Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T16:22:17.269Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE STABILITY OF SWIRLING FLOWS WITH RADIUS-DEPENDENT DENSITY WITH RESPECT TO INFINITESIMAL AZIMUTHAL DISTURBANCES

Published online by Cambridge University Press:  26 March 2015

H. DATTU  and
M. SUBBIAH
H. DATTU
Affiliation:
Department of Mathematics, Pondicherry University, India email dattuhalle@gmail.com, malaisubbiah@gmail.com
M. SUBBIAH*
Affiliation:
Department of Mathematics, Pondicherry University, India email dattuhalle@gmail.com, malaisubbiah@gmail.com
*
✉malaisubbiah@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the stability of inviscid, incompressible swirling flows of variable density with respect to azimuthal, normal mode disturbances. We prove that the wave velocity of neutral modes is bounded. A further refinement of Fung’s semi-elliptical instability region is given. This new instability region depends not only on the minimum Richardson number, and the lower and upper bounds for the angular velocity like Fung’s semi-ellipse, but also on the azimuthal wave number and the radii of the inner and outer cylinders. An estimation for the growth rate of unstable disturbances is obtained and it is compared to some of the recent asymptotic results.

Keywords

hydrodynamic stabilityswirling flowsincompressible variable densityazimuthal disturbances
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 3 , January 2015 , pp. 209 - 232
Copyright
© 2015 Australian Mathematical Society 

References

Davalos-Orozco, L. A. and Vazquez-Luis, E., “Instability of the interface between two inviscid fluids inside a rotating annulus in the absence of gravity”, Phys. Fluids 15 (2003) 2728–2739 doi:10.1063/1.1597682.CrossRefGoogle Scholar
Dixit, H. N. and Govindarajan, R., “Stability of vortex in radial density stratification: role of wave interactions”, J. Fluid Mech. 679 (2011) 582–615; doi:10.1017/jfm.2011.156.CrossRefGoogle Scholar
Drazin, P. G. and Reid, W. H., Hydrodynamic stability (Cambridge University Press, Cambridge, UK, 1981); doi:10.1017/CBO9780511616938.Google Scholar
Fung, Y. T., “Non-axisymmetric instability of a rotating layer of fluid”, J. Fluid Mech. 127 (1983) 83–90; doi:10.1017/S00221120830026218.CrossRefGoogle Scholar
Fung, Y. T. and Kurzweg, U. H., “Stability of swirling flows with radius-dependent density”, J. Fluid Mech. 72 (1975) 243–255; doi:10.1017/S0022112075003321.Google Scholar
Howard, L. N., “Note on a paper of John W. Miles”, J. Fluid Mech. 10 (1961) 509–512 ; doi:10.1017/S0022112061000317.Google Scholar
Howard, L. N. and Gupta, A. S., “On the hydrodynamic and hydromagnetic stability of swirling flows”, J. Fluid Mech. 14 (1962) 463–476; doi:10.1017/S0022112062001366.Google Scholar
Kochar, G. T. and Jain, R. K., “Note on Howard’s semicircle theorem”, J. Fluid Mech. 91 (1979) 489–491; doi:10.1017/S0022112079000276.Google Scholar
Makov, Yu. N. and Stepanyants, A., “Note on the paper of Kochar and Jain on Howard’s semicircle theorem”, J. Fluid Mech. 140 (1984) 1–10; doi:10.1017/S0022112084000471.CrossRefGoogle Scholar
Maslowe, S. A. and Nigam, N., “The nonlinear critical layer for Kelvin modes on a vortex with a continuous velocity profile”, SIAM J. Appl. Math. 68 (2008) 825–843; doi:10.1137/060658515.Google Scholar
Pierro, B. D. and Abid, M., “Instabilities of variable-density swirling flows”, Phys. Rev. E 82 (2010) 046312; doi:10.1103/PhysRevE.82.046312.Google Scholar
Pierro, B. D. and Abid, M., “Rayleigh–Taylor instability in variable density swirling flows”, Eur. Phys. J. B 85 (2012) 1–8; doi:10.1140/epjb/e2012-20540-6.Google Scholar
Shukhman, I. G., “Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid”, J. Fluid Mech. 200 (1989) 425–450; doi:10.1017/S0022112089000728.CrossRefGoogle Scholar
Spalart, P. R., “Airplane trailing vortices”, Annu. Rev. Fluid Mech. 30 (1998) 107–138 doi:10.1146/annurev.fluid.30.1.107.Google Scholar
Subbiah, M., “On the two-dimensional stability of inviscid incompressible swirling flows”, J. Anal. 17 (2009) 23–29 ; https://sites.google.com/site/journalofanalysis/volume17.Google Scholar