# Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition
## 4 Results
### 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 \(0^{\circ}\) under two turbulence intensities (\(I_{u}=12.7\%\) and 0.6%). [PERSON] [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 \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Mean Attachment Length (\(X_{r}\))**} & \multicolumn{2}{c}{**Turbulence Intensity**} \\ \cline{2-3}  & \(I_{u}=12.7\%\) & \(I_{u}=0.6\%\) \\ \hline Maximum (mm) & 410 & 694 \\ Minimum (mm) & 290 & 637 \\ Average (mm) & 326 & 670 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the estimated mean attachment length (\(X_{r}\)) using the POD (\(\alpha=0^{\circ}\)). Figure 11: The 1 st POD eigenmode and mean reattachment lengths (\(X_{r}\)), \(\alpha=0^{\circ}\), \(I_{u}=12.7\%\). location where the mean reattachment occurs has a specific value depending on the level of turbulence intensity (\(I_{u}\)), as shown in Table 3.  
\[C^{*}{}_{p}=\frac{\big{(}C_{p,mean}-C_{p,\ min}\big{)}}{(1-C_{p,\ min})} \tag{7}\]  
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 [PERSON]'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 (\(C^{*}{}_{p}\)) 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 \(C^{*}{}_{p}\) values obtained by [PERSON] et al. [21] and [PERSON] [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, associ  
Figure 12: Variation of reduced pressure coefficients (\(C^{*}{}_{p}\)) at the mean reattachment location (\(X/X_{r}=1\)) with turbulence intensity (\(I_{u}\)) [17; 21]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Turbulence Intensity** & **Reduced Pressure Coefficient** & **Reference** \\ (\(I_{u}\), \%)** & **(\(C^{*}{}_{p}\))** & **Reference** \\ \hline
4 & 0.35 & [PERSON] et al. [21] \\
13 & 0.28 & [PERSON] [17] \\
17 & 0.25 & [PERSON] [17] \\
18 & 0.24 & [PERSON] et al. [26] \\
26 & 0.21 & [PERSON] [17] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reduced pressure coefficients (\(C^{*}{}_{p}\)) at the reattachment location (\(X/X_{r}=1\)).