Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
Hostname: page-component-5c569c448b-gctlb Total loading time: 0.258 Render date: 2022-07-05T13:11:56.470Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true
  • English
Journal of Fluid Mechanics Journal of Fluid Mechanics

Article contents

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
Get access Rights & Permissions[Opens in a new window]

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.

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

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
70
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Nonlinear waves in a Kelvin-Helmholtz flow
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Nonlinear waves in a Kelvin-Helmholtz flow
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Nonlinear waves in a Kelvin-Helmholtz flow
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *