Conditions for validity of mean flow stability analysis

By Samir Beneddine, Denis Sipp, A. Arnault, J. Dandois, Lutz Lesshafft. Journal of Fluid Mechanics, 2016

192 citations recorded by the source corpus.

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

Subfields: Computational Mechanics, Environmental Engineering

View Paper: https://www.semanticscholar.org/paper/2c581616d7581718c3a4f9dc8878f4d3d858f2cc

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Citations in the corpus

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, methodology): “In some flows, reduced-order models have been obtained by retaining a limited number of terms in the expansion when the resolvent operator is low rank (e.g., Beneddine et al., 2016; Gómez et al., 2016b).”
Spectral analysis of jet turbulence (2017, background): “This spatial separation between optimal forcing and response characterizes a convective non-normality (Marquet et al. 2009) in the presence of a spatial instability mechanism (Alizard et al. 2009; Dergham et al. 2013; Beneddine et al. 2016).”
The engine behind (wall) turbulence: perspectives on scale interactions (2017, methodology): “The latter approach has been developed subsequently by, e.g., Gómez Carrasco et al. (2014), Lu & Papadakis (2014) and Beneddine et al. (2016), and while significantly more flexible in terms of geometries that can be considered, it is also considerably more expensive to compute than the simple…”