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Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces

  • R. KRECHETNIKOV (a1)

Abstract

In this work we discuss a non-trivial effect of the interfacial curvature on the stability of uniformly and suddenly accelerated interfaces, such as liquid rims. The new stability analysis is based on operator and boundary perturbation theories and allows us to treat the Rayleigh–Taylor and Richtmyer–Meshkov instabilities as a single phenomenon and thus to understand the interrelation between these two fundamental instabilities. This leads, in particular, to clarification of the validity of the original Richtmyer growth rate equation and its crucial dependence on the frame of reference. The main finding of this study is the revealed and quantified influence of the interfacial curvature on the growth rates and the wavenumber selection of both types of instabilities. Finally, the systematic approach taken here also provides a generalization of the widely accepted ad hoc idea, due to Layzer (Astrophys. J., vol. 122, 1955, pp. 1–12), of approximating the potential velocity field near the interface.

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Corresponding author

Email address for correspondence: rkrechet@engineering.ucsb.edu

References

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Arnett, D. 2000 The role of mixing in astrophysics. Astrophys. J. 127 (Suppl.), 213217.
Arons, J. & Lea, S. M. 1976 Accretion onto magnetized neutron stars: structure and interchange instability of a model magnetosphere. Astrophys. J. 207, 914936.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Birkhoff, G. 1954 Note on Taylor instability. Quart. Appl. Math. 12, 306309.
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445468.
Carlés, P. & Popinet, S. 2002 The effect of viscosity, surface tension and non-linearity on Richtmyer–Meshkov instability. Euro. J. Mech. B 21, 511526.
Cattaneo, F. & Hughes, D. W. 1988 The nonlinear breakup of a magnetic layer: instability to interchange modes. J. Fluid Mech. 196, 323344.
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic stability. Cambridge University Press.
Dyke, M. Van 1975 Perturbation Methods in Fluid Mechanics. Parabolic Press.
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave-density discontinuity interaction. Phys. Fluids 29, 376386.
Frieman, E. A. 1954 On elephant-trunk structures in the region of O-associations. Astrophys. J. 120, 1821.
Grove, J. W., Holmes, R., Sharp, D. H., Yang, Y. & Zhang, Q. 1993 Quantitative theory of Richtmyer–Meshkov instability. Phys. Rev. Lett. 71, 34733476.
Hazak, G. 1996 Lagrangian formalism for the Rayleigh–Taylor instability. Phys. Rev. Lett. 76, 41674170.
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6, 40194030.
Jones, M. A. & Jacobs, J. W. 1997 A membraneless experiment for the study of Richtmyer–Meshkov instability of a shock-accelerated gas interface. Phys. Fluids 9, 30783085.
Kato, T. 1966 Perturbation Theory for Linear Operators. Springer.
Khokhlov, A. M., Oran, E. S. & Thomas, G. O. 1999 Numerical simulation of deflagration-to-detonation transition: the role of shock-flame interactions in turbulent flames. Combust. Flames 117, 323329.
Krechetnikov, R. & Homsy, G. M. 2009 Crown-forming instability phenomena in the drop splash problem. J. Colloid Interface Sci. 331, 555559.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.
Lawrentjew, M. A. & Schabat, B. V. 1967 Methoden der komplexen Funktionentheorie. Deutscher Verlag der Wissenschaften.
Layzer, D. 1955 On the instability of superimposed fluids in a gravitational field. Astrophys. J. 122, 112.
Lewis, D. J. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Part 2. Proc. R. Soc. A 202, 8196.
Lindl, J. D. & Mead, W. C. 1975 2-dimensional simulation of fluid instability in laser-fusion pellets. Phys. Rev. Lett. 34, 12731276.
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid. Dyn. 4, 101108.
Mikaelian, K. O. 1990 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified spherical shells. Phys. Rev. A 42, 34003420.
Mikaelian, K. O. 1994 Freeze-out and the effect of compressibility in the Richtmyer–Meshkov instability. Phys. Fluids 6, 356368.
Mikaelian, K. O. 1998 Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80, 508511.
Mikaelian, K. O. 2005 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17, 094105.
Ott, E. 1972 Nonlinear evolution of the Rayleigh–Taylor instability of a thin layer. Phys. Rev. Lett. 20, 14291432.
Plesset, M. S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25, 9698.
Rayleigh, L. 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170177.
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Math. XIII, 297319.
Sazonov, S. V. 1991 Dissipative structures in the f-region of the equatorial ionosphere generated by Rayleigh–Taylor instability. Planet. Space Sci. 39, 16671671.
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 318.
Sirignano, W. A. & Mehring, C. 2000 Review of theory of distortion and disintegration of liquid streams. Prog. Energy Combust. Sci. 26, 609655.
Spivak, M. 1999 A Comprehensive Introduction to Differential Geometry. Publish or Perish Press.
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Part 1. Waves on fluid sheets. Proc. R. Soc. Lond. A 201, 192196.
Velikovich, A. L. & Dimonte, G. 1996 Nonlinear perturbation theory of the incompressible Richtmyer–Meshkov instability instability. Phys. Rev. Lett. 76, 31123115.
Wilcock, W. S. D. & Whitehead, J. A. 1991 The Rayleigh–Taylor instability of an embedded layer of low-viscosity fluid. J. Geophys. Res. 96, 1219312200.
Wouchuk, J. G. & Nishihara, K. 1996 Linear perturbation growth at a shocked interface. Phys. Plasmas 3, 37613776.
Yang, Y., Zhang, Q. & Sharp, D. H. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6, 18561873.
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Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces

  • R. KRECHETNIKOV (a1)

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