Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T23:32:21.090Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  22 April 2020

Paul Billant Open the ORCID record for Paul Billant [Opens in a new window]  and
Julien Bonnici
Paul Billant*
Affiliation:
LadHyX, CNRS, École polytechnique, 91128Palaiseau CEDEX, France
Julien Bonnici
Affiliation:
LadHyX, CNRS, École polytechnique, 91128Palaiseau CEDEX, France
*
†Email address for correspondence: billant@ladhyx.polytechnique.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

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 $Re$, based on the circulation $\unicode[STIX]{x1D6E4}$ and radius $a_{0}$ of the vortex, have been varied in the ranges: $0.1\leqslant F_{h}\leqslant 0.5$ and $3000\leqslant Re\leqslant 10\,000$. The shear flow amplitude $\hat{U} _{S}$ and vertical wavenumber $\hat{k}_{z}$ lie in the ranges: $0.02\leqslant 2\unicode[STIX]{x03C0}a_{0}\hat{U} _{S}/\unicode[STIX]{x1D6E4}\leqslant 0.4$ and $0.1\leqslant \hat{k}_{z}a_{0}\leqslant 2\unicode[STIX]{x03C0}$. 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 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.

JFM classification

Geophysical and Geological Flows: Stratified flows Instability: Transition to turbulence Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 893 , 25 June 2020 , A18
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Augier, P. & Billant, P. 2011 Onset of secondary instabilities on the zigzag instability in stratified fluids. J. Fluid Mech. 682, 120–131.CrossRefGoogle Scholar
Augier, P., Billant, P. & Chomaz, J.-M. 2015 Stratified turbulence forced with columnar dipoles: numerical study. J. Fluid Mech. 769, 403–443.CrossRefGoogle Scholar
Augier, P., Billant, P., Negretti, M. E. & Chomaz, J.-M. 2014 Experimental study of stratified turbulence forced with columnar dipoles. Phys. Fluids 26, 046603.CrossRefGoogle Scholar
Augier, P., Chomaz, J.-M. & Billant, P. 2012 Spectral analysis of the transition to turbulence from a dipole in stratified fluid. J. Fluid Mech. 713, 86–108.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Bonnici, J.2018 Décorrélation verticale d’un tourbillon soumis à un champ de cisaillement dans un fluide fortement stratifié. PhD thesis, LadHyX, Université Paris-Saclay.Google Scholar
Bonnici, J. & Billant, P. 2020 Evolution of a vortex in a strongly stratified shear flow. Part 1: Asymptotic analysis. J. Fluid Mech. 893, A17.Google Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Deloncle, A., Billant, P. & Chomaz, J.-M. 2008 Nonlinear evolution of the zigzag instability in stratified fluids : a shortcut on the route to dissipation. J. Fluid Mech. 599, 299–239.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68 (3), 036308.Google ScholarPubMed
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.2.0.CO;2>CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Otheguy, P., Chomaz, J.-M. & Billant, P. 2006 Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid. J. Fluid Mech. 553, 253–272.CrossRefGoogle Scholar
Pradeep, D. S. & Hussain, F. 2004 Effects of boundary condition in numerical simulations of vortex dynamics. J. Fluid Mech. 516, 115–124.CrossRefGoogle Scholar
Rennich, S. C. & Lele, S. K. 1997 Numerical method for incompressible vortical flows with two unbounded directions. J. Comput. Phys. 137 (1), 101–129.CrossRefGoogle Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.CrossRefGoogle Scholar
Waite, M. L. 2013 The vortex instability pathway in stratified turbulence. J. Fluid Mech. 716, 1–4.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.CrossRefGoogle Scholar
Waite, M. L. & Smolarkiewicz, P. K. 2008 Instability and breakdown of a vertical vortex pair in a strongly stratified fluid. J. Fluid Mech. 606, 239–273.CrossRefGoogle Scholar