# Mean Reattachment Length of Roof Separation Bubbles Using Proper Orthogonal Decomposition
###### Abstract
POD, first introduced by [PERSON] [31] as the Karhunen-Loeve 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. [PERSON] 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. [PERSON] 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, [PERSON] 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, [PERSON] et al. [41] employed POD to interpret the aerodynamic interference effects between tall buildings in various configurations, advancing our comprehension of building interactions. Additionally, [PERSON] 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.