Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: application to turbulent cylinder flow

Abstract

Oftentimes, turbulent flows exhibit a high-frequency turbulent component developing on a strong low-frequency periodic motion. In such cases, the low-frequency motion may strongly influence the spatio-temporal features of the high-frequency component. A typical example of such behaviour is the flow around bluff bodies, for which the high-frequency turbulent component, characterized by Kelvin–Helmholtz structures associated with thin shear layers, depends on the phase of the low-frequency vortex-shedding motion. In this paper, we propose extended versions of spectral proper orthogonal decomposition (SPOD) and of resolvent analysis that respectively extract and reconstruct the high-frequency turbulent fluctuation field as a function of the phase of the low-frequency periodic motion. These approaches are based on a quasi-steady (QS) assumption, which may be justified by the supposedly large separation between the frequencies of the periodic and turbulent components. After discussing their relationship to more classical Floquet-like analyses, the new tools are illustrated on a simple periodically varying linear Ginzburg–Landau model, mimicking the overall characteristics of a turbulent bluff-body flow. In this simple model, we in particular assess the validity of the QS approximation. Then, we consider the case of turbulent flow around a squared-section cylinder at a Reynolds number of $Re=22\,000$ , for which we show reasonable agreement between the extracted spatio-temporal fluctuation field and the prediction of QS resolvent analysis at the various phases of the periodic vortex-shedding motion.