Identification of coexisting dynamics in boundary layer flows through proper orthogonal decomposition with weighting matrices

A different version of the classic proper orthogonal decomposition (POD) procedure introducing spatial and temporal weighting matrices is proposed. Furthermore, a newly defined non-Euclidean (NE) inner product that retain similarities with the POD is introduced in the paper. The aim is to emphasize fluctuation events localized in spatio-temporal regions with low kinetic energy magnitude, which are not highlighted by the classic POD. The different variants proposed in this work are applied to numerical and experimental data, highlighting analogies and differences with respect to the classic and other normalized variants of POD available in the literature. The numerical test case provides a noise-free environment of the strongly organized vortex shedding behind a cylinder. Conversely, experimental data describing transitional boundary layers are used to test the capability of the procedures in strongly not uniform flows. By-pass and separated flow transition processes developing with high free-stream disturbances have been considered. In both cases streaky structures are expected to interact with other vortical structures (i.e. free-stream vortices in the by-pass case and Kelvin–Helmholtz rolls in the separated type) that carry a significant different amount of energy. Modes obtained by the non-Euclidean POD (NE-POD) procedure (where weighted projections are considered) are shown to better extract low energy events sparse in time and space with respect to modes extracted by other variants. Moreover, NE-POD modes are further decomposed as a combination of Fourier transforms of the related temporal coefficients and the normalized data ensemble to isolate the frequency content of each mode.