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Author: Ek, Hanna Maria × Item Type: Publication × search.filters.applied.f.isSeriesOfPublicationTitle: Doctor of Philosophy with a Major in Aerospace Engineering × Type: Dissertation ×

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Permuted proper orthogonal decomposition for analysis of advecting structures

2024-04-27 , Ek, Hanna Maria

This work is motivated by the large and ever-increasing amounts of data from studies in experimental and computational fluid dynamics, and the desire to extract and analyze coherent structures from such datasets. Specifically, this thesis is concerned with vortex patterns in turbulent shear flows, which appear as advecting structures in planar measurements or slices through three-dimensional computational domains. Space-only proper orthogonal decomposition (POD) is one of the most widely used techniques for the analysis of coherent structures and decomposes mean-subtracted data into the space-time separated form q^' (x,t)=∑_j =〖a_j (t) ϕ_j (x) 〗. This method is optimal in the spatial inner product and targets high energy spatial structures, but it is sensitive to input data alignment and cannot effectively handle translations. This work applies a re-orientation of the space-time coordinates in the POD framework, and the modified POD method, referred to as permuted POD (PPOD), is the focus of this thesis. PPOD decomposes data as q^' (x,t)=∑_j =〖a_j (n) ϕ_j (s,t) 〗, where x=(s,n) is a general spatial coordinate system, s is the coordinate along the bulk advection direction in curvilinear space, and n=(n_1,n_2 ) are the mutually-orthogonal directions normal to s. PPOD is optimal in the s,t inner product and, thus, targets advecting structures via their s,t correlations. Specifically, the PPOD modes, ϕ_j (s,t), portray advection as diagonal features in s,t space, where the slope of the features corresponds to the phase speed. Hence, these speeds are a natural output of the decomposition and can vary in an arbitrary and dispersive manner along the s coordinate. Generally, the PPOD modes have arbitrary s,t dependences, and a single mode can describe a broadband or multi-frequency disturbance, as well as time-varying characteristics, such as transient and intermittent dynamics. Additionally, one- and two-dimensional Fourier transforms of the PPOD modes provide useful alternative ways to portray the modal characteristics. For example, the wavenumber-frequency spectrum provides a compact visualization of disturbance advection velocity or dispersion. The PPOD properties are considered through the analysis of data from three high Reynolds number advection-dominated flows: an acoustically forced reacting wake, a swirling annular jet, and a jet in cross flow (JICF), and the results are compared with those from space-only POD. In the wake and swirling jet cases, the leading PPOD and space-only POD modes focus on similar features: advecting shear layer structures. However, low-rank approximations of the wake flow, which is characterized by a broad range of spectral and wavenumber content, show clear differences in the methods’ ability to capture the spatial and temporal information. For equal low-rank approximations, space-only POD provides higher-fidelity spatial reconstructions, while PPOD provides higher-order frequency content. In contrast, the leading PPOD and space-only POD modes for the JICF datasets capture different types of flow structures: advecting shear layer vortices (SLVs) and bulk jet flapping, respectively, while the SLVs are spread over lower energy modes in the case of space-only POD. This shows that the s,t inner product allows the PPOD method to directly target the SLVs, despite them containing a smaller fraction of the energy compared to the jet flapping. Additionally, the leading PPOD mode captures key characteristics of the SLV dynamics for each of the JICF cases, including those typical of convectively and globally unstable JICF, as well as intermittent characteristics and minor time-dependent differences or shifts in the dynamics. On the other hand, higher-order space-only POD approximations are required for comparable descriptions of these dynamics, and the rank depends on the operating conditions and stability characteristics of the JICF.

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