intensity ($I_u$) at the model height decreased. In flow conditions with lower turbulence intensity ($I_u$ = 0.6%), there is less variation in wind pressure within the separation bubble compared to conditions with an intermediate turbulence intensity ($I_u$ = 12.7%). This is attributed to the more stable nature of the flow and the extended reattachment length across the roof. Such stable flow in lower-turbulence conditions leads to a more uniform reattachment of the flow on the roof surface, consequently resulting in reduced fluctuating wind pressure.

4.2. Interpretation of the POD Eigenmode

In this study, the first three eigenmodes were used to interpret the flow characteristics and pressure distribution on the roof surface. These were identified via POD analysis of the wind pressure data obtained under intermediate turbulence intensity ($I_u$ = 12.7%) with wind angles ($\alpha$) of 0° and 5°.

4.2.1. Eigenvalues, Wind Angles ($\alpha$) of 0°

The eigenvalues, computed under intermediate turbulence intensity ($I_u$ = 12.7%) and with a wind angle ($\alpha$) of 0°, are presented in Figure 8 and

Table 1. First three eigenvalues of the fluctuation component of roof pressure: $\alpha$ = 0°, $I_u$ = 12.7%.

Mode	Eigenvalue	Normalized Eigenvalue
1	0.91	0.14
2	0.77	0.11
3	0.49	0.08

. The cumulative contribution of each mode to the total energy (sum of all eigenvalues) is depicted in Figure 8. The first three of the 169 computed eigenvalues, along with normalized eigenvalues calculated using the total energy, are shown in Table 1.

4.2.2. Interpretation of the POD Eigenmodes, Wind Angles ($\alpha$) of 0°

The first three POD eigenmodes at a wind angle ($\alpha$) of 0° are shown in Figure 9. The first POD eigenmode, depicted in Figure 9a, displays distinct positive (red) and negative (blue) regions. When comparing this eigenmode with the standard deviation values shown in Figure 7a, it is clear that the areas with high positive values in the POD eigenmode closely match the regions of high standard deviation. This correlation suggests that the first POD eigenmode plays a significant role in influencing the intense fluctuating wind pressures within the separation bubble.

In contrast to the broader spread of high standard deviation values, the concentrated high values of the first POD eigenmode toward its center suggest the contribution of higher-order POD eigenmodes to the intense fluctuating wind pressure within the separation bubble. In the first POD eigenmode, the small-magnitude negative region (blue), which was not observed in the area of low standard deviation in Figure 7a, carries a distinct physical significance compared to the positive (red) regions, as different signs in the POD eigenmode correspond to different directions of correlation [44] in wind pressure. In the flow separation area, the instability of the flow separation bubble, along with the vortices formed inside it and the high turbulence intensity ($I_u$), increases the complexity and variability of the fluid flow. This may result in high positive values in the POD eigenmode. Conversely, in the reattachment area, where the fluid reattaches to the roof surface, the flow stabilizes, and the turbulence intensity ($I_u$) decreases, which may lead to relatively low negative values in the POD eigenmode. From this examination, it can be inferred that the positive (red) region represents the area affected by flow separation, whereas the negative (blue) region corresponds to the area influenced by flow reattachment phenomena.

The second POD eigenmode is shown in Figure 9b. The eigenmode contour lines, which exhibit relatively high magnitudes of positive and negative values at their centers, stretch from the front to the midpoint of the roof. This indicates an inherent rotational flow [45] on the roof, particularly suggesting the presence of vortices within the roof separation bubble. The incongruity between the vortex region of the second POD eigenmode and the region of the separation bubble estimated by the first POD eigenmode is believed to arise from the irregularities and intermittency of the separation bubbles that form on the roof surface, as well as from the complex vortex phenomena they contain.

The third POD eigenmode, illustrated in Figure 9c, exhibits high magnitudes of values at the front left and right corners of the roof. This can be interpreted as a result of the influence exerted by the corners on the flow dynamics. This distribution of significant magnitudes correlates with the high standard deviations of wind pressure at the corresponding corners observed in Figure 7a, indicating that these variations are likely contributions from the third POD eigenmode.

Noteworthy are the contributions of the eigenvalues of the first three POD eigenmodes to the total fluctuating wind pressure energy. As shown in Table 1, the first POD eigenmode accounts for 14% of the total energy, the second POD eigenmode contributes 11%, and the third POD eigenmode 8%. Together, these modes constitute 33% of the total energy, which underscores the importance of their combined impact on the pressure fluctuation of the roof surface.

4.2.3. Effects of Wind Direction on POD Eigenmodes, Wind Angles ($\alpha$) of 5°

Figure 10 depicts the first three POD modes at a wind angle ($\alpha$) of 5°. The region of the separation bubble, estimated by the first POD eigenmode (depicted in Figure 10a), reveals the pronounced influence of the wind direction, as it tilts and extends noticeably to the left side of the roof. Simultaneously, the center of the high-value region in the POD eigenmode, demarcated by a white circular contour line within the separation bubble, seems to shift slightly inward and to the left on the roof surface compared to its position in the first POD eigenmode with a wind angle ($\alpha$) of 0° (as seen in Figure 9a). Consequently, the region affected by the vortex inside the separation bubble, as highlighted by the second POD eigenmode (presented in Figure 10b), also moves slightly to the left and inward on the roof, relative to the results (depicted in Figure 9b) from the wind angle ($\alpha$) of 0°. Moreover, a closer examination of the third POD eigenmode in Figure 10c clearly reveals a broader distribution of the high magnitudes of the POD eigenmode values at the front right corner of the roof compared to the front left side. This pattern distinctly indicates the impact resulting from a change in the wind angle ($\alpha$) to 5°. This asymmetric effect, which is more pronounced on the right corner than on the left, is also observed via the differences in the standard deviation values of the pressure coefficients at the corresponding front corners of the roof, as illustrated in Figure 7b.

4.3. Evaluation of the Mean Attachment Length ($X_r$) Using the POD Eigenmode

Among the dominant POD eigenmodes previously analyzed, as observed in Figure 9 and Figure 10, only the first POD eigenmode clearly exhibits the general characteristic patterns of flow separation and reattachment phenomena. Although the first mode contributes only 14% to the total fluctuating wind pressure energy, as shown in Table 1, this value is considered substantial, given the complexity of the fluctuating wind pressures. Based on these observations, the first POD eigenmode was identified as the mode that most significantly contributed to the roof separation and reattachment phenomena.

The position of mean reattachment, which serves as the boundary between flow separation and reattachment, was determined using the first eigenmode of the POD. Specifically, the contour line with a POD eigenmode value of zero, following the contour line corresponding to the maximum value of the first POD eigenmode, was chosen to represent the mean reattachment location. In this context, the mean reattachment length ($X_r$) is defined as the distance from the leading edge of the roof, where flow separation begins, to the contour line where the POD eigenmode value is zero, as illustrated in Figure 11.

The mean reattachment lengths ($X_r$) for the intermediate ($I_u$ = 12.7%) and low turbulence intensities ($I_u$ = 0.6%), with a wind angle ($\alpha$) of 0°, were determined by identifying the contour line where the first POD eigenmode value is 0, as illustrated in Figure 11.

Table 2. Summary of the estimated mean attachment length ($X_r$) using the POD ($\alpha$ = 0°).

$\mathbf{Mean}\mathbf{Attachment}\mathbf{Length}\mathbf{\left(}{\mathit{X}}_{\mathit{r}}\mathbf{\right)}$	Turbulence Intensity
	${\mathit{I}}_{\mathit{u}}$ = 12.7%	${\mathit{I}}_{\mathit{u}}$ = 0.6%
Maximum (mm)	410	694
Minimum (mm)	290	637
Average (mm)	326	670

presents the assessed maximum, minimum, and average mean reattachment lengths ($X_r$) across the roof, derived from the first POD eigenmodes under the two flow conditions.

An evaluation of the mean reattachment lengths ($X_r$) using the first POD eigenmodes reveals significant variations as the turbulence intensity ($I_u$) varies from 12.7% to 0.6%. Specifically, the maximum mean reattachment length ($X_r$) increases approximately 1.7 times, the minimum mean reattachment length ($X_r$) increases about 2.2 times, and the average mean reattachment length ($X_r$) undergoes a growth of approximately 2.1 times. These findings emphasize the influence of turbulence intensity ($I_u$) on reattachment phenomena, as documented in numerous previous related studies [3,17,46,47].

4.4. Validation of the POD-Based Mean Reattachment Length ($X_r$)

To validate the contour line representing the mean reattachment lengths ($X_r$) determined via POD analysis, comparisons were made between the mean reattachment lengths ($X_r$) along the roof centerline obtained from the aerodynamic database and those identified via POD analysis. These validations and comparisons were conducted for a wind angle ($\alpha$) of 0° under two turbulence intensities ($I_u$ = 12.7% and 0.6%).

Akon [17] confirmed that when the mean pressure coefficient of the roof surface is normalized, as shown in Equation (7), the reduced pressure coefficient (${C^{*}}_p$) at the location where the mean reattachment occurs has a specific value depending on the level of turbulence intensity ($I_u$), as shown in

Table 3. Reduced pressure coefficients (${C^{*}}_p$) at the reattachment location (${X/X}_r$ = 1).

Turbulence Intensity (${\mathit{I}}_{\mathit{u}}$ , %)	Reduced Pressure Coefficient (${{\mathit{C}}^{\mathbf{*}}}_{\mathit{p}}$)	Reference
4	0.35	Hudy et al. [21]
13	0.28	Akon [17]
17	0.25	Akon [17]
18	0.24	Ho et al. [26]
26	0.21	Akon [17]

.

$${C^{*}}_p=\frac{\left(C_{p,mean}-C_{p,min}\right)}{(1-C_{p,min})}$$

(7)

where $C_{p,mean}$ is the mean pressure coefficient and $C_{p,min}$ is the minimum value of the mean pressure coefficient on the surface under the separation bubble.

Based on Akon’s study [17], the reduced pressure coefficient (${C^{*}}_p$) at the mean reattachment location (${X/X}_r$ = 1) was approximately 0.28 for a turbulence intensity ($I_u$) of 13%. Since this intensity ($I_u$ = 13%) closely aligned with the intermediate turbulence condition ($I_u$ = 12.7%) in the current study, the reduced pressure coefficient (${C^{*}}_p$) of 0.28 was adopted as the coefficient at the reattachment location (${X/X}_r$ = 1) for the present study’s intermediate turbulence condition.

In this study, the reduced pressure coefficient (${C^{\mathrm{*}}}_p$) at the mean reattachment location (${X/X}_r$ = 1) on the centerline of the building model’s roof under low turbulence intensity ($I_u$ = 0.6%) was estimated by linearly extrapolating the ${C^{\mathrm{*}}}_p$ values obtained by Hudy et al. [21] and Akon [17] for turbulence intensities ($I_u$) of 4% and 13%, respectively, as shown in Figure 12. The choice to use linear extrapolation was made due to a lack of available data in the literature regarding the application of nonlinear extrapolation methods, limiting the analysis to these two data points.

Figure 13 illustrates the distributions of the calculated reduced pressure coefficients (${C^{*}}_p$) using roof centerline wind pressure coefficients under conditions of the intermediate and low turbulence intensities ($I_u$ = 12.7% and 0.6%, respectively). In the figures, open circles represent the distributions of reduced pressure coefficients (${C^{*}}_p$). From these distributions, it can be observed that the reduced pressure coefficient (${C^{*}}_p$) of 0.28, associated with the intermediate turbulence intensity ($I_u$ = 12.7%), occurs at a distance of about 340 mm from the leading edge of the roof surface. On the other hand, in the case of the low turbulence intensity ($I_u$ = 0.6%), the reduced pressure coefficient (${C^{*}}_p$) of 0.38 occurs at a distance of about 657 mm from the leading edge of the roof surface due to the influence of low turbulence effects.

Table 4. Comparison of the mean attachment lengths ($X_r$).

Turbulence Intensity (${\mathit{I}}_{\mathit{u}}$, %)	12.7	0.6
Mean reattachment length according to the current POD-based method ($X_r$, mm)	324	689
Mean reattachment length according to Akon’s method [17] ($X_r$, mm)	340	657
Relative error (%) in the mean reattachment length ($X_r$) between the current POD-based method and Akon’s method [17]	4.9	4.6

compares the mean reattachment lengths ($X_r$) along the roof centerline as determined via the current POD-based method with those proposed by Akon [17]. The results in Table 4 indicate that the mean reattachment lengths ($X_r$) derived from the proposed first eigenmode of the POD closely match those from Akon’s [17] methodology, with a relative error of less than 5%. Discrepancies in this comparison may arise from the differences in flow characteristics between the datasets, variations in model configurations, the interpolation methods employed for generating the first POD eigenmode contour lines, and other influencing factors.

As the analysis of our results is concluded, it is crucial to recognize the inherent limitations of this study. Our focus was on the centerline of the roof with a wind angle ($\alpha$) of 0°, a choice that provided significant insights but also came with specific constraints. This particular location and angle were selected primarily due to the availability of comparative data, enabling a robust validation of the proposed methodology.

However, our validation process was limited when considering other locations on the roof and additional wind angles ($\alpha$) (beyond 0°), such as a wind angle ($\alpha$) of 5°. This limitation predominantly stemmed from the lack of detailed empirical data for these other locations and wind angles in the existing literature, thereby constraining the scope of the validation.

These limitations highlight the need for further research that encompasses various locations on the roof and a broader range of wind angles. Future efforts in this area, especially those involving the acquisition of more comprehensive datasets, will help in validating and enhancing the applicability of the proposed methodology to diverse locations and wind angles. Such advancements will not only strengthen the robustness of our findings but also contribute to a more comprehensive understanding of flow separation and reattachment dynamics in low-rise building roofs.

5. Conclusions

This research utilized proper orthogonal decomposition (POD) to evaluate the mean reattachment length ($X_r$) of roof separation bubbles on a three-dimensional rectangular bluff body model, offering key insights:

• Primary POD eigenmode: This mode effectively characterizes the major global mechanisms influencing the roof pressure field. The regions of high magnitude in the eigenmodes are closely associated with intense pressure fluctuations, reflecting the complex and variable fluid flow within the separation bubble. Conversely, areas of low magnitudes in the eigenmode are indicative of low-pressure fluctuations, characteristic of the more stable flow found in the reattachment area. Notably, these regions are distinguished by reversed signs in the eigenmode. This comprehensive characterization captures the dynamics associated with flow separation and reattachment, demonstrating the detailed nature of the pressure field’s variability in these regions.
• Secondary POD eigenmode: This mode reveals the presence of vortex structures within the separation bubbles. This finding offers a deeper understanding of the complex flow behaviors and the underlying mechanics within the roof separation bubble.
• Third POD eigenmode: This mode exhibits pronounced effects at the windward front corners of the roof caused by the wind flow. The distribution of this eigenmode strongly correlates with the pressure fluctuations at these corners, indicating the impact of roof corners on flow dynamics.
• Estimation of the mean reattachment length ($X_r$) using the first POD eigenmode: By evaluating the dominant POD eigenmodes and eigenvalues, it was deduced that the mean reattachment length ($X_r$) can be represented as the distance from the leading edge to the first zero-valued contour line after the first POD eigenmode reaches its maximum value. This offers a clear, quantifiable measure of the mean reattachment length ($X_r$) on the roof.
• Validation of the POD-based mean reattachment length ($X_r$): The mean reattachment length ($X_r$) determined via the proposed POD-based method closely aligns with those obtained from an established aerodynamic database, with observed discrepancies being less than 5%. This underscores the robustness and precision of the POD approach in estimating the mean reattachment length ($X_r$) using the first POD eigenmode.
• Methodological advantages: The technique’s capacity to precisely pinpoint the flow reattachment location across the entire separation region, using solely pressure data, is a significant advancement. This negates the need for specialized aerodynamic databases, making the approach adaptable to a model regardless of its shapes and flow conditions.

Based on the results of this study, it is deemed necessary to conduct additional research into the evaluation of pressure characteristics and mean reattachment lengths ($X_r$) for various bluff body geometries. Future studies should focus on evaluating the applicability of the proposed POD-based methodology to a diverse range of bluff body shapes and sizes. This is crucial for understanding how different geometric configurations influence flow separation and reattachment phenomena. Additionally, building upon the research conducted in this study, further extending the investigation to include the effects of varying turbulence intensities and wind directions is considered important for enhancing the generalization of the proposed method.

In conclusion, the use of the first POD eigenmode to identify the flow reattachment location offers numerous advantages. It provides a streamlined, accurate, and intuitive approach for experimental researchers and holds significant implications for the realms of wind engineering and fluid dynamics.