Frequency prediction from exact or self-consistent mean flows

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.

Global linear stability analysis of a flame anchored to a cylinder

“…In contrast, the frequencies predicted from the base flow analysis deviate with increasing distance from the critical Reynolds number. The base flow eigenmode is unstable above Re = 1150, while the mean flow mode has a growth rate of nearly zero for all supercritical conditions, which is consistent with the classical results for non-reacting cylinder wakes (Barkley 2006) and the RZIF criterion (Turton et al 2015;Bengana & Tuckerman 2021).…”

“…The base flow eigenmode is unstable above , while the mean flow mode has a growth rate of nearly zero for all supercritical conditions, which is consistent with the classical results for non-reacting cylinder wakes (Barkley 2006) and the RZIF criterion (Turton et al. 2015; Bengana & Tuckerman 2021).…”

“…The present work represents one of the first attempts to include a linearised chemical scheme in the analysis of time-averaged reacting flows, and the results are in excellent agreement with the RZIF criterion (Turton et al. 2015; Bengana & Tuckerman 2021). Modal mean-flow instability analysis therefore holds promise for the investigation of flame instability phenomena, also in more complex configurations.…”

“…Similarly, as in classical literature on non-reacting wake flows, it is demonstrated that the leading linear eigenmode, characterised by a zero growth rate, accurately matches the frequency of the nonlinear limit-cycle oscillations, under the condition that the equations are linearised around the mean flow obtained from the nonlinear simulations. The present work represents one of the first attempts to include a linearised chemical scheme in the analysis of time-averaged reacting flows, and the results are in excellent agreement with the RZIF criterion (Turton et al 2015;Bengana & Tuckerman 2021). Modal mean-flow instability analysis therefore holds promise for the investigation of flame instability phenomena, also in more complex configurations.…”

“…The growth rate of the least stable mean flow eigenmode was found to be almost exactly zero. Similar observations have been made in many other flow cases, prompting the postulation of a 'real-zero-imaginary-frequency' (RZIF) criterion (Turton, Tuckerman & Barkley 2015;Bengana & Tuckerman 2021), which states that perturbations inducing mean flow modifications saturate when the mean flow becomes neutrally stable. Making use of this criterion, Mantič-Lugo, Arratia & Gallaire (2014) proposed a self-consistent model to predict the frequency and amplitude of vortex shedding without prior knowledge of the mean flow.…”