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Non-linear Kelvin–Helmholtz instability

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh  and
William S. Saric
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California
William S. Saric
Affiliation:
Sandia Laboratories, Albuquerque, New Mexico
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Abstract

A non-linear analysis is presented for the stability of a liquid film adjacent to a compressible gas and under the influence of a body force directed either outward from or toward the liquid. The effects of the liquid's surface tension are taken into account. The non-linear Rayleigh–Taylor instability is included as a special case. The analysis considers the case of an inviscid liquid adjacent to a subsonic flow and the case of a very viscous liquid adjacent to a subsonic or a supersonic flow. For a subsonic external flow, it is found that the cut-off wave-number is amplitude dependent in the inviscid case whereas it is amplitude independent in the viscous case. It is found that the non-linear motion of the gas may be stabilizing or destabilizing, whereas the non-linear motion of the liquid is found to be stabilizing in the viscous case. For a supersonic external flow and a viscous liquid, the cut-off wave-number is amplitude dependent. Moreover, unstable disturbances with wave-numbers near the cut-off wave-number do not grow indefinitely with time but achieve a steady-state amplitude.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 46 , Issue 2 , 29 March 1971 , pp. 209 - 231
Copyright
© 1971 Cambridge University Press

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References

Barakat, R. & Houston, A. 1968 Non-linear periodic capillary-gravity waves on a fluid of finite depth J. Geophys. Res. 73, 6545.Google Scholar
Benjamin, T. Brooke 1959 Shearing flow over a wavy boundary J. Fluid Mech. 6, 161.Google Scholar
Benney, D. J. 1962 Non-linear gravity wave interactions J. Fluid Mech. 14, 577.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chang, I. D. & Russell, P. E. 1965 Stability of a liquid layer adjacent to a high-speed gas stream Phys. Fluids, 8, 1018.Google Scholar
Craik, A. D. D. 1966 Wind-generated waves in thin liquid films J. Fluid Mech. 26, 369.Google Scholar
Drazin, P. G. 1970 Kelvin—Helmholtz instability of finite amplitude J. Fluid Mech. 42, 321.Google Scholar
Liepmann, H. W. & Roshko, A. 1957 Elements of Gasdynamics. New York: John Wiley.
McGoldrick, L. F. 1965 Resonant interactions among capillary-gravity waves J. Fluid Mech. 21, 305.Google Scholar
McGoldrick, L. F. 1970 On Wilton's ripples J. Fluid Mech. 42, 193.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows J. Fluid Mech. 3, 185.Google Scholar
Miles, J. W. 1962 On the generation of surface waves by shear flows. Part 4 J. Fluid Mech. 13, 433.Google Scholar
Nayfeh, A. H. 1965 A perturbation method for treating nonlinear oscillation problems J. Math & Phys. 44, 368.Google Scholar
Nayfeh, A. H. 1968 Forced oscillations of the van der Pol oscillator with delayed amplitude limiting IEEE Trans. on Circuit Theory, 15, 192.Google Scholar
Nayfeh, A. H. 1969 On the non-linear Lamb—Taylor instability J. Fluid Mech. 38, 619.Google Scholar
Nayfeh, A. H. 1970a Finite amplitude surface waves in a liquid layer. J. Fluid Mech. 40, 671.Google Scholar
Nayfeh, A. H. 1970b Second harmonic resonance in the interaction of capillary and gravity waves. J Fluid Mech, submitted for publication.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions J. Fluid Mech. 9, 193.Google Scholar
Pierson, W. J. & Fife, P. 1961 Some non-linear properties of long-crested periodic waves with length near 2·44 centimeters J. Geophys. Res. 66, 163.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of an incompressible heavy fluid of variable density Proc. Roy. Soc. 14, 170.Google Scholar
Schooley, A. H. 1960 Double, triple, and higher-order dimples in the profiles of wind-generated water waves in the capillary-gravity transition region J. Geophys. Res. 65, 4075.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. Roy. Soc. A 201, 192.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. New York: Academic.
Willson, A. J. & Chang, I. D. 1967 Comments on ‘Stability of a liquid layer adjacent to a high-speed gas stream’. Phys. Fluids, 10, 2285.Google Scholar
Wilton, J. R. 1915 On ripples Phil. Mag. 29, 688.Google Scholar