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

Abstract

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.

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

“…Our results, however, being obtained in the highly idealized model of point vortices, are not expected to be quantitatively valid. Still, experimental observations of a confined Bénard-von Kármán vortex street [34] in the absence of advection have recently vindicated old predictions for the stability of these confined vortex streets based on von Kármán's point vortex model [50]. Also, our previous results reconciling Kármán's point vortex model with ubiquitous observations of vortex streets [31] further supports the utility of models of point vortices for studying hydrodynamic instabilites.…”

“…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.…”

“…We have previously shown [31] that this is the case for the Kármán street of point vortices [33] modelling vortex shedding phenomena: the well known (secondary) instability [22] was shown to be strongly convective when applied to wakes [31], thus reconciling the intrinsic instability of Kármán's point-vortex model with ubiquitous observations of vortex shedding behind obstacles. It has been recently confirmed experimentally [34] that this instability can be also stabilized by strong confinement, even in the absence of mean advection.…”

“…But ultimately, it should be the failure or success of experimental tests which shall eventually confirm or falsify our predictions. In this regard, soap films or an adaptation of the recent experiments [34] proving the stability of confined vortex streets seem promising, Appendix A: Legendre transform method applied to Kármán's street of point vortices…”

“…As hinted above, and given that our criterion favors long domains, it appears essential that the different streams be drag-and not pressure-driven, for otherwise the flow will likely result in a large stagnant region. The recent experiments testing the stabilization by confinement of stationary Kármán streets [34] could be adapted to test our prediction.…”