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Variety of unsymmetric multibranched logarithmic vortex spirals

Published online by Cambridge University Press:  22 December 2017

V. ELLING  and
M. V. GNANN
V. ELLING
Affiliation:
Department of Mathematics, Academia Sinica, 6F Astronomy-Mathematics Bldg., No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan email: velling@math.sinica.edu.tw
M. V. GNANN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching near Munich, Germany email: gnann@ma.tum.de
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Abstract

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

Keywords

76B4735Q3135C06
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 23 - 38
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

The authors' research was partially supported by the National Science Foundation under Grant NSF DMS-1054115 and by Taiwan MOST grant 105-2115-M-001-007-MY3.

References

[1] Alexander, R. C. (1971) Family of similarity flows with vortex sheets. Phys. Fluids 14 (2), 231–239. URL: http://scitation.aip.org/content/aip/journal/pof1/14/2/10.1063/1.1693419.Google Scholar
[2] Elling, V. (2012) Existence of algebraic vortex spirals. In: Li, Tatsien & Jiang, Song (editors), Hyperbolic problems. Theory, Numerics and Applications, Ser. Contemp. Appl. Math. CAM, 17, Vol. 1, World Sci. Publishing, Singapore, pp. 203–214. URL: http://www.worldscientific.com/worldscibooks/10.1142/8550.Google Scholar
[3] Elling, V. (2016) Self-similar 2d euler solutions with mixed-sign vorticity. Comm. Math. Phys. 348 (1), 27–68.Google Scholar
[4] Kaden, H. (1931) Aufwicklung einer unstabilen Unstetigkeitsfläche. Ingenieur-Archiv 2, 140–168.Google Scholar
[5] Kaneda, Y. (1989) A family of analytical solutions of the motions of double-branched spiral vortex sheets. Phys. Fluids A 1, 261–266.Google Scholar
[6] Lopes-Filho, M. C., Nussenzveig Lopes, H. J. & Schochet, S. (2007) A criterion for the equivalence of the Birkhoff–Rott and Euler descriptions of vortex sheet evolution. Trans. Amer. Math. Soc. 359 (9), 4125–4142.Google Scholar
[7] Majda, A. J. & Bertozzi, A. L. (2002) Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Vol. 27, Cambridge University Press, Cambridge. ISBN 0-521-63057-6; 0-521-63948-4.Google Scholar
[8] Moore, D. W. (1975) The rolling-up of a semi-infinite vortex sheet. Proc. Roy. Soc. London A 345, 417–430.Google Scholar
[9] Prandtl, L. (1924) Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben. In: v. Kármán T., Levi-Civita T. (editors), Vorträge aus dem Gebiet der Hydro- und Aerodynamik, Springer, Springer, Berlin Heidelberg. ISBN 978-3-662-00260-5. URL: https://doi.org/10.1007/978-3-662-00280-3_2.Google Scholar
[10] Pullin, D. (1989) On similarity flows containing two-branched vortex sheets. In: Caflisch, R. (editor), Mathematical Aspect of Vortex Dynamics, SIAM, Philadelphia, PA, pp. 97–106.Google Scholar
[11] Rott, N. (1956) Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1, 111–128.Google Scholar
[12] Saffman, P. G. (1992) Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York. ISBN 0-521-42058-X.Google Scholar
[13] van Dyke, M. (1982) An Album of Fluid Motion, The Parabolic Press, Stanford, California.Google Scholar