Adaptive spectral proper orthogonal decomposition of broadband-tonal flows

An adaptive algorithm for spectral proper orthogonal decomposition (SPOD) of mixed broadband-tonal turbulent flows is developed. Sharp peak resolution at tonal frequencies is achieved by locally minimizing bias of the spectrum. Smooth spectrum estimates of broadband regions are achieved by locally reducing variance of the spectrum. The method utilizes multitaper estimation with sine tapers. An iterative criterion based on modal convergence is introduced to enable the SPOD to adapt to spectral features. For tonal flows, the adaptivity is controlled by a single user input; for broadband flows, a constant number of sine tapers is recommended without adaptivity. The discrete version of Parseval’s theorem for SPOD is stated. Proper normalization of the tapers ensures that Parseval’s theorem is satisfied in expectation. Drastic savings in computational complexity and memory usage are facilitated by two aspects: (i) sine tapers, which permit post hoc windowing of a single Fourier transform; and (ii) time-domain lossless compression using a QR or eigenvalue decomposition. Sine-taper SPOD is demonstrated on time-resolved particle image velocimetry (TR-PIV) data from an open cavity flow (Zhang et al. in Exp Fluids 61(226):1–12, https://doi.org/10.1007/s00348-020-03057-8, 2020) and high-fidelity large-eddy simulation (LES) data from a round jet (Brès et al. in J. Fluid Mech. 851:83–124, https://doi.org/10.1017/jfm.2018.476, 2018), with and without adaptivity. For the tonal cavity flow, the adaptive algorithm outperforms Slepian-based multitaper SPOD in terms of variance and local bias of the spectrum, mode convergence, and memory usage. The tonal frequencies associated with the Rossiter instability are accurately identified. For both the tonal cavity and the broadband jet flows, results comparable to or better than those from standard SPOD based on Welch’s overlapped segment averaging are obtained with up to 75% fewer snapshots, including similar convergence of the Rossiter modes and Kelvin-Helmholtz wavepacket structures for the cavity and jet examples, respectively. Drawing from these examples, we establish best practices.