Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T13:41:55.519Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularity formation during Rayleigh–Taylor instability

Published online by Cambridge University Press:  26 April 2006

Gregory Baker ,
Russel E. Caflisch  and
Michael Siegel
Gregory Baker
Affiliation:
Mathematics Department, The Ohio State University, Columbus, OH 43210, USA
Russel E. Caflisch
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90024, USA
Michael Siegel
Affiliation:
Mathematics Department, The Ohio State University, Columbus, OH 43210, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore's approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using a form fit for the Fourier components at high wavenumber. Excellent agreement between the theoretical predictions and the numerical results is demonstrated for small to medium values of the Atwood number A, i.e. for A between 0 and approximately 0.9. For A near 1, however, the singularities actually slow down when close to the real axis. In particular, for A = 1, the numerical evidence suggests that the singularities do not reach the real axis in finite time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 51 - 78
Copyright
© 1993 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

Baker, G. R. 1983 Generalized vortex methods for free-surface flows. In Waves on Fluid Interfaces (ed. R. E. Meyer). Academic.
Baker, G. R. 1990 Singularities in the complex physical plane. In Hyperbolic Problems (ed. B. Engquist & B. Gustafsson). Lund: Studentlitteratur.
Baker, G. R., McCrory, C. P., Verdon, C. P. & Orszag, S. A. 1987 Rayleigh—Taylor instability of fluid layers. J. Fluid Mech. 178, 161–175.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulations of the Rayleigh—Taylor instability. Phys. Fluids 23, 1485–1490.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1984 Boundary integral methods for axisymmetric and three-dimensional Rayleigh—Taylor instability problems. Physica D 12, 19–31.Google Scholar
Baker, G. R. & Moore, D. W. 1989 The rise and distortion of a two-dimensional gas bubble in an inviscid liquid. Phys. Fluids A 1, 1451–1459.Google Scholar
Baker, G. R. & Shelley, M. J. 1990 On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161–194.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor instabilities. In Proc. Sympos. Appl. Maths, vol. XII. Providence, R.I.: AMS.
Caflisch, R. & Orellana, O. 1986 Long time existence for a slightly perturbed vortex sheet. Commun. Pure Appl. Maths 39, 807–816.Google Scholar
Caflisch, R., Orellana, O. & Siegel, M. 1990 A localized approximation method for vortical flows. SIAM J. Appl. Maths 50, 1517–1532.Google Scholar
Cowley, S. J., Baker, G. R., Tanveer, S. & Page, M. 1993 An asymptotic description of the formation of a Moore singularity in a vortex sheet. J. Fluid Mech. (submitted).Google Scholar
Delort, J. 1991 Existence de nappes de tourbillon en dimension deux. C.R. Acad. Sci. Paris I Maths 312, 85–88.Google Scholar
Diperna, R. & Majda, A. 1987 Concentrations in regularizations for 2-d incompressible flow. Commun. Pure Appl. Maths 60, 301–345.Google Scholar
Duchon, J. & Robert, O. 1988 Global vortex sheet solutions of euler equations in the plane. J. Diff Equat. 73, 215–224.Google Scholar
Ely, J. & Baker, G. R. 1993 High-precision calculations of vortex sheet motion. J. Comput. Phys. (submitted.)Google Scholar
Kelvin, Lord 1871 Hydro-kinetic solutions and observations. Phil. Mag. 42(4), 362–377.Google Scholar
Kerr, R. M. 1986 Analysis of Rayleigh—Taylor flows using vortex blobs. Rep. UCID-20915. Lawrence Livermore National Laboratory.
Krasny, R. 1986a Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313.Google Scholar
Krasny, R. 1986b On singularity formation in a vortex sheet and the point vortex approximation. J. Fluid Mech. 167, 65–93.Google Scholar
Meiron, D. I., Baker, G. R. & Orszag, S. A. 1982 Analytic structure of vortex sheet dynamics. Part 1. Kelvin—Helmholtz instability. J. Fluid Mech. 114, 283–298.Google Scholar
Moore, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A 365, 105–119.Google Scholar
Moore, D. W. 1982 A vortex method applied to interfacial waves. In Vortex Motion (ed. H. G. Hornung & E. A. Muller). Vieweg & Sons.
Moore, D. W. 1985 Numerical and analytical aspects of Helmholtz instability. In Theoretical and Applied Mechanics (ed. F. I. Niordson & N. Olhoff). Elsevier.
Pugh, D. A. 1989 Development of vortex sheets in Boussinesq flows – formation of singularities. PhD thesis, Imperial College.
Pugh, D. & Cowley, S. 1993 On the formation of an interface singularity in the rising two-dimensional Boussinesq bubble. J. Fluid Mech. (to appear).Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95–122.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh—Taylor instability. Physica D 12, 3–18.Google Scholar
Shelley, M. J. 1992 A study of singularity formation in vortex sheet motion by a spectrally accurate method. J. Fluid Mech. 244, 493–526.Google Scholar
Soegel, M. 1989 An analytical and numerical study of singularity formation in the Rayleigh—Taylor problem. PhD thesis, New York University.
Tanveer, S. 1992 Singularities in the classical Rayleigh—Taylor flow: formation and subsequent motion. Proc. R. Soc. Lond. A (submitted).Google Scholar
Tryggvason, G. 1988 Numerical simulations of the Rayleigh—Taylor instability. J. Comput. Phys. 75, 253–282.Google Scholar