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Person# Vladislav Mantic Lugo

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François Gallaire, Vladislav Mantic Lugo

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 d-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.

Cristobal Manuel Arratia Martinez, Edouard Boujo, François Gallaire, Vladislav Mantic Lugo, Philippe Meliga

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-c approximate to 46. After having reviewed Stuart's weakly nonlinear multiple-scale expansion method, we first demonstrate the crucial importance of the choice of the critical parameter. For the wake behind a cylinder considered in this paper, choosing. is an element of(2) = Re-c(1)-Re-c(-1) instead of is an element of'(2) = Re-Re-c/Re-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 is an element of, they have plotted their results adding an additional linearization, which amounts to using. is an element of' as approximation to is an element of'. 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 approximate to 100. We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuart's 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.

François Gallaire, Vladislav Mantic Lugo

Certain flows denominated as amplifiers arc characterized 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 mean flow computed by direct numerical simulations 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 wan flow equation together with a linear perturbation equation around the mean flow, which arc 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. The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalized, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.