Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T06:34:31.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification

Published online by Cambridge University Press:  09 February 2010

HARISH N DIXIT  and
RAMA GOVINDARAJAN
HARISH N DIXIT
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
RAMA GOVINDARAJAN*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
*
†Email address for correspondence: rama@jncasr.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

A vortex placed at a density interface winds it into an ever-tighter spiral. We show that this results in a combination of a centrifugal Rayleigh–Taylor (CRT) instability and a spiral Kelvin–Helmholtz (SKH) type of instability. The SKH instability arises because the density interface is not exactly circular, and dominates at large times. Our analytical study of an inviscid idealized problem illustrates the origin and nature of the instabilities. In particular, the SKH is shown to grow slightly faster than exponentially. The predicted form lends itself for checking by a large computation. From a viscous stability analysis using a finite-cored vortex, it is found that the dominant azimuthal wavenumber is smaller for lower Reynolds number. At higher Reynolds numbers, disturbances subject to the combined CRT and SKH instabilities grow rapidly, on the inertial time scale, while the flow stabilizes at low Reynolds numbers. Our direct numerical simulations are in good agreement with these studies in the initial stages, after which nonlinearities take over. At Atwood numbers of 0.1 or more, and a Reynolds number of 6000 or greater, both stability analysis and simulations show a rapid destabilization. The result is an erosion of the core, and breakdown into a turbulence-like state. In studies at low Atwood numbers, the effect of density on the inertial terms is often ignored, and the density field behaves like a passive scalar in the absence of gravity. The present study shows that such treatment is unjustified in the vicinity of a vortex, even for small changes in density when the density stratification is across a thin layer. The study would have relevance to any high-Péclet-number flow where a vortex is in the vicinity of a density-stratified interface.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 415 - 439
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Bajer, K., Bassom, A. P. & Gilbert, A. D. 2001 Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411.CrossRefGoogle Scholar
Bajer, K., Bassom, A. P. & Gilbert, A. D. 2004 Vortex motion in a weak background shear flow. J. Fluid Mech. 509, 281–304.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167–188.CrossRefGoogle Scholar
Brandt, L. K. & Nomura, K. K. 2007 The physics of vortex merger and the effects of ambient stable stratification. J. Fluid Mech. 592, 413–446.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Coquart, L., Sipp, D. & Jacquin, L. 2005 Mixing induced by Rayleigh–Taylor instability in a vortex. Phys. Fluids 17, 021703.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duff, R. E., Harlow, F. H. & Hirt, C. W. 1962 Effects of diffusion on interface instability between gases. Phys. Fluids 5, 417–425.CrossRefGoogle Scholar
Flohr, P. & Vassilicos, J. C. 1997 Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, 295–317.CrossRefGoogle Scholar
Fontane, J. & Joly, L. 2008 The stability of the variable-density Kelvin–Helmholtz billow. J. Fluid Mech. 612, 237–260.CrossRefGoogle Scholar
Fung, Y. T. 1983 Non-axisymmetric instability of a rotating layer of fluid. J. Fluid Mech. 127, 83–90.CrossRefGoogle Scholar
Fung, Y. T. & Kurzweg, U. H. 1975 Stability of swirling flows with radius-dependent density. J. Fluid Mech. 72, 243–255.CrossRefGoogle Scholar
Gilbert, A. D. 1988 Spiral structures and spectra in two-dimensional turbulence. J. Fluid Mech. 193, 475–497.Google Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic Press.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Itano, T. 2004 Stability of an elliptic flow with a horizontal axis under stable stratification. Phys. Fluids 16, 1164–1167.CrossRefGoogle Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh–Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415–431.Google Scholar
Josserand, Ch. & Rossi, M. 2007 The merging of two co-rotating vortices: a numerical study. Eur. J. Mech. B Fluids 26, 779–794.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.Google Scholar
Leibovich, S. 1969 Stability of density stratified rotating flows. AIAA J. 7, 177–178.CrossRefGoogle Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Miyazaki, T. & Fukumoto, Y. 1992 Three dimensional instability of strained vortices in a stably stratified fluid. Phys. Fluids A 4, 2515–2522.CrossRefGoogle Scholar
Moffatt, H. K. & Kamkar, H. 1983 The time-scale associated with flux expulsion. In Stellar and Planetary Magnetism (ed. Soward, A. M.), pp. 91–97. Gordon and Breach.Google Scholar
Neilsen, A. H., He, X., Rasmussen, J. J. & Bohr, T. 1996 Vortex merging and spectral cascade in two-dimensional flows. Phys. Fluids 8, 2263–2265.CrossRefGoogle Scholar
Reinaud, J., Joly, L. & Chassaing, P. 2000 The baroclinic secondary instability of the two-dimensional shear layer. Phys. Fluids 12, 2489–2505.CrossRefGoogle Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.CrossRefGoogle Scholar
Saunders, P. M. 1973 The instability of a baroclinic vortex. J. Phys. Oceanogr. 3, 61–65.2.0.CO;2>CrossRefGoogle Scholar
Sipp, D., Fabre, D., Michelin, S. & Jacquin, L. 2005 Stability of a vortex with a heavy core. J. Fluid Mech. 526, 67–76.CrossRefGoogle Scholar
Villermaux, E. 1998 On the role of viscosity in shear instabilities. Phys. Fluids 10, 368–373.CrossRefGoogle Scholar
Yih, C. S. 1961 Dual role of viscosity in the instability of revolving fluids of variable density. Phys. Fluids 4, 806–811.CrossRefGoogle Scholar