Band-Ensemble Spectral Proper Orthogonal Decomposition with Frequency Attribution

TL;DR

Band-ensemble SPOD (bSPOD) addresses spectral leakage and fixed-frequency limitations of traditional SPOD by computing cross-spectral density from a full-record Fourier spectrum assembled from $N_f$ consecutive narrow-band modes, and by attributing data-driven frequencies to modes via expansion coefficients and weights. This yields reduced variance in broadband regions while keeping bias near tonal components, and provides a flexible, frequency-adaptive framework with comparable computational cost to Welch-based SPOD. The approach is validated on an artificial broadband–tonal signal and a broadband–tonal cavity-flow PIV dataset, showing improved spectral localization and explicit in-band frequency attribution for tonal components. The work offers a practical method for spectral modal analysis of turbulent flows with broadband-tonal characteristics and supports adaptive strategies for choosing frequency-band widths based on convergence criteria.

Abstract

This study presents band-ensemble Spectral Proper Orthogonal Decomposition (bSPOD). The approach is inspired by frequency smoothing, a method used to reduce estimator variance in power spectral density estimates, and is here extended to SPOD. The algorithm estimates SPOD modes from consecutive Fourier coefficients obtained from a single Fourier transform of the full time record and thus avoids time segmentation. In this study, bSPOD is applied to artificial test data and to a PIV data set of a broadband-tonal cavity flow. Compared to the more commonly used Welch-based SPOD formulation, bSPOD reduces spectral leakage, permits increased frequency resolution, and retains frequency information of tonal components at comparable computational cost. These features enable reduced estimator variance while maintaining low bias for tonal components, making bSPOD particularly effective for broadband-tonal flows.

Figures (9)

• Figure 1: Schematic comparison between Welch-based and Band-ensemble SPOD.
• Figure 2: Mach 0.6 cavity-flow mono-PIV dataset of zhang2020. Left: time-averaged axial (top) and vertical (bottom) velocity field; thick black lines mark the cavity boundaries. Right: snapshot of the fluctuating (mean-subtracted) vertical velocity (top) and the PSD (bottom) of the fluctuating vertical velocity at the green dot. Red vertical lines indicate the three Rossiter modes (940, 1540, and 2130 Hz).
• Figure 3: Modal power spectral densities and tonal power for the artificial data set. Welch-based SPOD and bSPOD are compared against the theoretical values for three configurations: $\Delta f=0.5,\mathrm{Hz}$ with $N_b=N_f=50$ (top), $\Delta f=2\,\mathrm{Hz}$ with $N_b=N_f=200$ (middle), and $\Delta f=2\,\mathrm{Hz}$ with $N_b=N_f=200$ using a Hann taper for Welch-based SPOD (bottom). Blue and orange markers denote the leading three eigenvalues from Welch-based SPOD and bSPOD, respectively. Markers enclosed by the red box represent tonal power (right axis), whereas all other markers and lines indicate power spectral densities (left axis). Black lines show the analytic broadband power spectral density ($g_m^2 S(\omega)L_x$), and black squares indicate the true tonal power ($B_r^2L_x$), see Section \ref{['sec:artdata']}. Dashed vertical lines mark the tonal frequencies.
• Figure 4: Leading three SPOD eigenvalues versus frequency for the Welch-based (blue) and band-ensemble (orange) formulations with $N_f=N_b$ and three frequency resolutions: $\Delta f=25~\mathrm{Hz}$ (top), $50~\mathrm{Hz}$ (middle), and $100~\mathrm{Hz}$ (bottom). Vertical red lines denote the Rossiter-mode frequencies identified from the single-point PSD (see Fig. \ref{['fig:cavity_fields']}). Yellow arrows and circles mark the eigenvalues whose corresponding mode shapes are shown in Section \ref{['sec:mode_convergence']}.
• Figure 5: Real part of the vertical velocity component of the Rossiter mode at $1540\,$Hz obtained with band-ensemble SPOD (left) and Welch-based SPOD (right), both using a frequency resolution of $\Delta f=25\,$Hz. The corresponding eigenvalues are marked by yellow crosses in the top panel of Fig. \ref{['fig:cavity1']}.