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A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method

Published online by Cambridge University Press:  26 April 2006

M. J. Shelley
M. J. Shelley
Affiliation:
Department of Mathematics, University of Chicago, Chicago, 1L 60637, USA
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Abstract

Moore's asymptotic analysis of vortex-sheet motion predicts that the Kelvin–Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron. Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff–Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 493 - 526
Copyright
© 1992 Cambridge University Press

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References

Baker G. R., McCrory R. L., 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 A. A. 1980 Vortex simulations of the Rayleigh–Taylor instability. Phys. Fluids 23, 1485–1496.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. & Shelley M. J. 1986 Boundary integral techniques for multi-connected domains. J. Comput. Phys. 64, 112–132.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 instability. Proc. Symp. Appl. Math. Am. Math. Soc. 13, 55–76.Google Scholar
Birkhoff, G. & Fisher J. 1959 Do vortex sheets roll up? Circ. Mat. Palermo 8, 77–90.Google Scholar
Caflisch, R. & Lowengrub J. 1989 Convergence of the vortex method for vortex sheets. SIAM J. Numer. Anal. 26, 1060–1080.Google Scholar
Caflisch, R. & Orellana O. 1986 Long time existence for a slightly perturbed vortex sheet. Commun. Pure Appl. Maths. 34, 807–838.Google Scholar
Caflisch, R. & Orellana O. 1989 Singular solutions and ill-posedness of the evolution of vortex sheets. SIAM J. Math. Anal. 20, 293–307.Google Scholar
Carrier G., Krook, M. & Pearson C. 1966 Functions of a Complex Variable. McGraw-Hill.
Conte R. 1979 Etude numerique des nappes tourbillonnaires. Thése, DPh-G/PSRM, CEN. Saclay.
Duchon, R. & Robert R. 1988 Global vortex sheet solutions to Euler equations in the plane. Commun. PDE 13, 215–224.Google Scholar
Diperna, R. & Majda A. 1987 Concentrations in regularizations for 2-d incompressible flow. Commun. Pure Appl. Maths. 40, 301–345.Google Scholar
Ebin D. 1988 Ill-posedness of the Rayleigh–Taylor and Helmholtz problems for incompressible fluids. Commun. PDE 13, 1265–1295.Google Scholar
Keasny R. 1986a A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93.Google Scholar
Krasny R. 1986b Desingularization of period vortex sheet roll-up. J. Comput. Phys. 65, 292–313.Google Scholar
Hou T., Lowengrub, J. & Krasny R. 1991 Convergence of a point vortex method for vortex sheets. SIAM J. Numer. Anal. 28, 308–320.Google Scholar
Lamb H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuett-Higgins, M. S. & Cokelet E. D. 1976 Proc. R. Soc. Lond. A 350, 1–26.
McGrath F. J. 1967 Nonstationary plane flow of viscous and ideal fluids. Arch. Rat. Mech. Anal. 27, 329–348.Google Scholar
Majda A. 1987 Vortex dynamics: numerical analysis, scientific computing, and mathematical theory. Proc. of the First Intern. Conf. on Industrial and Applied Maths. Paris.
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. 1985 Numerical and analytical aspects of Helmholtz instability. In Theoretical and Applied Mechanics, Proc. 16th IUTAM (eds F. I. Niordson & N. Olhoff), pp. 263–274.
Pugh D. A. 1989 Development of vortex sheets in Boussinesq flows. PhD thesis, Imperial College, London.
Rosenhead L. 1931 The formation of vortices from a surface of discontinuity Proc. R. Soc. Lond. A 134, 170–191.Google Scholar
Sidi, A. & Israeli M. 1988 Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3, 201–231.Google Scholar
Sulem C., Sulem P. L., Bardos C & Frisch U. 1981 Finite time analyticity for the two and three dimensional Kelvin–Helmholtz instability. Commun. math. Phys. 80, 485–516.Google Scholar
Sulem C., Sulem, P. L. & Frisch H. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50, 138–161.Google Scholar
Van der Vooren A. I. 1980 A numerical investigation of the rolling up of vortex sheets Proc. R. Soc. Lond. A 373, 67–91.Google Scholar
Van Dyke M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.