Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Bounded and unbounded solutions of the von Kármán vortex trail

Authors: Chjan Lim and Lawrence Sirovich
Journal: Quart. Appl. Math. 47 (1989), 447-458
MSC: Primary 76C05
DOI: https://doi.org/10.1090/qam/1012269
MathSciNet review: MR1012269
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Abstract: The general initial value problem for the linearized von Karman vortex trail is solved. In particular, the fundamental differences between von Karman's normal mode solutions and aperiodic solutions with compact support are discussed. Within the natural space $ {\ell _2}$, it is shown that the von Karman trail at its special aspect ratio (neutrally stable case) supports unbounded solutions. Two invariants of the equations determine whether a solution is bounded or unbounded. The asymptotic behaviour of aperiodic solutions is discussed. It is found that unbounded solutions grow at the rate $ O\left( {ln \tau } \right)$ in $ {\ell _\infty }$ while bounded solutions decay as $ O\left( {1/\sqrt \tau } \right)$. In addition, a Loschmidt's demon is constructed for the reversible dispersive equations which demonstrates the focussing effect in the von Karman trail.

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  • [1] Th. von Karman, Göttingen Nach. Math. Phys. Kl., 509-519 1911
  • [2] Th. von Karman, Göttingen Nach, Math. Phys. Kl., 547-556 1911
  • [3] Th. von Karman and Rubach, Phys Zeitschrift 13, 49-59 (1912)
  • [4] H. Lamb, Hydrodynamics, Dover, new York, 1945
  • [5] L. Sirovich, Phys. Fluids 28, 2723-2726 (1985)
  • [6] L. Sirovich and C. Lim, Comparison of experiment with the dynamics of the von Karman vortex trail, Vortex Dominated Flows, eds. M.Y. Hussaini and M. Salas, Springer-Verlag, New York, 1986, pp. 44-60
  • [7] C. Lim and L. Sirovich, Phys. Fluids 29 12, 3910-3911 (1986)
  • [8] Chjan C. Lim and Lawrence Sirovich, Nonlinear vortex trail dynamics, Phys. Fluids 31 (1988), no. 5, 991–998. MR 942400, https://doi.org/10.1063/1.866719
  • [9] V. K. Tkachenko, Sov. Phys. JETP 23, 1049 (1966)
  • [10] D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, 2nd Ed., Adam Hilger, 1986
  • [11] U. Domm, Über die Wirbelstrassen von geringster Instabilität, Z. Angew. Math. Mech. 36 (1956), 367–371 (German, with English, French and Russian summaries). MR 0084297, https://doi.org/10.1002/zamm.19560360906
  • [12] N. E. Kochin, I. A. Kibel, and N. V. Roze, Theor. Hydromechanics, Interscience, 1964
  • [13] L. Sirovich, Techniques of asymptotic analysis, Applied Mathematical Sciences, Vol. 2, Springer-Verlag, New York-Berlin, 1971. MR 0275034
  • [14] D. J. Tritton, J. Fluid Mech. 6, 547 (1959)
  • [15] J. S. Waugh, W. K. Rhim, and A. Pines, Spin echoes and Loschmidt's paradox, 4th Int. Symp. on Magnetic Resonance, Rehovot and Jerusalem (1971), ed. D. Fiat, Butterworth's Scientific Publication, London, 1971, pp. 317-324
  • [16] S. K. Ma, Statistical Mechanics, World Scientific, Philadelphia, 1985

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DOI: https://doi.org/10.1090/qam/1012269
Article copyright: © Copyright 1989 American Mathematical Society

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