Bénard-von Kármán instability: transient and forced regimes

Provansal, Mireille; Mathis, Christian; Boyer, L.

doi:10.1017/s0022112087002222

Cited by 501 publications

(350 citation statements)

References 24 publications

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“…One could imagine to estimate all 15 coefficients using transient dynamics computed by DNS. However, such a procedure, although it has been used successfully for codimension one bifurcations (Provansal et al 1987), is a formidable task no one has ever attempted for multiple codimension problems. In the present study, we carry out a thoroughly analytical asymptotic expansion of the flow field based on the global modes destabilizing the axisymmetric wake.…”

Section: Introductionmentioning

confidence: 99%

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Méliga¹,

Chomaz

Sipp³

2009

J. Fluid Mech.

115

175

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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 Re A c = 116.9, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers m ± 1 at Re B c = 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.

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Section: Introductionmentioning

confidence: 99%

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Méliga¹,

Chomaz

Sipp³

2009

J. Fluid Mech.

115

175

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“…In the presence of a locally absolutely unstable flow region, in contrast, the flow may bifurcate to a global mode. Prominent examples of flows exhibiting global instability triggered by local absolute instability include the cylinder wake, [4][5][6][7] counterflowing shear layers, 8 swirling jets, 9 and jets with counterflow. 10,11 In the hot jet experiments of Monkewitz et al, 1 selfsustained synchronized oscillations were found to set in as the ambient-to-jet temperature ratio S = T ϱ / T c was lowered below a critical value of 0.73.…”

Section: Introductionmentioning

confidence: 99%

Frequency selection in globally unstable round jets

2007

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The 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.

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“…These results are confirmed by laboratory experiments. [10][11][12] Experimental 13 and numerical studies 14,15 showed that von Karman streets are a manifestation of a self-sustained oscillation ͑i.e., a global mode͒, associated to a local absolute instability region in the base flow. Hence a first step in order to understand the possible change in the nature of large-scale wake instability is to investigate the absolute versus the convective stability of local profiles.…”

Section: Introductionmentioning

confidence: 99%

Stability of parallel wake flows in quasigeostrophic and frontal regimes

et al. 2006

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International audienceRecent laboratory experiments [G. Perret, A. Stegner, M. Farge, and T. Pichon, Phys. Fluids 18, 036603 (2006)] have shown that the vortex-street formed in the wake of a towed cylinder in a rotating shallow-water layer could present a strong cyclone-anticyclone asymmetry. In extreme cases, only large-scale anticyclones were observed in the far wake. This asymmetry occurs in the so-called frontal regime when the Rossby number is small and the surface deviation is large. This asymmetry may have various origins and in particular may be attributed to the asymmetry of the flow around the cylinder, to the linear stability property of the wake, or to its nonlinear evolution. To discriminate between these mechanisms, we study the stability of two idealized parallel flows in the quasigeostrophic and in the frontal regimes. These parallel flows correspond to two velocity profiles measured just behind the cylinder in a region where the perturbations are negligible. According to our linear stability analysis, the most unstable mode, in the frontal regime, is localized in the anticyclonic shear region whether the base flow profile is symmetric or not. On a linear basis, it is thus more the instability that imposes the asymmetry than the base flow. Direct numerical simulations of the synthetic parallel wake flows show that nonlinearity exacerbates the dominance of the anticyclonic mode linearly selected. By numerically studying the spatio-temporal evolution of a small perturbation localized in space, we show that, unlike incompressible two-dimensional wake flows and the symmetric wake in the quasigeostrophic regime, the parallel asymmetric wake is strongly convectively unstable in the frontal regime, and not absolutely unstable. When the surface deformation becomes large, the wake instability changes from the absolute instability in the quasi-geostrophic regime to the strongly convective instability of the frontal regime. This explains well the changes. © 2006 American Institute of Physics

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Bénard-von Kármán instability: transient and forced regimes

Cited by 501 publications

References 24 publications

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Frequency selection in globally unstable round jets

Stability of parallel wake flows in quasigeostrophic and frontal regimes

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