Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T10:51:50.007Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  14 April 2009

R. KRECHETNIKOV
R. KRECHETNIKOV*
Affiliation:
University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 625 , 25 April 2009 , pp. 387 - 410
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Arnett, D. 2000 The role of mixing in astrophysics. Astrophys. J. 127 (Suppl.), 213–217.Google Scholar
Arons, J. & Lea, S. M. 1976 Accretion onto magnetized neutron stars: structure and interchange instability of a model magnetosphere. Astrophys. J. 207, 914–936.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. 1954 Note on Taylor instability. Quart. Appl. Math. 12, 306–309.CrossRefGoogle Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445–468.CrossRefGoogle Scholar
Carlés, P. & Popinet, S. 2002 The effect of viscosity, surface tension and non-linearity on Richtmyer–Meshkov instability. Euro. J. Mech. B 21, 511–526.CrossRefGoogle Scholar
Cattaneo, F. & Hughes, D. W. 1988 The nonlinear breakup of a magnetic layer: instability to interchange modes. J. Fluid Mech. 196, 323–344.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic stability. Cambridge University Press.CrossRefGoogle Scholar
Dyke, M. Van 1975 Perturbation Methods in Fluid Mechanics. Parabolic Press.Google Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave-density discontinuity interaction. Phys. Fluids 29, 376–386.CrossRefGoogle Scholar
Frieman, E. A. 1954 On elephant-trunk structures in the region of O-associations. Astrophys. J. 120, 18–21.Google Scholar
Grove, J. W., Holmes, R., Sharp, D. H., Yang, Y. & Zhang, Q. 1993 Quantitative theory of Richtmyer–Meshkov instability. Phys. Rev. Lett. 71, 3473–3476.CrossRefGoogle ScholarPubMed
Hazak, G. 1996 Lagrangian formalism for the Rayleigh–Taylor instability. Phys. Rev. Lett. 76, 4167–4170.CrossRefGoogle ScholarPubMed
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6, 4019–4030.CrossRefGoogle Scholar
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, 3078–3085.CrossRefGoogle Scholar
Kato, T. 1966 Perturbation Theory for Linear Operators. Springer.Google Scholar
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, 323–329.CrossRefGoogle Scholar
Krechetnikov, R. & Homsy, G. M. 2009 Crown-forming instability phenomena in the drop splash problem. J. Colloid Interface Sci. 331, 555–559.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lawrentjew, M. A. & Schabat, B. V. 1967 Methoden der komplexen Funktionentheorie. Deutscher Verlag der Wissenschaften.Google Scholar
Layzer, D. 1955 On the instability of superimposed fluids in a gravitational field. Astrophys. J. 122, 1–12.CrossRefGoogle Scholar
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, 81–96.Google Scholar
Lindl, J. D. & Mead, W. C. 1975 2-dimensional simulation of fluid instability in laser-fusion pellets. Phys. Rev. Lett. 34, 1273–1276.Google Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid. Dyn. 4, 101–108.CrossRefGoogle Scholar
Mikaelian, K. O. 1990 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified spherical shells. Phys. Rev. A 42, 3400–3420.CrossRefGoogle ScholarPubMed
Mikaelian, K. O. 1994 Freeze-out and the effect of compressibility in the Richtmyer–Meshkov instability. Phys. Fluids 6, 356–368.CrossRefGoogle Scholar
Mikaelian, K. O. 1998 Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80, 508–511.Google Scholar
Mikaelian, K. O. 2005 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17, 094105.CrossRefGoogle Scholar
Ott, E. 1972 Nonlinear evolution of the Rayleigh–Taylor instability of a thin layer. Phys. Rev. Lett. 20, 1429–1432.CrossRefGoogle Scholar
Plesset, M. S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25, 96–98.Google Scholar
Rayleigh, L. 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Math. XIII, 297–319.CrossRefGoogle Scholar
Sazonov, S. V. 1991 Dissipative structures in the f-region of the equatorial ionosphere generated by Rayleigh–Taylor instability. Planet. Space Sci. 39, 1667–1671.CrossRefGoogle Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.Google Scholar
Sirignano, W. A. & Mehring, C. 2000 Review of theory of distortion and disintegration of liquid streams. Prog. Energy Combust. Sci. 26, 609–655.CrossRefGoogle Scholar
Spivak, M. 1999 A Comprehensive Introduction to Differential Geometry. Publish or Perish Press.Google Scholar
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, 192–196.Google Scholar
Velikovich, A. L. & Dimonte, G. 1996 Nonlinear perturbation theory of the incompressible Richtmyer–Meshkov instability instability. Phys. Rev. Lett. 76, 3112–3115.Google Scholar
Wilcock, W. S. D. & Whitehead, J. A. 1991 The Rayleigh–Taylor instability of an embedded layer of low-viscosity fluid. J. Geophys. Res. 96, 12193–12200.Google Scholar
Wouchuk, J. G. & Nishihara, K. 1996 Linear perturbation growth at a shocked interface. Phys. Plasmas 3, 3761–3776.CrossRefGoogle Scholar
Yang, Y., Zhang, Q. & Sharp, D. H. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6, 1856–1873.CrossRefGoogle Scholar