Hostname: page-component-6bf8c574d5-qdpjg Total loading time: 0 Render date: 2025-02-22T07:15:40.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability

Published online by Cambridge University Press:  20 April 2006

Daniel I. Meiron ,
Gregory R. Baker  and
Steven A. Orszag
Daniel I. Meiron
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139
Gregory R. Baker
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139
Steven A. Orszag
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

The instability of an initially flat vortex sheet to a sinusoidal perturbation of the vorticity is studied by means of high-order Taylor series in time t. All finite-amplitude corrections are retained at each order in t. Our analysis indicates that the sheet develops a curvature singularity at t = tc < ∞. The variation of tc with the amplitude a of the perturbation vorticity is in good agreement with the asymptotic results of Moore. When a is O(1), the Fourier coefficient of order n decays slightly faster than predicted by Moore. Extensions of the present prototype of Kelvin-Helmholtz instability to other layered flows, such as Rayleigh-Taylor instability, are indicated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 114 , January 1982 , pp. 283 - 298
Copyright
© 1972 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. A. 1975 Essentials of Padé Approximants. Academic.
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulations of the Rayleigh — Taylor instability. Phys. Fluids 23, 1485.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free surface flow problems. (To appear.)
Birkhoff, G. 1962 Helmholtz and Taylor instability in hydrodynamic instability. Proc. Symp. on Applied Math. XIII. A.M.S.
Gaunt, D. S. & Guttmann, A. J. 1974 Asymptotic analysis of coefficients. In Phase Transitions and Critical Phenomena, vol. 3. Academic.
Menikoff, R. & Zemach, C. 1980 Methods for numerical conformal mapping. J. Comp. Phys. 36, 366.Google Scholar
Moore, D. W. 1976 The stability of an evolving two-dimensional vortex sheet. Mathematika 23, 35.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.Google Scholar
Moore, D. W. & Griffith-Jones, R. 1974 The stability of an expanding circular vortex sheet. Mathematika 21, 128.Google Scholar
Morf, R. H., Orszag, S. A. & Frisch, U. 1980 Spontaneous singularity in three-dimensional, inviscid, incompressible flow. Phys. Rev. Lett. 44, 572.Google Scholar
Nickel, B. G. 1980 In Cargèse Lectures in Physics. Plenum.
Orszag, S. A. 1970 Transform method for the evaluation of vector-coupled sums: Application to the spectral form of the vorticity equation. J. Atmos. Sci. 27, 890.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95.Google Scholar
Sulem, C., Sulem, P. L., Bardos, C. & Frisch, U. 1982 Finite time analyticity for the two and three dimensional Kelvin — Helmholtz instability. (To appear.)
Van De Vooren, A. T. 1965 A numerical investigation of the rollup of vortex sheets. Dept. Math. Groningen Univ., The Netherlands, Rep. TW-21.Google Scholar