Spectral Proper Orthogonal Decomposition via Dynamic Mode Decomposition for Non-Sequential Pairwise Data

Spectral proper orthogonal decomposition (SPOD) is a valuable tool to identify spatio-temporally coherent structures in statistically stationary turbulence. It can be used with particle image velocimetry (PIV) flow data, but current algorithms require sequential data with uniform time step, which is thus limited by camera speeds. Dynamic mode decomposition (DMD), a method that estimates structures that optimally capture the time dynamics of the data, relaxes the requirement for sequential data, but does not produce orthogonal, energy-ordered modes. The present work attempts to compute SPOD modes from pairs of PIV snapshots with a small time step but with large gaps between pairs, by using the DMD to estimate, segment-wise, sequential series of data that can then be used to estimate the SPOD modes, with the aim of dealiasing/resolving frequencies between the gap and pair Nyquist limits. This method is tested on several examples, including a numerical simulation of a turbulent jet. Results show this method can accurately estimate the SPOD mode shapes and dealias the SPOD spectrum with a resolved frequency that is 40 times higher than the Nyquist limit associated with the longer time step between pairs.