Bifurcations and symmetry breaking in the wake of axisymmetric bodies

Fabre, David; Auguste, Franck; Magnaudet, Jacques

doi:10.1063/1.2909609

Cited by 160 publications

(226 citation statements)

References 7 publications

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“…2. These results are in agreement with the global stability analysis carried out by Natarajan and Acrivos [7] in the context of the incompressible §ow past disks and spheres, for which mode B dominates the dynamics of the fully 3D §ow at large Reynolds numbers, and triggers the occurrence of a fully 3D periodic state [8,9]. Consequently, in the following, let focus on the e¨ect of open-loop control on the growth rate σ B of the oscillating mode B.…”

Section: Global Stability Analysissupporting

confidence: 73%

Adjoint-based control of afterbody flow unsteadiness in the compressible regime

Ph.

Sipp

Chomaz

2012

Progress in Flight Physics

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The theoretical study of a compressible afterbody §ow in the subsonic regime is presented. It relies on sensitivity analyses developed in the framework of the linear global stability theory and aims at predicting beforehand the e¨ect of a steady bulk and wall forcing on the growth rate of linear global modes. Such an analysis stands as a step in the perspective of a full control of afterbody §ow unsteadiness. The sensitivity functions are derived analytically using adjoint methods and presented for the global mode responsible for the onset of a periodic regime. Various control methods are investigated, including an additional control device in the lee of the main body, heat sources, and boundary forcing. The obtained results show that the global mode is sensitive to momentum forcing along the separation line, to a localized heating in the core of the recirculating bubble, and to a blowing velocity close to the edge of the base.

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Section: Global Stability Analysissupporting

confidence: 73%

Adjoint-based control of afterbody flow unsteadiness in the compressible regime

Ph.

Sipp

Chomaz

2012

Progress in Flight Physics

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“…In their study, Fabre et al (2008) have deduced, from symmetry breaking considerations, a normal form of structure similar to (1.2) and estimated all coefficients by a trial and error procedure based on their DNS calculations. However, for the present codimension two bifurcation, the determination of the exact normal form (1.2) requires to compute 15 real coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…We consider here the wake past a flat circular disk of diameter D, placed orthogonally with respect to a uniform flow of velocity U ∞ , at low Reynolds numbers Re < 150, where Re = DU ∞ /ν and ν is the kinematic viscosity of the fluid. This problem has been recently addressed using DNS calculations by Fabre, Auguste & Magnaudet (2008). When the Reynolds number is increased from sufficiently low values, the flow undergoes several successive bifurcations.…”

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.

111

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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“…Following Ref. 28, we normalize each mode in such a way that the lift force it exerts on the bubble is unity.…”

Section: B Perturbationsmentioning

confidence: 99%

Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble

2013

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The stability properties of the wake past an oblate spheroidal bubble held fixed in a uniform stream are studied in the framework of a global linear analysis. In line with previous studies, provided the geometric aspect ratio of the bubble, χ, is large enough, the wake is found to be unstable only within a finite range of Reynolds number, Re. The neutral curves corresponding to the occurrence of the first two unstable modes are determined over a wide range of the (χ, Re) domain and the structure of the modes encountered along the two branches of each neutral curve is discussed. Then, using an adjoint-based approach, a series of sensitivity analyses of the flow past the bubble is carried out in the spirit of recent studies devoted to twodimensional and axisymmetric rigid bodies. The regions of the flow most sensitive to an external forcing are found to be concentrated in the core or at the periphery of the standing eddy, as already observed with bluff bodies at the surface of which the flow obeys a no-slip condition. However, since the shear-free condition allows the fluid to slip along the bubble surface, the rear half of this surface turns out to be also significantly sensitive to disturbances originating in the normal and shear stresses, a finding which may be related to the well-known influence of surfactants on the structure and stability properties of the flow past bubbles rising in water.

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Bifurcations and symmetry breaking in the wake of axisymmetric bodies

Cited by 160 publications

References 7 publications

Adjoint-based control of afterbody flow unsteadiness in the compressible regime

Adjoint-based control of afterbody flow unsteadiness in the compressible regime

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

Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble

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