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On the ‘δ-equations’ for vortex sheet evolution

Published online by Cambridge University Press:  26 April 2006

James W. Rottman and
Peter K. Stansby
James W. Rottman
Affiliation:
Universities Space Research Association. NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
Peter K. Stansby
Affiliation:
Department of Engineering, Simon Building, The University, Manchester M13 9PL, UK
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Abstract

We use a set of equations, sometimes referred to as the ‘δ-equations’, to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with δ2 = 0 (for which the equations are exact) and for δ2 > 0. For small values of the smoothing parameter δ a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when δ2 = 0 at a critical time tc After tc the solutions do not converge as the computational mesh is refined. With δ2 > 0, converged solutions were found for all values of δ2 when t < tc, and for all but the two smallest values of δ2 used when t > tc. Our results show that when δ2 > 0 the vortex sheet deforms into two doubly branched spirals some time after tc The limiting solution as δ→0 clearly exists and equals the δ = 0 solution when t < tc. For t > tc, the limiting solution appears to exist if only the converged solutions are used, but it is unclear what relation this limiting solution has to any δ2 = 0 solution for these times.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993, pp. 527 - 549
Copyright
© 1993 Cambridge University Press

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References

Anderson, C. 1985 A vortex method for flows with slight density variations. J. Comput. Phys. 61, 417– 444.Google Scholar
Caflisch, R. E. & Lowengrub, J. S. 1989 Convergence of the vortex method for vortex sheets. SIAM J. Numer. Anal. 26, 1060– 1080.Google Scholar
Caflisch, R. E. & Orellana, O. F. 1986 Long time existence for a slightly perturbed vortex sheet. Commun. Pure Appl. Maths 39, 807– 838.Google Scholar
Caflisch, R. E. & Orellana, O. F. 1989 Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20, 293– 307.Google Scholar
Carrier, G. F., Krook, M. & Pearsen, C. E. 1966 Functions of a Complex Variable. McGraw Hill.
Chorin, A. J. & Bernard, P. S. 1973 Discretisation of a vortex sheet with an example of roll-up. J. Comput. Phys. 13, 423– 429.Google Scholar
Ebin, D. 1988 III-posedness of the Rayleigh–Taylor and Helmholtz problems for incompressible fluids. Commun. Partial Diff. Equat. 13, 1265– 1295.Google Scholar
Krasny, 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 periodic vortex sheet roll-up. J. Comput. Phys. 65, 292– 313.Google Scholar
Krasny, R. 1987 Computation of vortex sheet roll-up in the Treffzt plane. J. Fluid Mech. 184, 123– 155.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. 1981 On the point vortex method. SIAM J. Sci. Statist. Comput. 2, 65– 84.Google Scholar
Pugh, D. A. 1989 Development of vortex sheets in Boussinesq flows - formation of singularities. Ph.D. dissertation, Imperial College, London.
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134, 170– 192.Google Scholar
Rottman, J. W., Simpson, J. E. & Stansby, P. K. 1987 The motion of a cylinder of fluid released from rest in a cross-flow. J. Fluid Mech. 177, 307– 337.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, U. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50, 138– 161.Google Scholar
Tryggvason, G., Dahn, W. J. A. & Sbeih, K. 1991 Fine structure of vortex sheet roll-up by viscous and inviscid simulation. Trans. ASME I: J. Fluids Engng 113, 31– 36.Google Scholar
Vooren, A. I. Van De 1980 A numerical investigation of the rolling up of vortex sheets. Proc. R. Soc. Lond. A 373, 67– 80.Google Scholar