Nonlinear vortex trail dynamics. II. Analytic solutions

Lim, Chjan C.; Sirovich, Lawrence

doi:10.1090/qam/1205942

Authors: Chjan C. Lim and Lawrence Sirovich
Journal: Quart. Appl. Math. 51 (1993), 129-146
MSC: Primary 76C05
DOI: https://doi.org/10.1090/qam/1205942
MathSciNet review: MR1205942
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Abstract: Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.

V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295

C. Basdevant, Y. Couder, and R. Sadourny, Vortices and vortex-couples in 2-D turbulence, Lecture Notes in Phys., vol. 157, Springer, 1984, pp. 327–345

Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773

N. E. Kochin, I. A. Kiebel, and N. F. Roze, Theoretical Hydrodynamics, Wiley Interscience, 1964

H. Lamb, Hydrodynamics, Dover, New York, 1945

Chjan C. Lim, Singular manifolds and quasi-periodic solutions of Hamiltonians for vortex lattices, Phys. D 30 (1988), no. 3, 343–362. MR 947905, DOI https://doi.org/10.1016/0167-2789%2888%2990025-5
Chjan C. Lim, Existence of KAM tori in the phase-space of lattice vortex systems, Z. Angew. Math. Phys. 41 (1990), no. 2, 227–244. MR 1045813, DOI https://doi.org/10.1007/BF00945109

Chjan C. Lim and L. Sirovich, Wave propagation on the vortex trail, Phys. Fluids 29, 3910–3911 (1986)

Chjan C. Lim and Lawrence Sirovich, Nonlinear vortex trail dynamics, Phys. Fluids 31 (1988), no. 5, 991–998. MR 942400, DOI https://doi.org/10.1063/1.866719

T. Matsui and M. Okude, XVth International Congress of Theoretical Applied Mechanics, University of Toronto, Aug. 1980, pp. 1–27.

Jürgen Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J; Annals of Mathematics Studies, No. 77. MR 0442980

L. Sirovich, The Karman vortex trail and flow behind a circular cylinder, Phys. Fluid 28, 2723–2726 (1985)

L. Sirovich and Chjan C. Lim, Comparison of experiment with the dynamics of the von Karman vortex trail, Studies in Vortex Dominated Flows, Y. Hussaini and M. Salas, eds., Springer-Verlag, New York, 1986, p. 4

S. Taneda, Downstream development of the wakes behind cylinders, J. Phys. Soc. Japan 14, 843–848 (1959)

Th. von Karman, Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt, Gottingen Nachr. Math Phys. K 1, 509–517 (1911)

Chjan C. Lim, A combinatorial perturbation method and Arnol′d whiskered tori in vortex dynamics, Phys. D 64 (1993), no. 1-3, 163–184. MR 1214551, DOI https://doi.org/10.1016/0167-2789%2893%2990254-X