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Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid

Published online by Cambridge University Press:  14 October 2021

Axel Deloncle ,
Jean-Marc Chomaz  and
Paul Billant
Axel Deloncle
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, [email protected]
Jean-Marc Chomaz
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, [email protected]
Paul Billant
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, [email protected]
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Abstract

This paper investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, Fh, is varied from ∞ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for Fh ≪ 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 570 , 10 January 2007 , pp. 297 - 305
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Basak, S. & Sarkar, S. 2006 Dynamics of a stratified shear layer with horizontal shear. J. Fluid Mech. 568, 19–54.CrossRefGoogle Scholar
Billant, P. & Chomaz, J. M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Blumen, W. 1970 Hydrostatic neutral waves in a parallel shear flow of a stratified fluid. J. Atmos. Sci. 28, 340–344.2.0.CO;2>CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria for instability for laminar flows and for the baroclinic circular vortex. Geofys. Publ. Oslo 17 (6), 1–52.Google Scholar
Grosch, C. E. & Orszag, S. A. 1977 Numerical solution of problems in unbounded regions: coordinates transforms. J. Comput. Phys. 25, 273–296.CrossRefGoogle Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed shear layers. Annu. Rev. Fluid. Mech. 16, 365–424.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech. 19, 543–556.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Pedlosky, J. 1982 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Praud, O., Fincham, A. M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratified fluid. J. Fluid Mech. 522, 1–33.CrossRefGoogle Scholar
Rayleigh, Lord 1887 On the stability of certain fluid motions. Proc. Math. Soc. Lond. 11, 57–70.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. Math. Soc. Lond. A 142, 621–628.Google Scholar