Fully Nonlinear Global Modes in Spatially Developing Media

International audienceGlobal modes on a doubly-infinite one-dimensional domain $-\infty < X < +\infty$ are studied in the context of the complex Ginzburg-Landau equation with slowly spatially varying coefficients. A fully nonlinear frequency selection criterion is derived for global-mode solutions under the assumption of weak inhomogeneity of the medium. The global mode is found to be governed by the fully nonlinear equations in a region of finite size, and by the linearized equations in the vicinity of $X=\pm\infty$. Asymptotic matching techniques are used to relate the WKB approximations in the linear and nonlinear regions through appropriate transition layers. The real global frequency is determined by requiring that spatial branches issuing from $X=-\infty$ and $X=+\infty$ be continuously connected at a saddle point of the local nonlinear dispersion relation $\omega=\Omega^{nl}(k,R,X)$ between the frequency $\omega$, the wavenumber $k$ and amplitude $R$ at a given station $X$. The results constitute a fully nonlinear generalization of the linear frequency selection criteria previously obtained by Chomaz et al. (1991), Monkewitz et al. (1993), and Le Dizès et al. (1996)

Nonlinear synchronization in open flows

International audienceThe selection criteria governing finite-amplitude synchronized oscillating states are discussed for model systems and real wake flows in a domain of infinite streamwise extent. Two types of nonlinear global modes are possible: hat modes with overall smoothly varying amplitude and elephant modes with a sharp front. The vortex street in wake flows is of elephant type, as observed in direct numerical simulations of a real spatially developing wake. Furthermore, the elephant frequency selection criterion is in excellent agreement with the numerically determined vortex shedding frequency

Linear impulse response in hot round jets

International audienceThe linear impulse response is retrieved from a numerical solution of the spatial eigenvalue problem, which is derived from the fully compressible equations of motion. Changes in the spatiotemporal stability of heated versus isothermal jets are shown to arise solely from the effect of the baroclinic torque. By considering the full linear impulse response, the competition between jet column modes and shear layer modes is characterized. Jet column modes are only found to occur for axisymmetric disturbances. In thin shear layer jets, the jet column mode is shown to prevail at low group velocities, whereas axisymmetric and helical shear layer modes dominate at high group velocities. The absolute mode of zero group velocity is found to always be of the jet column type. Although only convectively unstable, the maximum growth rates of the shear layer modes greatly exceed those of the jet column modes in thin shear layer jets. In thick shear layer jets, axisymmetric modes of mixed jet column/shear layer type arise. The weakened maximum growth rate of mixed modes accounts for the dominance of helical modes in temporal stability studies of thick shear layer jets. © 2007 American Institute of Physics

Nonlinear self-sustained structures and fronts in spatially developing wake flows

International audienceA family of slowly spatially developing wakes with variable pressure gradient is numerically demonstrated to sustain a synchronized finite-amplitude vortex street tuned at a well defined frequency. This oscillating state is shown to be described by a steep global mode exhibiting a sharp Dee--Langer type front at the streamwise station of marginal absolute instability. The front acts as a wavemaker which sends out nonlinear travelling waves in the downstream direction, the global frequency being imposed by the real absolute frequency prevailing at the front station. The nonlinear travelling waves are determined to be governed by the local nonlinear dispersion relation resulting from a temporal evolution problem on a local wake profile considered as parallel. Although the vortex street is fully nonlinear, its frequency is dictated by a purely linear marginal absolute instability criterion applied to the local linear dispersion relation

Low-frequency sound radiated by a nonlinearly modulated wavepacket of helical modes on a subsonic circular jet

International audienceA possible fundamental physical mechanism by which instability modes generate sound waves in subsonic jets is presented in the present paper. It involves a wavepacket of a pair of helical instability modes with nearly the same frequencies but opposite azimuthal wavenumbers. As the wavepacket undergoes simultaneous spatialtemporal development in a circular jet, the mutual interaction between the helical modes generates a strong three-dimensional, slowly modulating mean-flow distortion. It is demonstrated that this mean field radiates sound waves to the far field. The emitted sound is of very low frequency, with characteristic time and length scales being comparable with those of the envelope of the wavepacket, which acts as a non-compact source. A matched-asymptotic-expansion approach is used to determine, in a self-consistent manner, the acoustic field in terms of the envelope of the wavepacket and a transfer factor characterizing the refraction effect of the background base flow. For realistic jet spreading rates, the nonlinear development of the wavepacket is found to be influenced simultaneously by non-parallelism and non-equilibrium effects, and so a composite modulation equation including both effects is constructed in order to describe the entire growthattenuationdecay cycle. Parametric studies pertaining to relevant experimental conditions indicate that the acoustic field is characterized by a single-lobed directivity pattern beamed at an angle about 4560 to the jet axis and a broadband spectrum centred at a Strouhal number St 0.070.2. As the nonlinear effect increases, the radiation becomes more efficient and the noise spectrum broadens, but the gross features of the acoustic field remain robust, and are broadly in agreement with experimental observations. © 2009 Copyright Cambridge University Press

Aerodynamic sound generation by global modes in hot jets

International audienceThe acoustic field generated by the synchronized vortex street in self-excited hot subsonic jets is investigated via direct numerical simulation of the compressible equations of motion in an axisymmetric geometry. The simulation simultaneously resolves both the aerodynamic near field and the acoustic far field. Self-sustained near-field oscillations in the present flow configurations have been described as nonlinear global modes in an earlier study. The associated acoustic far field is found to be that of a compact dipole, emanating from the location of vortex roll-up. A far-field solution of the axisymmetric Lighthill equation is derived, on the basis of the source term formulation of Lilley (AGARD-CP, vol. 131, 1974, pp. 13.1-13.12). With the near-field source distributions obtained from the direct numerical simulations, the Lighthill solution is in good agreement with the far-field simulation results. Fluctuations of the enthalpy flux within the jet are identified as the dominant aeroacoustic source. Superdirective effects are found to be negligible. © 2010 Cambridge University Press

Global linear stability of a model subsonic jet

The global stability of a subsonic jet is investigated using a model base flow designed to fit experimental results for turbulent mean flows. Eigenmodes are computed for axisymmetric perturbations in order to investigate the nature of typically observed large-scale coherent oscillations ("preferred mode"). We do not find evidence that this preferred mode corre- sponds to the least damped global mode. Non-modal stability is also considered through the computation of optimal perturbations. Although non-axisymmetric perturbations (in particular for azimuthal wavenumber m = 1) are subject to larger transient growth, these reach their peak amplitude far downstream of the potential core, and therefore they are less likely to be observed

Yves Couder (1941–2019): A life in search of the beauty of fluid motion

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Optimal disturbances in swept Hiemenz flow

International audienceThe initial perturbation with the largest transient energy growth is computed in the context of the swept leading-edge boundary layer. The highest energy amplification is found for perturbations which are homogeneous in the spanwise z-direction, although on shorter time scales the most amplified disturbances have a finite spanwise wavenumber. In both cases the production term associated with the shear of the spanwise velocity is responsible for the energy amplification in the perturbation energy equation. A connection is made with the amplification mechanism exhibited by optimal perturbations in streaky boundary layers (Hoepffner et al. J. Fluid Mech. vol. 537, 2005, p.91) and the results are compared to the optimal Görtler-Hämmerlin disturbances computed by Guégan et al. (J. Fluid Mech. vol. 566, 2006, p. 11). © Cambridge University Press 2007

Frequency selection in globally unstable round jets

International audienceThe self-sustained formation of synchronized ring vortices in hot subsonic jets is investigated by direct numerical simulation of the axisymmetric equations of motion. The onset of global instability and the global frequency of synchronized oscillations are examined as functions of the ambient-to-jet temperature ratio and the initial jet shear layer thickness. The numerical results are found to follow the predictions from nonlinear global instability theory; global instability sets in as the unperturbed flow is absolutely unstable over a region of finite streamwise extent at the inlet, and the global frequency near the global instability threshold corresponds to the absolute frequency of the inlet profile. In strongly supercritical thin shear layer jets, however, the simulations display global frequencies well above the absolute frequency, in agreement with experimental results. The inner structure of rolled-up vortices in hot jets displays fine layers of positive and negative vorticity that are produced and maintained by the action of the baroclinic torque. © 2007 American Institute of Physics