Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T06:50:05.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial instability of coupled-rotating inviscid fluids

Published online by Cambridge University Press:  30 July 2013

Malek Ghantous  and
John A. T. Bye
Malek Ghantous*
Affiliation:
Centre for Ocean Engineering, Science and Technology, Swinburne University of Technology, VIC 3122, Australia
John A. T. Bye
Affiliation:
School of Earth Sciences, University of Melbourne, VIC 3010, Australia
*
†Email address for correspondence: mghantous@swin.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the three-dimensional, cylindrical equivalent to the problem of instability between two inviscid fluids due to a velocity shear between them, known as Kelvin–Helmholtz instability. We begin by developing the solution to the linearized equations for a rotating fluid. While this solution is not in itself new, we carry the analysis further than previous treatments by including non-zero modes and considering the effect of the surface tension, particularly on the dispersion relation. We then consider a system of two fluids rotating at different rates and derive its dispersion relation, which is rather more complicated than that for a single rotating fluid. While a general analytic solution is unattainable, by investigating some special cases we show that the fundamental mode is always stable, and that Kelvin–Helmholtz instability also exists for the system. We compare our results with experiments and conclude by suggesting some hypothetical links between the theory and nature.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 730 , 10 September 2013 , pp. 343 - 363
Copyright
©2013 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

Abramowitz, M. & Stegun, I. A. (Eds) 1964 Handbook of Mathematical Functions. National Bureau of Standards, United States Department of Commerce.Google Scholar
Bachelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bradford, J., Berman, A. S. & Lundgren, T. S. 1981 Nongeostrophic baroclinic instability in a two-fluid layer rotating system. J. Atmos. Sci. 38, 1376–1389.Google Scholar
Bye, J. A. T. & Ghantous, M. 2012 Observations of Kelvin–Helmholtz instability at the air–water interface in a circular domain. Phys. Fluids 24.Google Scholar
Bye, J. A. T., Hughes, R. L. & Vladusic, J. C. 2005 A nice cup of tea. Weather 60 (3), 83.Google Scholar
Chandrasekhar, S. 1961 Introduction to Hydrodynamic Stability. Oxford University Press.Google Scholar
Gauthier, G., Gondret, P., Moisy, F. & Rabaud, M. 2002 Instabilities in the flow between co- and counter-rotating disks. J. Fluid Mech. 473, 1–21.Google Scholar
Ghantous, M. 2010 Modelling of oceanic and laboratory surface waves. PhD thesis, The University of Melbourne.Google Scholar
Gula, J., Zeitlin, V. & Plougonven, R. 2009 Instabilities of two-layer shallow-water flows with vertical shear in the rotating annulus. J. Fluid Mech. 638 (1), 27–47.Google Scholar
Hart, J. E. 1972 A laboratory study of baroclinic instability. Geophys. Astrophys. Fluid Dyn. 3 (1), 181–209.CrossRefGoogle Scholar
von Helmholtz, H. 1868a On discontinuous movements of fluids. Phil. Mag. 36 (244), 337–346; translated from the German by Frederick Guthrie.CrossRefGoogle Scholar
von Helmholtz, H. 1868b Über discontinuierliche Flüssigkeits–Bewegungen. Monatsberichte der Kniglichen Preussische Akademie der Wissenschaften zu Berlin (23), 215–228.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Le Méhauté, B. 1976 An Introduction to Hydrodynamics and Water Waves. Springer.Google Scholar
Miles, J. W. 1963 Free-surface oscillations in a slowly rotating liquid. J. Fluid Mech. 18, 187–194.CrossRefGoogle Scholar
Moisy, F., Doaré, O., Pasutto, T., Daube, O. & Rabaud, M. 2004 Experimental and numerical study of the shear layer instability between two counter-rotating disks. J. Fluid Mech. 507, 175–202.Google Scholar
Nore, C., Tartar, M., Daube, O. & Tuckerman, L. S. 2004 Survey of instability thresholds of flow between exactly counter-rotating disks. J. Fluid Mech. 511, 45–65.Google Scholar
Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1 : 2 mode interaction in exactly counter-rotating von Kármán swirling flow. J. Fluid Mech. 477, 51–88.Google Scholar
Phillips, O. M. 1957 On the generation of waves by turbulent wind. J. Fluid Mech. 2, 417–445.Google Scholar
Proudman, J. 1953 Dynamical Oceanography. John Wiley & Sons.Google Scholar
Thomson, W. 1871 The influence of wind on waves in water supposed frictionless. Phil. Mag. 42, 368–370.Google Scholar
Vladusic, J. C. 2001 Wind wave growth in a circular tank. BSc Honours thesis, The University of Melbourne.Google Scholar