Permuted proper orthogonal decomposition for analysis of advecting structures

Abstract

Flow data are often decomposed using proper orthogonal decomposition (POD) of the space-time separated form, which targets spatially correlated flow structures in an optimal manner. This paper analyses permuted POD (PPOD), which decomposes data as, where is a general spatial coordinate system, is the coordinate along the bulk advection direction and are along mutually orthogonal directions normal to the advection characteristic. This separation of variables is associated with a fundamentally different inner product space for which PPOD is optimal and targets correlations in space. This paper presents mathematical features of PPOD, followed by analysis of three experimental datasets from high-Reynolds-number, turbulent shear flows: a wake, a swirling annular jet and a jet in cross-flow. In the wake and swirling jet cases, the leading PPOD and space-only POD modes focus on similar features but differ in convergence rates and fidelity in capturing spatial and temporal information. In contrast, the leading PPOD and space-only POD modes for the jet in cross-flow capture completely different features - advecting shear layer structures and flapping of the jet column, respectively. This example demonstrates how the different inner product spaces, which order the PPOD and space-only POD modes according to different measures of variance, provide unique 'lenses' into features of advection-dominated flows, allowing complementary insights.