The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.

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GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

The stability properties of the flow past an infinitely long circular cylinder are studied in the context of linear theory. An immersed-boundary technique is used to represent the cylinder surface on a Cartesian mesh. The characteristics of both direct and adjoint perturbation modes are studied and the regions of the flow more sensitive to momentum forcing and mass injection are identified. The analysis shows that the maximum of the perturbation envelope amplitude is reached far downstream of the separation bubble, where as the highest receptivity is attained in the near wake of the cylinder, close to the body surface. The large difference between the spatial structure of the two-dimensional direct and adjoint modes suggests that the instability mechanism cannot be identified from the study of either eigenfunctions separately. For this reason a structural stability analysis of the problem is used to analyse the process which gives rise to the self-sustained mode. In particular, the region of maximum coupling among the velocity components is localized by inspecting the spatial distribution of the product between the direct and adjoint modes. Results show that the instability mechanism is located in two lobes placed symmetrically across the separation bubble, confirming the qualitative results obtained through a locally plane-wave analysis. The relevance of this novel technique to the development of effective control strategies for vortex shedding behind bluff bodies is illustrated by comparing the theoretical predictions based on the structural perturbation analysis with the experimental data of Strykowski & Sreenivasan (J. Fluid Mech. vol. 218, 1990, p. 71).

Structural sensitivity of the first instability of the cylinder wake

This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Karman 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 e = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = e λA − eμ′ 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.

Global stability of base and mean flows : a general approach and its applications to cylinder and open cavity flows

A highly accurate 2D linear stability analysis is performed on the mean flow of laminar vortex shedding from a circular cylinder for Reynolds numbers between 46 and 180. Consistent with past studies of mean profiles, the analysis shows that the eigenfrequency of the mean flow tracks almost exactly the Strouhal number of vortex shedding. The linear growth rate reveals that the wake mean flow is a marginally stable state over the whole range of Reynolds numbers for stable 2D vortex shedding. This is contrasted with 2D stability analysis about the unstable steady base flow. The relevance to nonlinear saturation and frequency selection are discussed.

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Linear analysis of the cylinder wake mean flow

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. © 2010 by ASME.

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Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

Numerical simulations of wake flow behind an equilateral triangular obstacle are presented. The form of global modes and their dependence on the Reynolds number found in this study are in accordance with recent experimental results of Goujon–Durand et al. [Phys. Rev. E 50, 308 (1994)]. A scaling law of the amplitude oscillating with the fundamental frequency corresponding to the maximum of the global mode is found to agree with the Landau model in a range of Reynolds Re numbers larger than in previous studies. The position of the maximum amplitude of the fundamental modes scales as (Re−Rec)−1/2. The amplitude of the second harmonic of the longitudinal component of the velocity as well as the correction to the mean flow have different critical behavior than the velocity components oscillating with fundamental frequency. During linear growth the position of the maximum of the global modes is constant and moves only in the nonlinear regime. The effects of the blockage and the boundary conditions on the side ...

On the spatial structure of global modes in wake flow

We present an experimental study of a planar jet confined in a rectangular cavity. In certain geometrical configurations and for sufficiently large Reynolds numbers, this system exhibits self-sustained oscillations characterized by a well-defined wavelength and frequency of the jet. We describe flow regimes observed by varying the Reynolds number and the cavity length. The self-sustained oscillation regime is studied in detail: we extract the fundamental frequency and determine the selection criterion for the wavelength using a method of visualization, which has the advantage of being non intrusive. We show the existence of a band of allowed wavelengths and establish the upper and lower limits for the wavelength selection criterion. We discuss the validity of the visualization method for the measurement of the wavelength and frequency using a simple analytical model of the streaklines. \textcopyright{} 1996 The American Physical Society.

Experimental study of self-sustained oscillations in a confined jet.

We perform experiments and numerical simulations on the mean streamwise velocity field of an unstable cylinder wake. We show that the mean velocity exhibits two weak secondary minima as the consequence of nonlinear interactions, resulting in a strong mean flow correction of the unstable mode. This correction, dominating the basic flow, governs the decrease in the length of the recirculation region $\ensuremath{\Delta}{L}_{r}$ in the supercritical regime. This explains the early classical observations of M. Nishioka and H. Sato [J. Fluid Mech. 89, 49 (1978)].

Strongly Nonlinear Effect in Unstable Wakes

We characterize experimentally and numerically the two-dimensional structure of the global mode of the longitudinal component of the velocity in flow behind obstacles of different geometrical shapes. Measurements in a low velocity water tunnel and numerical simulations of the longitudinal component V x of the oscillating velocity are carried out in the near wake region of the flow where vortex shedding is regular. We find that in the range of Reynolds numbers studied, the position (X max , Y max ) of the maximum of the global mode of V x shifts very little away from the central line in the y direction. The amplitude A max of the maximum of the global mode scales as (R-R c ) 1/2 in a range of R-R c larger then in previous studies. We also confirm that X max of the global mode scales as (R-R c ) -1/2 in the same range of Reynolds numbers. An analysis of the initial slope of the global mode is presented.

https://hal.archives-ouvertes.fr/jpa-00248374/document

Global Mode Behavior of the Streamwise Velocity in Wakes

Fluidic oscillator : application to single-phase flows flowmetry The authors have experimentally and numerically analysed self-sustained oscillations in a cavity confined stream. They have studied the dynamics of the oscillator which is at the origin of flowmeters fluidics prototypes. They determined how to choose the frequency when the fluid velocity of the flow configuration and the geometrical characteristics of the cavity are known. They also analysed the linear and the nonlinear stability.

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B. J. A. Zielinska

Papers

On the spatial structure of global modes in wake flow

Experimental study of self-sustained oscillations in a confined jet.

Strongly Nonlinear Effect in Unstable Wakes

Global Mode Behavior of the Streamwise Velocity in Wakes

Oscillateur fluidique : application à la débitmétrie des écoulements monophasiques