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Desingularization of periodic vortex sheet roll-up. (English) Zbl 0591.76059

Summary: The equations governing periodic vortex sheet roll-up from analytic initial data are desingularized. Linear stability analysis shows that this diminishes the vortex sheet model’s short wavelength instability, yielding a numerically more tractable set of equations. Computational evidence is presented which indicates that this approximation converges, beyond the critical time of singularity formation in the vortex sheet, if the mesh is refined and the smoothing parameter is reduced in the proper order. The results suggest that the vortex sheet rolls up into a double branched spiral past the critical time. It is demonstrated that either higher machine precision or a spectral filter can be used to maintain computational accuracy as the smoothing parameter is decreased. Some conjectures on the model’s long time asymptotic state are given.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76B47 Vortex flows for incompressible inviscid fluids
76M99 Basic methods in fluid mechanics
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[1] Anderson, C., J. Comput. Phys., 61, 417 (1985)
[2] Anderson, C.; Greengard, C., SIAM J. Numer. Anal., 22, 413 (1985)
[3] Aref, H., Annu. Rev. Fluid Mech., 15 (1983)
[4] Batchelor, G. K., An Introduction to Fluid Mechanics (1967), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0152.44402
[5] Beale, J. T.; Majda, A., J. Comput. Phys., 58, 188 (1985)
[6] Bellman, R.; Pennington, R. H., Quart. Appl. Math., 12, 151 (1954)
[7] Birkhoff, G.; Fisher, J., Rend. Circ. Mat. Palermo Ser. 2, 8, 77 (1959)
[8] Birkhoff, G., Helmholtz and Taylor Instability, (Proceedings of the Symposium on Applied Mathematics, Vol. XIII (1962), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0107.42702
[9] Chorin, A. J.; Bernard, P. S., J. Comput. Phys., 13, 423 (1973)
[10] Corcos, G. M.; Sherman, F. S., J. Fluid Mech., 139, 29 (1984)
[11] Garabedian, P. R., Partial Differential Equations (1964), Wiley: Wiley New York · Zbl 0124.30501
[12] Hald, O. H., SIAM J. Numer. Anal., 16, 726 (1979)
[13] Higdon, J. J.L.; Pozrikidis, C., J. Fluid Mech., 150, 203 (1985)
[14] Ho, C.-H.; Huerre, P., Annu. Rev. Fluid Mech., 16 (1984)
[15] Krasny, R., J. Fluid Mech., 167, 65 (1986)
[16] Leonard, A., J. Comput. Phys., 37, 289 (1980)
[17] Meiron, D. I.; Baker, G. R.; Orszag, S. A., J. Fluid Mech., 114, 283 (1982)
[18] Moore, D. W., Stud. Appl. Math., 58, 119 (1978)
[19] Moore, D. W., (Proc. R. Soc. London Ser. A, 365 (1979)), 105
[20] Moore, D. W., SIAM J. Sci. Stat. Comput., 2, 65 (1981)
[21] Pozrikidis, C.; Higdon, J. J.L., J. Fluid Mech., 157, 225 (1985)
[22] Pullin, D. I.; Phillips, W. R.C., J. Fluid Mech., 104, 45 (1981)
[23] Pullin, D. I., J. Fluid Mech., 119, 507 (1982)
[24] Richtmeyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502
[25] Rosenhead, L., (Proc. R. Soc. London Ser. A, 134 (1931)), 170
[26] Saffman, P. G.; Baker, G. R., Annu. Rev. Fluid Mech., 11 (1979)
[27] Sethian, J., Commun. Math. Phys., 101, 4 (1985)
[28] Sulem, C.; Sulem, P. L.; Bardos, C.; Frisch, U., Commun. Math. Phys., 80, 485 (1981)
[29] Thompson, C. J., Mathematical Statistical Mechanics (1979), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0417.60096
[30] van de Vooren, A. I., (Proc. R. Soc. London Ser. A, 373 (1980)), 67
[31] Van Dyke, M., An Album of Fluid Motion (1982), Parabolic Press: Parabolic Press Stanford, CA
[32] Zabusky, N. J.; Overman, E. A., J. Comput. Phys., 52, 351 (1983)
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