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Nonlinear waves in a Kelvin-Helmholtz flow

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh  and
William S. Saric
Ali Hasan Nayfeh
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
William S. Saric
Affiliation:
Sandia Laboratories, Albuquerque, New Mexico
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Abstract

Nonlinear waves on the interface of two incompressible in viscid fluids of different densities and arbitrary surface tension are analysed using the method of multiple scales. Third-order equations are presented for the space and time variation of the wavenumber, frequency, amplitude and phase of stable waves. A third-order expansion is also given for wavenumbers near the linear neutrally stable wave-numbers. A second-order expansion is presented for wavenumbers near the second harmonic resonant wavenumber, for which the fundamental and its second harmonic have the same phase velocity. This expansion shows that this resonance does not lead to instabilities.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 55 , Issue 2 , 26 September 1972 , pp. 311 - 327
Copyright
© 1972 Cambridge University Press

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References

Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. J. Math. & Phys. 46, 133.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chang, I. D. & Russell, P. E. 1965 Stability of a liquid adjacent to a high-speed gas stream. Phys. Fluids, 8, 1018.Google Scholar
Davey, A. 1972 The propagation of a weak nonlinear wave. J. Fluid Mech. 53, 769.Google Scholar
DiPrima, R. C., Eckhaus, W. & Segel, L. A. 1971 Nonlinear wavenumber interaction in near-critical two-dimensional flows. J. Fluid Mech. 49, 705.Google Scholar
Drazin, P. G. 1970 Kelvin-Helmholtz instability of finite amplitude. J. Fluid Mech. 42, 321.Google Scholar
Gater, R. L. & L'’cuyer, M. R. 1969 A fundamental investigation of the phenomena that characterize liquid film cooling. Purdue University Rep. no. Tm-69–1.Google Scholar
Gold, H., Otis, J. H. & Schlier, R. E. 1971 Surface liquid film characteristics: an experimental study. A.I.A.A. Paper, no. 71–623.CrossRefGoogle Scholar
Kadomtsev, B. B. & Karpman, V. I. 1971 Nonlinear waves. Soviet Phys. Uspekhi, 14, 40.Google Scholar
Mcgoldrick, L. F. 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193.Google Scholar
Maslowe, S. A. & Kelly, R. E. 1970 Finite amplitude oscillations in a Kelvin-Helmholtz flow. Int. J. Non-Linear Mech. 5, 427.Google Scholar
Nayfeh, A. H. 1972a Perturbation Methods. Wiley.
Nayfeh, A. H. 1972b Second-harmonic resonance in the interaction of capillary and gravity waves. To be published.
Nayfeh, A. H. & Hassan, S. D. 1971 The method of multiple scales and nonlinear dispersive waves. J. Fluid Mech. 48, 463.Google Scholar
Nayfeh, A. H. & Saric, W. S. 1971 Nonlinear Kelvin-Helmholtz instability. J. Fluid Mech. 46, 209.Google Scholar
Saric, W. S. & Marshall, B. W. 1971 An experimental investigation of the stability of a thin liquid layer adjacent to a supersonic stream. A.I.A.A. J. 9, 1546.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529.Google Scholar
Taniuti, T. & Washimi, H. 1968 Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma. Phys. Rev. Lett. 21, 209.Google Scholar
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693.Google Scholar
Thorpe, S. A. 1969 Experiments on the instability of stratified shear flows: immiscible fluids. J. Fluid Mech. 39, 25.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Watanabe, T. 1969 A nonlinear theory of two-stream instability. J. Phys. Soc. Japan, 27, 1341.Google Scholar