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 ...

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.

Certain flows denominated as amplifiers are characterised by their global linear stability while showing large linear amplifications to sustained perturbations. As the forcing amplitude increases, a strong saturation of the response appears when compared to the linear prediction. However, a predictive model that describes the saturation of the response to higher amplitudes of forcing in stable laminar flows is still missing.While an asymptotic analysis based on the weakly nonlinear theory shows qualitative agreement only for very small forcing amplitudes, the linear response to harmonic forcing around the DNS mean flow presents a good prediction of the saturation also at higher forcing amplitudes. These results suggest that the saturation process is governed by the Reynolds stress and thus motivate the introduction of a simple self-consistent model.The model consists of a decomposition of the full nonlinear Navier-Stokes equations in a mean flow equation together with a linear perturbation equation around the mean flow, which are coupled through the Reynolds stress. The full fluctuating response and the resulting Reynolds stress are approximated by the first harmonic calculated from the linear response to the forcing around the aforementioned mean flow. This closed set of coupled equations is solved in an iterative manner as partial nonlinearity is still preserved in the mean flow equation despite the assumed simplifications.The results show an accurate prediction of the response energy when compared to Direct Numerical Simulations (DNS). The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalised, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.

Selective noise amplifiers are characterized by large linear amplification to external perturbations in a particular frequency range despite their global linear stability. Applying a stochastic forcing with increasing amplitude, the response undergoes a strong nonlinear saturation when compared to the linear estimation. Building upon our previous work, we introduce a predictive model that describes this nonlinear dynamics, and we apply it to a canonical example of selective noise amplifiers: the backward-facing step flow. Rewriting conveniently the stochastic forcing and response in the frequency domain, the model consists in a mean flow equation coupled to the linear response to forcing at each frequency. This coupling is attained by the Reynolds stress, which is constructed by the integral in frequency of the independent responses. We generalize the model for a response to a white noise forcing δ-correlated in space and time restricting the flow dynamics to its most energetic patterns calculated from the optimal harmonic forcing and response of the flow. The model estimates accurately the response saturation when compared to direct numerical simulations, and it correctly approximates the structure of the response and the mean flow modification. It also shows that the response undergoes a selective process governed by the nonlinear gain, which promotes a response structure with an approximately single frequency and wavelength in the whole domain. These results suggest that the mean flow modification by the Reynolds stress is the key nonlinearity in the saturation process of the response to white noise.

The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the Stuart-Landau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number » Re 46 c. After having reviewed 2 c 1 1 instead of  ¢ =-Re Re Re 2 c c 2 considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech 593 333-58) correctly identified the adequate bifurcation parameter ò, they have plotted their results adding an additional linearization, which amounts to using  ¢ as approximation to ò. We then illustrate the risks of calculating 'running' Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to » Re 100. We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuartʼs multiple-scale expansion method. We then briefly review a self-consistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure.