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Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability

Pullin, D. I. (1982) Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability. Journal of Fluid Mechanics, 119 . pp. 507-532. ISSN 0022-1120. http://resolver.caltech.edu/CaltechAUTHORS:20181120-153100022

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Abstract

We consider the behaviour of an interface between two immiscible inviscid incompressible fluids of different density moving under the action of gravity, inertial and interfacial tension forces. A vortex-sheet model of the exact nonlinear two-dimensional motion of this interface is formulated which includes expressions for an appropriate set of integral invariants. A numerical method for solving the vortex-sheet initial-value equations is developed, and is used to study the nonlinear growth of finite-amplitude normal modes for both Kelvin-Helmholtz and Rayleigh-Taylor instability. In the absence of an interfacial or surface-tension term in the integral-differential equation that describes the evolution of the circulation distribution on the vortex sheet, it is found that chaotic motion of, or the appearance of curvature singularities in, the discretized interface profiles prevent the simulations from proceeding to the late-time highly nonlinear phase of the motion. This unphysical behaviour is interpreted as a numerical manifestation of possible ill-posedness in the initial-value equations equivalent to the infinite growth rate of infinitesimal-wavelength disturbances in the linearized stability theory. The inclusion of an interfacial tension term in the circulation equation (which stabilizes linearized short-wavelength perturbations) was found to smooth profile irregularities but only for finite times. While coherent interfacial motion could then be followed well into the nonlinear regime for both the Kelvin-Helmholtz and Rayleigh-Taylor modes, locally irregular behaviour eventually reappeared and resisted subsequent attempts at numerical smoothing or suppression. Although several numerical and/or physical mechanisms are discussed that might produce irregular behaviour of the discretized interface in the presence of an interfacial-tension term, the basic cause of this instability remains unknown. The final description of the nonlinear interface motion thus awaits further research.


Item Type:Article
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https://doi.org/10.1017/s0022112082001463DOIArticle
Additional Information:© 1982 Cambridge University Press. (Received 5 January 1981 and in revised form 9 November 1981) The author would like to thank Cr J. Y. Holyer of Topexpress Limited, who kindly supplied him with detailed solutions for interfacial gravity waves, and also Dr R. H. J. Grimshaw of the Department of Mathematics, University of Melbourne, who provided useful comments on the original manuscript.
Record Number:CaltechAUTHORS:20181120-153100022
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20181120-153100022
Official Citation:Pullin, D. (1982). Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability. Journal of Fluid Mechanics, 119, 507-532. doi:10.1017/S0022112082001463
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:91098
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:21 Nov 2018 15:30
Last Modified:21 Nov 2018 15:30

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