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Frequency prediction from exact or self-consistent mean flows

Yacine Bengana and Laurette S. Tuckerman
Phys. Rev. Fluids 6, 063901 – Published 7 June 2021

Abstract

A number of approximations have been proposed to estimate basic hydrodynamic quantities, in particular, the frequency of a limit cycle. One of these, real zero imaginary frequency (RZIF), calls for linearizing the governing equations about the mean flow and estimating the frequency as the imaginary part of the leading eigenvalue. A further reduction, the self-consistent model (SCM), approximates the mean flow as well, as resulting only from the nonlinear interaction of the leading eigenmode with itself. Both RZIF and SCM have proven dramatically successful for the archetypal case of the wake of a circular cylinder. Here, the SCM is applied to thermosolutal convection, for which a supercritical Hopf bifurcation gives rise to branches of standing waves and traveling waves. The SCM is solved by means of a full Newton method coupling the approximate mean flow and leading eigenmode. Although the RZIF property is verified for the traveling waves, the SCM reproduces the nonlinear frequency only very near the onset of the bifurcation and for another isolated parameter value. Thus, the nonlinear interaction arising from the leading mode is insufficient to reproduce the nonlinear mean field and frequency.

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  • Received 14 February 2021
  • Accepted 20 May 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.063901

©2021 American Physical Society

Physics Subject Headings (PhySH)

Fluid DynamicsNonlinear Dynamics

Authors & Affiliations

Yacine Bengana*

  • Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom

Laurette S. Tuckerman

  • Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université de Paris, 75005 Paris, France

  • *b.y.bengana@gmail.com
  • laurette.tuckerman@espci.fr

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Vol. 6, Iss. 6 — June 2021

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