Evolution of a vortex in a strongly stratified shear flow. Part 2. Numerical simulations

Paul Billant (1) , Julien Bonnici (1)

1 LadHyX - Laboratoire d'hydrodynamique

We conduct direct numerical simulations of an initially vertical Lamb-Oseen vortex in an ambient shear flow varying sinusoidally along the vertical in a stratified fluid. The Froude number $F_h$ and the Reynolds number $\Rey$, based on the circulation $\Gamma$ and radius $a_0$ of the vortex, have been varied in the ranges: $0.1 \leqslant F_h \leqslant 0.5$ and $3000 \leqslant \Rey \leqslant 10000$. The shear flow amplitude $\hat{U}_S$ and vertical wavenumber $\hat{k}_z$ lie in the ranges: $0.02 \leqslant 2\upi a_0\hat{U}_S/\Gamma \leqslant 0.4$ and $0.1 \leqslant \hat{k}_z a_0 \leqslant 2\upi$. The results are analysed in the light of the asymptotic analyses performed in part $1$. The vortex is mostly advected in the direction of the shear flow but also in the perpendicular direction owing to the self-induction. The decay of potential vorticity is strongly enhanced in the regions of high shear. The long-wavelength analysis for $\hat{k}_z a_0 F_h \ll 1$ predicts very well the deformations of the vortex axis. The evolutions of the vertical shear of the horizontal velocity of the vortex and of the vertical gradient of the buoyancy at the location of maximum shear are also in good agreement with the asymptotic predictions when $\hat{k}_z a_0 F_h$ is sufficiently small. As predicted by the asymptotic analysis, the minimum Richardson number never goes below the critical value $1/4$ when $\hat{k}_z a_0 F_h \ll 1$. The numerical simulations show that the shear instability is triggered only when $\hat{k}_z a_0 F_h \gtrsim 1.6$ for sufficiently high buoyancy Reynolds number $ReF_h^2$. There is also a weak dependence of this threshold on the shear flow amplitude. In agreement with the numerical simulations, the long-wavelength analysis predicts that the minimum Richardson number goes below $1/4$ when $\hat{k}_z a_0 F_h \gtrsim 1.7$ although this is beyond its expected range of validity.

Domaines: Sciences de l'ingénieur [physics]

Paul Billant, Julien Bonnici. Evolution of a vortex in a strongly stratified shear flow. Part 2. Numerical simulations. Journal of Fluid Mechanics, 2020, 893, ⟨10.1017/jfm.2020.227⟩. ⟨insu-03025142⟩