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Projection method for mean resolvent analysis of periodic flows

A. Bongarzone*, C. Content, D. Sipp, and C. Leclercq

  • *Contact author: alessandro.bongarzone@onera.fr
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Phys. Rev. Fluids 11, 083903 – Published 17 August, 2026

DOI: https://doi.org/10.1103/xrsr-v8bl

Abstract

The mean resolvent operator predicts the mean linear response to forcing in the frequency domain and provides the optimal linear time-invariant approximation of statistically steady, time-varying flows [C. Leclercq et al., J. Fluid Mech. 968, A13 (2023)]. We first leverage the harmonic resolvent framework [N. M. Wereley and S. R. Hall (IEEE, Piscataway, NJ, 1990); A. Padovan et al., J. Fluid Mech. 900, A14 (2020)] to propose an algorithm for performing mean resolvent analysis of a periodic flow. Next, we propose an alternative approach which does not explicitly rely on the harmonic resolvent framework. The approach leverages the fact that the mean-flow resolvent approximates the mean resolvent operator, therefore the optimal forcing modes of the latter operator may be sought in a subspace spanned by optimal modes of the former. This projection approach does not require computing the adjoint dynamics about the attractor, which may be convenient for future extensions to more chaotic and turbulent flows. The present paper is however focused on periodic flows, where the convergence of the projection approach can be checked in comparison to the “ground truth” provided by the harmonic resolvent framework. This test is performed on a nearly incompressible axisymmetric laminar jet forced harmonically at the inlet. For the weakly unsteady case, the mean-flow resolvent captures the dominant receptivity peak but misses a secondary one present in the mean resolvent gain. For the strongly unsteady case, the mean-flow resolvent fails to predict the frequency of the vortex-pairing, while the mean resolvent correctly locates the corresponding gain peak. The projection method converges with a subspace dimension of 10 in the weakly unsteady case, while about 100 modes are required for accurate predictions in the strongly unsteady regime. Nonetheless, even a one-dimensional subspace correctly identifies the dominant receptivity peak.

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Phys. Rev. Fluids 11, 083903– Published 17 August, 2026
  • Received 18 January 2026
  • Accepted 27 July 2026
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