Spatio-temporal stability of the Kármán vortex street and the effect of confinement

Mowlavi, Saviz; Arratia, Cristóbal; Gallaire, François

doi:10.1017/jfm.2016.195

Cited by 11 publications

(23 citation statements)

References 33 publications

(46 reference statements)

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“…Nevertheless, point-vortex models are a good first approximation for 2D vortex dynamics at large scales; these large scales are the least affected by viscous effects, and they are indeed the ones involved in the 2VPI. Regarding the stability of viscous shear flows, the pertinence of point-vortex models is further supported by our previous results [31] on the spatiotemporal stability of the Kármán street, and by recent experiments [34] showing the stabilizing effect of confinement on vortex streets. These experiments have validated results from the 1920's [50] on the stability of streets of point vortices.…”

Section: Shortcomings and Applicationssupporting

confidence: 70%

“…In this appendix, we present the application of the method developed in Section III A to a system with a more complicated dispersion relation than that of the confined single row. Specifically, we consider the dispersion relation of the unconfined and inviscid Kármán street of point vortices [22,31,33] given by…”

Section: Discussionmentioning

confidence: 99%

“…In the words of Huerre [28], "primary and secondary instabilities arising in fluid flows need not have the same absolute/convective character". Some absolute/convective analyses of secondary instabilities include those of the Ekhaus and zig-zag instabilities [29], the subharmonic instability of a periodic array of vortex rings [30] or the von Karman street of alternating point vortices [31]. There are no major difficulties when the secondary instability is convective.…”

Section: Absolute/convective Secondary Instabilitiesmentioning

confidence: 99%

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Absolute/convective secondary instabilities and the role of confinement in free shear layers

2018

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We study the linear spatiotemporal stability of an infinite row of equal point-vortices under symmetric confinement between parallel walls. These rows of vortices serve to model the secondary instability leading to the merging of consecutive (Kelvin-Helmholtz) vortices in free shear layers, allowing us to study how confinement limits the growth of shear layers through vortex pairings. Using a geometric construction akin to a Legendre transform on the dispersion relation, we compute the growth rate of the instability in different reference frames as a function of the frame velocity with respect to the vortices. This novel approach is verified and complemented with numerical computations of the linear impulse response, fully characterizing the absolute/convective nature of the instability. Similar to results by Healey on the primary instability of parallel tanh profiles [J. Fluid Mech. 623, 241 (2009)], we observe a range of confinement in which absolute instability is promoted. For a parallel shear layer with prescribed confinement and mixing length, the threshold for absolute/convective instability of the secondary pairing instability depends on the separation distance between consecutive vortices, which is physically determined by the wavelength selected by the previous (primary or pairing) instability. In the presence of counterflow and moderate to weak confinement, small (large) wavelength of the vortex row leads to absolute (convective) instability. While absolute secondary instabilities in spatially developing flows have been previously related to an abrupt transition to a complex behavior, this secondary pairing instability regenerates the flow with an increased wavelength, eventually leading to a convectively unstable row of vortices. We argue that since the primary instability remains active for large wavelengths, a spatially developing shear layer can directly saturate on the wavelength of such a convectively unstable row, by-passing the smaller wavelengths of absolute secondary instability. This provides a wavelength selection mechanism, according to which the distance between consecutive vortices should be sufficiently large in comparison with the channel width in order for the row of vortices to persist. We argue that the proposed wavelength selection criteria can serve as a guideline for experimentally obtaining plane shear layers with counterflow, which has remained an experimental challenge.

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Section: Shortcomings and Applicationssupporting

confidence: 70%

Section: Discussionmentioning

confidence: 99%

Section: Absolute/convective Secondary Instabilitiesmentioning

confidence: 99%

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Absolute/convective secondary instabilities and the role of confinement in free shear layers

2018

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“…The study of incompressible vortex dynamics is a vital subfield of fluid mechanics, especially important for the study of wakes [1], and it plays a crucial role in aerodynamics, geophysics, astrophysics, turbulence theory [2], and biofluids [3][4][5]. A surprising circumstance, remarked upon by previous authors [6], is that the study of compressible vortex dynamics has received markedly less attention even though gaining an understanding of compressible vortex flows is of fundamental interest for compressible wakes [7,8] and the theory of vortex sound and aeroacoustics [9][10][11]. There have been a few fundamental studies of compressible vortices [12][13][14][15], but many basic theoretical questions remain to be answered.…”

Section: Introductionmentioning

confidence: 99%

Speed of a von Kármán point vortex street in a weakly compressible fluid

Crowdy

Krishnamurthy

2017

Phys. Rev. Fluids

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Analytical expressions are obtained for the change in speed of translation of the von Kármán point vortex streets of given aspect ratios due to the effects of weak compressibility in subsonic flow of an isentropic fluid. We also clarify the nature of the force-free condition on a weakly compressible point vortex in equilibrium. For staggered streets, it is found that the speed of a compressible point vortex street can both increase and decrease relative to its incompressible counterpart of the same aspect ratio. Compressibility increases the speeds of streets with aspect ratios less than the critical value of 0.38187, at which no change in speed occurs to first order in the (squared) Mach number. In particular, the compressible counterpart to the neutrally stable incompressible point vortex street of aspect ratio 0.28056 is found to propagate with increased speed. Streets with aspect ratios larger than 0.38187 slow down under the effects of compressibility, with the slowdown becoming maximal at a street aspect ratio of 0.52630. On the other hand, the speed of unstaggered streets always increases, with the first-order correction in speed relative to its incompressible value increasing maximally at aspect ratios around κ = 0.36216

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“…Monkewitz [10]. The whole vortex arrangement would be in fact intrinsically unstable in its own framework [11].…”

mentioning

confidence: 99%

Absolute stability of a Bénard-von Kármán vortex street in a confined geometry

Boniface¹,

Lebon²,

Limat³

et al. 2017

EPL

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We have investigated the stability of a double vortex street, induced in a rectangular container by a tape, or a rope, moving at high speed on its free surface. Depending on the tape velocity and on the geometrical aspect ratios, three patterns of flows are observed: (1) a vortex street with recirculation of the liquid along the lateral sides of the container, (2) the same recirculation but with no stable vortex array, (3) recirculation along the bottom of the container. We have investigated the spatial structure of the vortex street and found that this system explores the phase space available inside a stability tongue predicted at the end of the 1920s by Rosenhead for point vortices in a perfect fluid. Although this very surprising result contrasts with the wellknown von Kármán unique stability condition for point vortex streets in an infinite domain, this complements the theory inside a channel of finite breadth. In this paper, we present the very first experimental confirmation of this 90-year old theory. The Bénard-Von Kármán (BVK) vortex street is a 1 structure that can be observed in the wake of an obstacle 2 immersed inside a stationary unidirectional flow of large 3 enough velocity [1]. It was investigated experimentally for 4 the first time by Bénard [2] and modeled by von Kármán 5 [3] [4] at the beginning of the XXth century and has since 6 been extensively investigated. This phenomenon is ubiq-7 uitous in meteorology, oceanography, naval engineering, 8 vehicle design, modeling of the air flows around buildings 9 or piles of bridges, etc [5]. According to Von Kármán [3], 10 a vortex street of point vortices in a plane perfect flow can 11 be stable only if the wavelength of the vortex street, i.e.

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Spatio-temporal stability of the Kármán vortex street and the effect of confinement

Cited by 11 publications

References 33 publications

Absolute/convective secondary instabilities and the role of confinement in free shear layers

Absolute/convective secondary instabilities and the role of confinement in free shear layers

Speed of a von Kármán point vortex street in a weakly compressible fluid

Absolute stability of a Bénard-von Kármán vortex street in a confined geometry

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