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Response modes computed via linear resolvent analysis of a turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such resolvent models predictive, the nonlinear forcing term must be closed. Strategies to do so include imposing self-consistent sets of triadic interactions, proposing various source models or through turbulence modelling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In this study, a data-driven approach is taken to quantitatively improve linear resolvent models by deducing an optimal eddy-viscosity field that maximizes the projection of the dominant resolvent mode to the energy-optimal coherent structure educed using spectral proper orthogonal decomposition (SPOD) of data from high-fidelity simulations. We use large-eddy simulation databases for round isothermal jets at subsonic, transonic and supersonic conditions and show that the optimal eddy viscosity substantially improves the agreement between resolvent and SPOD modes, reaching over 90 % agreement at those frequencies where the jet exhibits a low-rank response. We then consider a fixed model for the eddy viscosity and show that with the calibration of a single constant, the results are generally close to the optimal one. In particular, the use of a standard Reynolds-averaged Navier–Stokes eddy-viscosity resolvent model, with a single coefficient, provides substantial agreement between SPOD and resolvent modes for three turbulent jets and across the most energetic wavenumbers and frequencies.

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.

The 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 input–output 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.

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 paper examines the three-dimensional cellular patterns appearing on wings in subsonic stall and transonic buffet conditions. Unsteady Reynolds-averaged Navier–Stokes simulations are carried out for three-dimensional infinite swept configurations closed by periodic boundary conditions in the spanwise direction. In both flow conditions the occurrence of stall/buffet cells is observed, as well as their convection at a speed proportional to the sweep angle. In transonic buffet conditions, this phenomenon is superimposed to the well-documented two-dimensional buffet instability. These results indicate that the discrepancies between two-dimensional and three-dimensional buffet are caused by the occurrence of buffet cells and that this phenomenon is similar to the one observed at low speed. These phenomena are then studied using global linear stability analysis with the assumption of a periodic flow in the spanwise direction. From these analyses a mode coherent with the two-dimensional buffet is obtained, as well as a mode coherent with two-dimensional vortex shedding in stall conditions. In addition, in both flow conditions an unstable mode reminiscent of stall/buffet cells is observed.

The transonic flow field around a supercritical airfoil is investigated. The objective of the present paper is to enhance the understanding of the physical mechanics behind two-dimensional transonic buffet. The paper is composed of two parts. In the first part, a global stability analysis based on the linearized Reynolds-averaged Navier–Stokes equations is performed. A recently developed technique, based on the direct and adjoint unstable global modes, is used to compute the local contribution of the flow to the growth rate and angular frequency of the unstable global mode. The results allow us to identify which zones are directly responsible for the existence of the instability. The technique is firstly used for the vortex-shedding cylinder mode, as a validating case. In the second part, in order to confirm the results of the first part, a selective frequency damping method is locally used in some regions of the flow field. This method consists of applying a low-pass filter on selected zones of the computational domain in order to damp the fluctuations. It allows us to identify which zones are necessary for the persistence of the instability. The two different approaches give the same results: the shock foot is identified as the core of the instability; the shock and the boundary layer downstream of the shock are also necessary zones while damping the fluctuations on the lower surface of the airfoil; and outside the boundary layer between the shock and the trailing edge or above the supersonic zone does not suppress the shock oscillation. A discussion on the several physical models, proposed until now for the buffet phenomenon, and a new model are finally offered in the last section.

The generation of discrete acoustic tones in the compressible flow around an aerofoil is addressed in this work by means of nonlinear numerical simulations and global stability analyses. The nonlinear simulations confirm the appearance of discrete tones in the acoustic spectrum and, for the chosen flow case, the global stability analyses of the mean-flow dynamics reveal that the linearized operator is stable. However, the flow response to incoming disturbances exhibits important transient growth effects that culminate in the onset of aeroacoustic feedback loops, involving instability processes on the suction- and pressure-surface boundary layers together with their cross-interaction by acoustic radiation at the trailing edge. The features of the aeroacoustic feedback loops and the appearance of discrete tones are then related to the features of the least-stable modes in the global spectrum: the spatial structure of the direct modes displays the coupled dynamics of hydrodynamic instabilities on the suction surface and in the near wake. Finally, different families of global modes will be identified and the dynamics that they represent will be discussed.

Direct numerical simulations (DNS) of the wake of a circular disk placed normal to a uniform flow show that, as the Reynolds number is increased, the flow undergoes a sequence of successive bifurcations, each state being characterized by specific time and space symmetry breaking or recovering (Fabre, Auguste & Magnaudet, Phys. Fluids, vol. 20 (5), 2008, p. 1). To explain this bifurcation scenario, we investigate the stability of the axisymmetric steady wake in the framework of the global stability theory. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers Re and characteristics of the destabilizing modes agree with the study of Natarajan & Acrivos (J. Fluid Mech., vol. 254, 1993, p. 323): the first destabilization occurs for a stationary mode of azimuthal wavenumber m = 1 at RecA = 116.9, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers m ± 1 at RecB = 125.3. Since these critical Reynolds numbers are close to one another, we use a multiple time scale expansion to compute analytically the leading-order equations that describe the nonlinear interaction of these three leading eigenmodes. This set of equations is given by imposing, at third order in the expansion, a Fredholm alternative to avoid any secular term. It turns out to be identical to the normal form predicted by symmetry arguments. Though, all coefficients of the normal form are here analytically computed as the scalar product of an adjoint global mode with a resonant third-order forcing term, arising from the second-order base flow modification and harmonics generation. We show that all nonlinear interactions between modes take place in the recirculation bubble, as the contribution to the scalar product of regions located outside the recirculation bubble is zero. The normal form accurately predicts the sequence of bifurcations, the associated thresholds and symmetry properties observed in the DNS calculations.

For moderate-to-high Reynolds numbers, aerofoils are known to produce substantial levels of acoustic radiation, known as tonal noise, which arises from a complex interplay between laminar boundary-layer instabilities, trailing-edge acoustic scattering and upstream receptivity of the boundary layers on both aerofoil surfaces. The resulting acoustic spectrum is commonly characterised by distinct equally spaced peaks encompassing the frequency range of convectively amplified instability waves in the pressure-surface boundary layer. In this work, we assess the receptivity and sensitivity of the flow by means of global stability theory and adjoint methods which are discussed in light of the spatial structure of the adjoint global modes, as well as the wavemaker region. It is found that for the frequency range corresponding to acoustic tones the direct global modes capture the growth of instability waves on the suction surface and the near wake together with acoustic radiation into the far field. Conversely, it is shown that the corresponding adjoint global modes, which capture the most receptive region in the flow to external perturbations, have compact spatial support in the pressure surface boundary layer, upstream of the separated flow region. Furthermore, we find that the relative spatial amplitude of the adjoint modes is higher for those modes whose real frequencies correspond to the acoustic peaks. Finally, analysis of the wavemaker region points at the pressure surface near 30 % of the chord as the preferred zone for the placement of actuators for flow control of tonal noise.

Columnar vortices are known to support a family of waves initially discovered by Lord Kelvin (1880) in the case of the Rankine vortex model. This paper presents an exhaustive cartography of the eigenmodes of a more realistic vortex model, the Lamb–Oseen vortex. Some modes are Kelvin waves related to those existing in the Rankine vortex, while some others are singular damped modes with a completely different nature. Several families are identified and are successively described. For each family, the underlying physical mechanism is explained, and the effect of viscosity is detailed. In the axisymmetric case (with azimuthal wavenumber $m\\,{=}\\,0$), all modes are Kelvin waves and weakly affected by viscosity. For helical modes ($m\\,{=}\\,1$), four families are identified. The first family, denoted D, corresponds to a particular wave called the displacement wave. The modes of the second family, denoted C, are cograde waves, except in the long-wave range where they become centre modes and are strongly affected by viscosity. The modes of the third family, denoted V, are retrograde, singular modes which are always strongly damped and do not exist in the inviscid limit. The modes of the last family, denoted L, are regular, counter-rotating waves for short wavelengths, but they become singular damped modes for long wavelengths. In an intermediate range of wavelengths between these two limits, they display a particular structure, with both a wave-like profile within the vortex core and a spiral structure at its periphery. This kind of mode is called a critical layer wave, and its significance is explained from both a physical and a mathematical point of view. Double-helix modes ($m\\,{=}\\,2$) can similarly be classified into the C, V and L families. One additional mode, called F, plays a particular role. For short wavelenghs, this mode corresponds to a helical flattening wave, and has a clear physical significance. However, for long wavelenghts, this mode completely changes its structure, and becomes a critical layer wave. Modes with larger azimuthal wavenumbers $m$ are all found to be substantially damped.