Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake

Abstract

The Bénard-von Karman vortex shedding instability in the wake of a cylinder is perhaps the best known example of a supercritical Hopf bifurcation in fluid dynamics. However, a simplified physical description that accurately accounts for the saturation amplitude of the instability is still missing. Here we present a simple self-consistent model that provides a clear description of the saturation mechanism and quantitatively predicts the saturated amplitude and flow fields. The model is formally constructed by a set of coupled equations governing the mean flow together with its most unstable eigenmode with finite size. The saturation amplitude is determined by requiring the mean flow to be neutrally stable. Without requiring any input from numerical or experimental data, the resolution of the model provides a good prediction of the amplitude and frequency of the vortex shedding, as well as the spatial structure of the mean flow and the Reynolds stress.Simple models are essential to our understanding of complex nonlinear phenomena. The van der Pol oscillator, for example, demonstrates how nonlinear oscillations can be described by the appearance of a limit cycle [1]. In large dimensional systems, however, these simple models do not entirely reveal the mechanisms that determine relevant parameters like the dominant frequency or saturation amplitude. For supercritical instabilities in fluid dynamics, the mean flow has been proposed as a key element to explain the origin of the dominant frequency [2-5] and the physical mechanism of the saturation process [5][6][7]. The physical picture thus invoked to understand the saturation is the following: perturbations feeding on an unstable flow induce mean flow modifications that increase while perturbations grow, up to the point where the mean flow becomes neutrally stable and perturbations stop growing and saturate. The present Letter aims at assessing this scenario.An early formulation of this concept of marginal stability of the mean flow was given by Malkus [8] in the context of turbulent flows. Shortly after, aiming for an equation describing the saturation of supercritical instabilities, Stuart [6] devised a simplified closed system wherein the mean flow was only affected by the Reynolds stress divergence of its leading eigenmode. By further assuming that the eigenmode was given by the unperturbed base flow, Stuart managed to obtain an equation for the saturation amplitude through the exact balance between the dissipation of the perturbation and the energy transfer from the mean flow. It wasn't until after two more years, through a more rigorous perturbative analysis close to threshold, that he mathematically derived an amplitude equation, the Stuart-Landau equation, directly from the Navier-Stokes equations [9].Despite the beauty and consistency of the multiplescale expansion method, its perturbative nature implies that the spatial structure of the growing unstable mode is in large part fixed by the unperturbed base flow. However, there are cases in which the spatial ...

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

“…As in related studies [7,8,33], this scalar can be determined by employing an iterative algorithm. Following Dempsey et al [34], we instead specify A and treat the streamwise wavenumber α as a scalar unknown that must be determined via the solution of a nonlinear eigenvalue problem.…”

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.…”

“…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].…”