# Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition
###### Abstract
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 [PERSON] and [PERSON] [23]. Subsequent research by [PERSON] 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, [PERSON] [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 mod  
Figure 1: Failure of roof components under extreme wind loads. Figure 2: Roof separation bubble and reattachment (sectional view along the building centerline). els 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. [PERSON] 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. [PERSON] 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.