# Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition
## 4 Results
### Interpretation of the POD Eigenmode
#### 4.2.2 Interpretation of the POD Eigenmodes, Wind Angles (\(\alpha\)) of 0\({}^{\circ}\)  
The first three POD eigenmodes at a wind angle (\(\alpha\)) of 0\({}^{\circ}\) 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  
Figure 9: The POD eigenmodes, \(\alpha=0^{\circ}\), \(I_{u}=12.7\%\): (**a**) Mode 1; (**b**) Mode 2; (**c**) Mode 3.  
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\({}^{\circ}\)  
Figure 10 depicts the first three POD modes at a wind angle (\(\alpha\)) of 5\({}^{\circ}\).