Permuted proper orthogonal decomposition for analysis of advecting structures

Flow data are often decomposed using proper orthogonal decomposition (POD) of the space–time separated form, $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (t\right )\boldsymbol {\phi }_j\left (\boldsymbol {x}\right )$, which targets spatially correlated flow structures in an optimal manner. This paper analyses permuted POD (PPOD), which decomposes data as $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (\boldsymbol {n}\right )\boldsymbol {\phi }_j\left (s,t\right )$, where $\boldsymbol {x}=(s,\boldsymbol {n})$ is a general spatial coordinate system, $s$ is the coordinate along the bulk advection direction and $\boldsymbol {n}=(n_1,n_2)$ 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 $s,t$ 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.