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.

Junction flow inside and around three-row cylindrical group on rigid flat surface

“…In work ( Rajagopalan and Antonia, 2005 ) it is indicated that the dependence is observed for the pairing process of shear layer vortices, which corresponded to the frequency , for Reynolds numbers . When a square cylinder was streamlined, the frequency ratio was changed from 20 to 30 in accordance with the results of ( Franceschini et al., 2022 ).…”

“…Hydrodynamic instabilities in the form Kelvin-Helmholtz instability and Karman instability are formed in the case of a supercritical transverse flow around cylinders of circular ( >1300) or square ( >1000) cross section ( Williamson, 1996 ; Prasad and Williamson, 1997 ; Brun et al., 2008 ; Franceschini et al., 2022 ). The Kelvin-Helmholtz instability is formed as a result of separation of the boundary layer from the streamlined surface of the cylinder and the formation of a shear layer, which is similar to the mixing layer of the jet flow ( Sadr and Klewicki, 2003 ; Vanierschot et al., 2021 ).…”

Mean resolvent operator of a statistically steady flow

“…2020; Franceschini et al. 2022) with respect to the relative initial time of forcing , or simply phase in the periodic case. Mean transfer functions based on the resulting mean resolvent operator , may be identified from input–output data without the need for averaging over several input realizations if harmonic forcings are used, even though the linearized system is not time-invariant.…”

“…Since we are considering the incompressible Navier–Stokes equations, the nonlinearity is quadratic, hence the mean Jacobian is equal to the Jacobian operator about the mean flow: Assuming , taking the Laplace transform of (1.4 a , b ) and plugging (3.23)-(3.24) yields which in turn leads, through harmonic balance (Khalil 2002), to an alternative form of the harmonic transfer operator, also referred to as the harmonic resolvent operator (Padovan, Otto & Rowley 2020; Franceschini et al. 2022): …”

Towards Real-Time Reconstruction of Velocity Fluctuations in Turbulent Channel Flow

“…Beneddine et al [34] found a similar result by separately capturing low-frequency and high-frequency regions in a backward-facing step flow configuration. Recently, a quasi-steady resolvent analysis has been used to reconstruct high-frequency fluctuations using the phase of a lower-frequency periodic motion [35]. Resolvent truncations have also recently been used to design 𝐻 2 -optimal estimators and controllers [36] and select optimal sensor and actuator locations [37] in cylinder flows.…”