A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake

Abstract

The supercritical instability leading to the Bénard-von Karman vortex street in a cylinder wake is a well known example of supercritical Hopf bifurcation: the steady solution becomes linearly unstable and saturates into a periodic limit cycle. Nonetheless, a simplified physical formulation accurately predicting the transition dynamics of the saturation process is lacking. Building upon our previous work, we present here a simple self-consistent model that provides a clear description of the saturation mechanism in a quasi-steady manner by means of coupling the instantaneous mean flow with its most unstable eigenmode and its instantaneous amplitude through the Reynolds stress. The system is coupled for different oscillation amplitudes, providing an instantaneous mean flow as function of an equivalent time. A transient physical picture is described, wherein a harmonic perturbation grows and changes in amplitude, frequency, and structure due to the modification of the mean flow by the Reynolds stress forcing, saturating when the flow is marginally stable. Comparisons with direct numerical simulations show an accurate prediction of the instantaneous amplitude, frequency, and growth rate, as well as the saturated mean flow, the oscillation amplitude, frequency, and the resulting mean Reynolds stresses.

Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

“…In the same spirit, our recent studies [19,32,33] seem to indicate that the nonlinear interaction of the fluctuation with itself gathered in the term −(u · ∇)u + (u · ∇)u has a negligible influence in the saturation process for certain flows. Therefore, this nonlinear interaction is also neglected in the present model while keeping the nonlinearity gathered in the Reynolds stress.…”

“…Along with the amplitude saturation, the response exhibits a change in structure with a migration upstream related to an increase in the forcing amplitude. This migration is connected to a shortening of the mean recirculation bubble, which is reminiscent the mean flow correction in the cylinder flow caused by the limit-cycle amplitude saturation [32,33,35]. …”

“…5 have been modeled by means of a coupled system of mean flow and linear fluctuation equations for the case of the response to harmonic forcing [19], and for the flow past a cylinder above linear instability onset [32,33]. Another related approach is undertaken in the SSST [23] where a time-varying ensemble averaged mean flow, rather than a time average mean flow, is coupled to a linear response to white noise.…”

“…The system (15) models the nonlinear behavior of the flow as a response to stochastic forcing. It is an extension of the SC model recently introduced for harmonic forcing [19], and it appears as the forced counterpart of the self-consistent model introduced first for unstable flows [32,33] where the linear equation corresponds to an eigenvalue problem and the amplitude is dictated by the marginal stability criterion.…”

“…In the present study, we reformulate the self-consistent model in the frequency domain to account for the stochastic nonlinear response to a band-limited δ-correlated white noise. The results aim at clarifying whether the nonlinear stochastic response can be well approximated linearly in the frequency domain and the role of the Reynolds stress and the mean flow distortion in the saturation mechanism, as discussed in the literature [29][30][31][32][33].…”