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Three-dimensional Rayleigh-Taylor instability Part 1. Weakly nonlinear theory

Published online by Cambridge University Press:  21 April 2006

J. W. Jacobs  and
I. Catton
J. W. Jacobs
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, University of California, Los Angeles, CA 90024, USA Present address: California Institute of Technology, Pasadena, CA 91125, USA.
I. Catton
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, University of California, Los Angeles, CA 90024, USA
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Abstract

Three-dimensional weakly nonlinear Rayleigh-Taylor instability is analysed. The stability of a confined inviscid liquid and an overlying gas with density much less than that of the liquid is considered. An asymptotic solution for containers of arbitrary cross-sectional geometry, valid up to order ε3 (where ε is the root-mean-squared initial surface slope) is obtained. The solution is evaluated for the rectangular and circular geometries and for various initial modes (square, hexagonal, axisymmetric, etc.). It is found that the hexagonal and axisymmetric instabilities grow faster than any other shapes in their respective geometries. In addition it is found that, sufficiently below the cutoff wavenumber, instabilities that are equally proportioned in the lateral directions grow faster than those with longer, thinner shape. However, near the cutoff wavenumber this trend reverses with instabilities having zero aspect ratio growing faster than those with aspect ratio near 1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 329 - 352
Copyright
© 1988 Cambridge University Press

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References

Baker, L. & Freeman, J. R. 1981 Heuristic model of the nonlinear Rayleigh-Taylor instability. Phys. Fluids 52, 655–663.Google Scholar
Baker, G. R., Meiron, D. I. & Orzag, S. A. 1980 Vortex simulations of the Rayleigh-Taylor instability. Phys. Fluids 23, 1485–1490.Google Scholar
Bellman, R. & Pennington, R. H. 1954 Effects of surface tension and viscosity on Taylor instability. Q. Appl. Maths 12, 151–162.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Daly, B. J. 1967 Numerical study of two fluid Rayleigh-Taylor instability. Phys. Fluids 10, 297–307.Google Scholar
Daly, B. J. 1969 Numerical study of the effect of surface tension on interface instability. Phys. Fluids 12, 1340–1354.Google Scholar
Davis, H. T. 1962 Introduction to Nonlinear Differential and Integral Equations. Dover.
Dienes, J. K. 1978 Method of generalized coordinates and an application to Rayleigh-Taylor Instability. Phys. Fluids 21, 736–744.Google Scholar
Emmons, H. W., Chang, C. T. & Watson, B. C. 1960 Taylor instability of finite surface waves. J. Fluid Mech. 7, 177–193.Google Scholar
Fermi, E. & von Neumann, J. 1963 Taylor instability at the boundary of two incompressible liquids. In Collected Works of John von Neumann, vol. 6, pp. 431–434. Pergamon.
Harlow, F. H. & Welch, J. E. 1966 Numerical study of large-amplitude free-surface motions. Phys. Fluids 9, 842–851.Google Scholar
Jacobs, J. W. 1986 Three-dimensional Rayleigh-Taylor instability: experiment and theory. Ph.D. dissertation, University of California, Los Angeles.
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1–12.Google Scholar
Lewis, D. J. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. II. Proc. R. Soc. Lond. A 202, 81–96.Google Scholar
Menikoff, R. & Zemach, C. 1983 Rayleigh-Taylor instability and the use of conformal maps for ideal fluid flow. J. Comput. Phys. 51, 28–64.Google Scholar
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419–448.Google Scholar
Nayfeh, A. H. 1969 On the non-linear Lamb-Taylor instability. J. Fluid Mech. 38, 619–631.Google Scholar
Pullin, D. J. 1982 Numerical studies of surface-tension effects in nonlinear Kelvin-Helmholtz and Rayleigh-Taylor instability. J. Fluid Mech. 119, 507–532.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Verma, G. R. & Keller, J. B. 1962 Three-dimensional standing waves of finite amplitude. Phys. Fluids 5, 52–56.Google Scholar