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Stability of a vortex street of finite vortices

Published online by Cambridge University Press:  20 April 2006

P. G. Saffman
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A.
J. C. Schatzman
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A.

Abstract

The stability of the finite-area Kármán ‘vortex street’ to two-dimensional disturbances is determined. It is shown that for vortices of finite size there exists a finite range of spacing ratio κ for which the array is stable to infinitesimal disturbances. As the vortex size approaches zero, the range narrows to zero width about the classical von Kármán value of 0·281.

Type
Research Article
Copyright
© 1982 Cambridge University Press

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References

Arnol'D, V. I. 1980 Mathematical Methods of Classical Mechanics, pp. 332337. Springer.
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263379.Google Scholar
Christiansen, J. P. & Zabusky, N. J. 1973 Instability, coalescence and fission of finite-area vortex structures. J. Fluid Mech. 61, 219243.Google Scholar
Domm, U. 1956 Über die Wirbelstraßen von geringster Instabilität. Z. angew. Math. Mech. 30, 367371.Google Scholar
Von Kármán, T. 1912 Über den Mechanismus des Widerstands, den ein bewegter Körper in einer Flüssigkeit erfährt. Gött. Nachr. Math. Phys. Kl. 00, 547556.Google Scholar
Kelvin, Lord 1910 Mathematical and Physical Papers, vol. IV. Cambridge University Press.
Kochin, N. J. 1939 On the instability of von Kármán's vortex street. C. R. Acad. Sci. U.R.S.S. 24, 1923.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lin, C. C. 1943 On the motion of vortices in two dimensions. University of Toronto Studies, Appl. Math. Series, vol. 5.
Onsager, L. 1949 Statistical hydrodynamics. Suppl. Nuovo Cim. 6, 279287.Google Scholar
Nekhoroshev, N. N. 1971 The behaviour of Hamiltonian systems that are close to integrable. Func. Anal. Appl. 5, 338339.Google Scholar
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129144.Google Scholar
Rosenhead, L. 1929 Double row of vortices with arbitrary stagger. Proc. Camb. Phil. Soc. 25, 132138.Google Scholar
Saffman, P. G. & Schatzman, J. C. 1981 Properties of a vortex street of finite vortices. SIAM J. Sci. Stat. Comp. (to be published).
Taneda, S. 1965 Experimental investigation of vortex streets. J. Phys. Soc. Japan 20, 17141721.Google Scholar