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A study of singularity formation in a vortex sheet by the point-vortex approximation

Published online by Cambridge University Press:  21 April 2006

Robert Krasny
Robert Krasny
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY 10012, USA
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Abstract

The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead's point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point-vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 167 , June 1986 , pp. 65 - 93
Copyright
© 1986 Cambridge University Press

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References

Anderson, C. 1985 A vortex method for flows with slight density variations. J. Camp. Phys. 61, 417.Google Scholar
Anderson, C. & Greengard, C. 1985 On vortex methods. SIAM J. Num. Anal. 22, 413.Google Scholar
Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345.Google Scholar
Baker, G. R. 1980 A test of the method of Fink and Soh for following vortex sheet motion. J. Fluid Mech. 100, 209.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulations of the Rayleigh-Taylor instability. Phya. Fluids 23, 1485.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.
Beale, J. T. & Majda, A. 1982 Vortex methods 2: Higher order accuracy in two and three dimensions. Maths. Camp. 39, 29.Google Scholar
Bikkhoff, G. & Fisher, J. 1959 Do vortex sheets roll up? Rend. Circ. Mat. Palermo ser. 2 8, 77.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor instability. Proc. Symp. Appl. Math. XIII A.M.S. 55.
Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B., Morf, R. & Frisch, U. 1983 Small-scale structure of the Taylor-Green vortex. J. Fluid Mech. 130, 411.Google Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable, p. 255. McGraw Hill.
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex sheet, with an example of roll-up. J. Comp. Phys. 13, 423.Google Scholar
Conte, R. 1979 Etude numerique des nappes tourbillonnaires. These, DPh-G/PSRM, CEN. Saclay.
Cottet, G. H. 1982 Ph.D. these. Université Pierre et Marie Curie, Paris.
Fink, P. T. & Soh, W. K. 1978 A new approach to roll-up calculations of vortex sheets. Proc. R. Soc. Lond. A 62, 195.Google Scholar
Garabedian, P. R. 1964 Partial Differential Equations, p. 109. John Wiley.
Gear, C. W. 1971 Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall.
Hald, O. 1979 The convergence of vortex methods, 2. SIAM J. Num. Anal. 16, 726.Google Scholar
Higdon, J. J. L. & Pozrikidis, C. 1985 The self-induced motion of vortex sheets. J. Fluid Mech. 150, 203.Google Scholar
John, F. 1959 Numerical solution of problems which are not well posed in the sense of Hadamard. Proc. Rome Symp. Prov. Int. Comp. Center, 103.
John, F. 1982 Partial Differential Equations, 4th ed. p. 8, Springer.
Krasny, R. 1983 A numerical study of Kelvin-Helmholtz instability by the point vortex method. Ph.D. thesis. Lawr. Berk. Lab. rep.#17092.
Krasny, R. 1986 Desingularization of periodic vortex sheet roll-up. J. Comp. Phys. (to appear.)Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lax, P. D. 1973 Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Reg. Conf. Ser. Appl. Math. SIAM 11.
Leonard, A. 1980 Vortex methods for flow simulations. J. Comp. Phys. 37, 289.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. L. 1976 The deformation of steep surface waves on water. 1. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1.Google Scholar
Meiron, D. I., Baker, G. R. & Orszag, S. A. 1982 Analytic structure of vortex sheet dynamics. 1. Kelvin-Helmholtz instability. J. Fluid Mech. 114, 283.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. 1981 On the point vortex method. SIAM J. Sci. Stat. Comput. 2, 65.Google Scholar
Moore, D. W. 1984 Numerical and analytical aspects of Helmholtz instability. IUTAM lecture.
Pozrikidis, C. & Higdon, J. J. L. 1985 Non-linear Kelvin-Helmholtz instability of a finite vortex layer. J. Fluid Mech. 157, 225.Google Scholar
Pullin, D. I. 1982 Numerical studies of surface-tension effects in nonlinear Kelvin-Helmholtz and Rayleigh-Taylor instability. J. Fluid Mech. 119, 507.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134, 170.Google Scholar
Richtmyer, R. D. & Morton, K. W. 1967 Difference Methods for Initial-Value Problems. Interscience.
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95.Google Scholar
Sethian, J. A. 1985 Curvature and the evolution of fronts. Comm. Math. Phys. 101, 4.Google Scholar
Sulem, C., Sulem, P. L., Bardos, C. & Frisch, U. 1981 Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80, 485.Google Scholar
Sulem, C., Sulem, P. L. & Frisch, U. 1983 Tracing complex singularities with spectral methods. J. Comp. Phys. 5O, 138.Google Scholar
Van de Vooren, A. I. 1980 A numerical investigation of the rolling up of vortex sheets. Proc. R. Soc. Land. A 373, 67.Google Scholar