Frequency prediction from exact or self-consistent meanflows

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.

References in corpus (14)

• A critical layer model for turbulent pipe flow
• Direct Statistical Simulation of Out-of-Equilibrium Jets
• The Generalized Quasilinear Approximation: Application to Zonal Jets
• Prediction of frequencies in thermosolutal convection from mean flows
• On non-normality and classification of amplification mechanisms in stability and resolvent analysis
• Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow
• Non-linear input/output analysis: application to boundary layer transition
• A tale of two airfoils: resolvent-based modelling of an oscillator vs. an amplifier from an experimental mean
• A Streamwise Constant Model of Turbulence in Plane Couette Flow
• Three-Dimensional Rotating Couette Flow Via The Generalised Quasilinear Approximation
• Thermosolutal and binary fluid convection as a 2 x 2 matrix problem
• Dimensional Reduction of Direct Statistical Simulation
• Spirals and ribbons in counter-rotating Taylor-Couette flow: frequencies from mean flows and heteroclinic orbits
• Nonlinear evolution of the centrifugal instability using a semi-linear model

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