Frequency prediction from exact or self-consistent meanflows

TL;DR

This paper evaluates the effectiveness of the Self-Consistent Model (SCM) and RZIF methods in predicting frequencies of limit cycles in hydrodynamic systems, specifically in thermosolutal convection, revealing limitations of SCM away from bifurcation points.

Contribution

The study applies the SCM to thermosolutal convection and compares its predictions with RZIF, highlighting the conditions under which each method accurately estimates nonlinear frequencies.

Findings

RZIF property holds for traveling waves in thermosolutal convection.

SCM accurately predicts frequency only near bifurcation points.

Nonlinear interactions from the leading mode are insufficient for accurate mean field prediction.

Abstract

A number of approximations have been proposed to estimate basic hydrodynamic quantities, in particular the frequency of a limit cycle. One of these, RZIF (for Real Zero Imaginary Frequency), 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 SCM (for Self-Consistent Model), 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.