Three-dimensional stability of vortex arrays in a stratified and rotating fluid

Abstract

International audienceThis paper investigates numerically and through an asymptotic approach the three-dimensional stability of steady vertical vortex arrays in a stratified and rotating fluid. Three classical vortex arrays are studied: the Kármán vortex street, the symmetric double row and the single row of co-rotating vortices. The asymptotic analysis assumes well-separated vortices and long-wavelength bending perturbations following Billant (J. Fluid Mech., vol. 660, 2010, p. 354) and Robinson & Saffman (J. Fluid Mech., vol. 125, 1982, p. 411). Very good agreement with the numerical stability analysis is found even for finite wavelength and relatively close vortices. For a horizontal Froude number Fh = 1 and for a non-rotating fluid, it is found that the Kármán vortex street for a street spacing ratio (the distance h between the rows divided by the distance b between vortices in the same row) ? = 0.41 and the symmetric double row for any spacing ratio are most unstable to a three-dimensional instability of zigzag type that vertically bends the vortices. The most amplified vertical wavenumber scales like 1/(bFh) and the growth rate scales with the strain (2pb2), where is the vortex circulation. For the Kármán vortex street, the zigzag instability is symmetric with respect to the midplane between the two rows while it is antisymmetric for the symmetric double row. For the Kármán vortex street with well-separated vortex rows ? > 0.41 and the single row, the dominant instability is two-dimensional and corresponds to a pairing of adjacent vortices of the same row. The main differences between stratified and homogeneous fluids are the opposite symmetry of the dominant three-dimensional instabilities and the scaling of their most amplified wavenumber. When Fh > 1, three-dimensional instabilities are damped by a viscous critical layer. In the presence of background rotation in addition to the stratification, symmetric and antisymmetric modes no longer decouple and cyclonic vortices are less bent than anticyclonic vortices. However, the dominant instability remains qualitatively the same for the three vortex arrays, i.e. quasi-symmetric or quasi-antisymmetric and three-dimensional or two-dimensional. The growth rate continues to scale with the strain but the most unstable wavenumber of three-dimensional instabilities decreases with rotation and scales like Ro/(bFh) for small Rossby number Ro, in agreement with quasi-geostrophic scaling laws. © 2011 Cambridge University Press

Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid

“…However, † Email address for correspondence: eunok@ladhyx.polytechnique.fr many studies have shown that vortices have a pancake or lenticular shape in stratified fluids rather than being columnar. For example, interacting columnar vortices are unstable to zigzag instability (Billant & Chomaz 2000;Otheguy, Chomaz & Billant 2006;Billant 2010;Billant et al 2010;Deloncle, Billant & Chomaz 2011) and evolve into pancake vortices with a small aspect ratio. Coherent vortices generated from wakes or turbulence in stratified fluids also have a pancake shape (Lin, Boyer & Fernando 1992;Chomaz et al 1993;Fincham, Maxworthy & Spedding 1996;Spedding, Browand & Fincham 1996;Bonnier, Eiff & Bonneton 2000).…”

Onset of secondary instabilities on the zigzag instability in stratified fluids

“…Layers can arise spontaneously through an instability, the zigzag instability, when several vertical vortices are interacting. This has been shown in the cases of pairs of counter-or co-rotating vortices (Billant & Chomaz 2000a;Otheguy, Chomaz & Billant 2006) and vortex arrays (Deloncle, Billant & Chomaz 2011). The study of such elementary flows is convenient to identify and understand some of the fundamental mechanisms at work in more complicated flows such as stratified turbulence.…”

Three-dimensional instabilities of a stratified cylinder wake

“…However, the small dimensions of the tank used in the experiments do not allow study of very small Froude numbers; it would require large cylinder diameters, which would reduce the aspect ratio below 10, leading to strong end effects. It is thus unclear what other instabilities would appear for very small Froude numbers (F < 0.1) where a zigzag instability might be observed (Deloncle, Billant & Chomaz 2011). We have tried to visualize the zigzag instability using larger cylinder diameters but it has not been observed with this experiment probably due to the small dimensions on the tank for that range of diameters (D > 32 mm).…”