This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Kármán vortex street regime that sets in for supercritical Reynolds numbers Re > 46. Barkley (Europhys. Lett. vol.75, 2006, p. 750) has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range 46 < Re < 180. The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number Rec based on the small parameter ε = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = ε λA − εμ′ A|A|2. Here A is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term μ′, we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

A general theoretical formalism is developed to assess how base-flow modifications may alter the stability properties of flows studied in a global approach of linear stability theory. It also comprises a systematic approach to the passive control of globally unstable flows by the use of small control devices. This formalism is based on a sensitivity analysis of any global eigenvalue to base-flow modifications. The base-flow modifications investigated are either arbitrary or specific ones induced by a steady force. This leads to a definition of the so-called sensitivity to base-flow modifications and sensitivity to a steady force. These sensitivity analyses are applied to the unstable global modes responsible for the onset of vortex shedding in the wake of a cylinder for Reynolds numbers in the range 47≤Re≤80. First, it is demonstrated how the sensitivity to arbitrary base-flow modifications may be used to identify regions and properties of the base flow that contribute to the onset of vortex shedding. Secondly, the sensitivity to a steady force determines the regions of the flow where a steady force acting on the base flow stabilizes the unstable global modes. Upon modelling the presence of a control device by a steady force acting on the base flow, these predictions are then extensively compared with the experimental results of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, p. 71). A physical interpretation of the suppression of vortex shedding by use of a control cylinder is proposed in the light of the sensitivity analysis.

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

International audienceThe control of separated fluid flow by reduced-order models is studied using the two-dimensional incompressible flow over an open square cavity at Reynolds numbers where instabilities are present. Actuation and measurement locations are taken on the upstream and downstream edge of the cavity. A bi-orthogonal projection is introduced to arrive at reduced-order models for the compensated problem. Global modes, proper orthogonal decomposition (POD) modes and balanced modes are used as expansion bases for the model reduction. The open-loop behaviour of the full and the reduced systems is analysed by comparing the respective transfer functions. This analysis shows that global modes are inadequate to sufficiently represent the inputoutput behaviour whereas POD and balanced modes are capable of properly approximating the exact transfer function. Balanced modes are far more efficient in this process, but POD modes show superior robustness. The performance of the closed-loop system corroborates this finding: while reduced-order models based on POD are able to render the compensated system stable, balanced modes accomplish the same with far fewer degrees of freedom. © 2009 Cambridge University Press

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier–Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows.