Hostname: page-component-6dbcb7884d-g628x Total loading time: 0 Render date: 2025-02-14T08:16:34.701Z Has data issue: false hasContentIssue false
  • English
  • Français

On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid

Published online by Cambridge University Press:  26 April 2006

V.V. Meleshko  and
G.J.F. van Heijst
V.V. Meleshko
Affiliation:
Institute of Hydromechanics of the Ukrainian Academy of Sciences, 252057Kiev, Ukraine
G.J.F. van Heijst
Affiliation:
J. M. Burgers Centre for Fluid Mechanics, Fluid Dynamics Laboratory, Department of Technical Physics, Eindhoven University of Technology, PO Box 513, 5600MBEindhoven, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper describes exact solutions of two-dimensional vortex structures that were published by Chaplygin (1899, 1903) at the turn of the last century, which seem to have escaped the attention of later investigators in this field. Chaplygin's solutions include that of an elliptical patch of uniform vorticity in an exterior field of pure shear and that of a (symmetric or non-symmetric) dipolar vortex with a continuous distribution of vorticity translating steadily along a straight path. In addition, a solution is presented for a non-symmetric vortex dipole moving along a circular trajectory. A concise account of Chaplygin's solutions is given, complemented by a more detailed analysis of some of their relevant properties.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 272 , 10 August 1994 , pp. 157 - 182
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aref, H. Rott, N. & Thomann, H. 1992 Gröbli's solution of the three-vortex problem: A case study of graduate work in the 1870's. Ann. Rev. Fluid Mech. 24, 1–25.Google Scholar
Arnol’d, V. I. 1965 Condition for nonlinear stability of stationary plane curvilinear flows of ideal fluid. Sov. Math. Dokl. 6, 773–777.Google Scholar
Auerbach, F. 1908 Wirbelbewegung. In Handbuch der Physik (2nd edn) (ed. Winkelmann, A.), vol. 1, pp. 1047–1074. Barth.Google Scholar
Basset, A. B. 1888 A Treatise on Hydrodynamics. Deighton Bell.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Chaplygin, S. A. 1899 On a pulsating cylindrical vortex. Trans. Phys. Sect. Imperial Moscow Soc. Friends of Natural Sciences 10, N 1, 13–22. Also In Collected Works, 1948, vol. 2, pp. 138–154 (in Russian). German abstract: Über einen pulsierenden cylindrischen Wirbel. Jahrbuch über die Fortschritte der Mathematik (Jahrgang 1899), 1901, 30, 683–684.Google Scholar
Chaplygin, S. A. 1903 One case of vortex motion in fluid. Trans. Phys. Sect. Imperial Moscow Soc. Friends of Natural Sciences 11, N 2, 11–14. Also In Collected Works, 1948, vol. 2, pp. 155–165 (in Russian).Google Scholar
Couder, Y. & Basdevant, C. 1986 Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid Mech. 173, 225–251.Google Scholar
Dhanak, M. R. & Marshall, M. P. 1993 Motion of an elliptic vortex under applied periodic strain. Phys. Fluids A 5, 1224–1230.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Dritschel, D. G. 1990 The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261.Google Scholar
Flierl, G. R. 1987 Isolated eddy models in geophysics. Ann. Rev. Fluid Mech. 19, 493–530.Google Scholar
Flierl, G. R., Stern, M. E. & Whitehead, J. A. 1983 The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans 7, 233–264.Google Scholar
Giacomelli, R. & Pistolesi, E. 1934 Historical sketch. In Aerodynamic Theory. A General Review of Progress (ed. Durand, W. F.), 1963, vol. 1, pp 305–394. Dover.CrossRefGoogle Scholar
Goldstein, S. 1969 Fluid mechanics in the first half of this century. Ann. Rev. Fluid Mech. 1, 1–28.Google Scholar
Griffiths, R. W. & Linden, P. F. 1981 The stability of vortices in a rotating, stratified fluid. J. Fluid Mech. 105, 283–316.Google Scholar
Grigoryan, A. T. 1965 Die Entwicklung der Hydrodynamik und Aerodynamik in den Arbeiten von N.J. Shukowski und S.A. Tschaplygin. Schriftenreihe für Geschichte der Naturwissensch., Technik und Medizin 2, 39–62.Google Scholar
Heijst van, G. J. F. & Flór, J. B. 1989a Dipole formation and collisions in a stratified fluid. Nature 340, 212–215.Google Scholar
Heijst van, G. J. F. & Flór, J. B. 1989b Laboratory experiments on dipole structures in a stratified fluid. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. Nihoul, J. C. J. & Jamart, B. M.), pp. 609–626. Elsevier.Google Scholar
Helmholtz, H. 1958 Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55.Google Scholar
Hill, M. J. M. 1984 On the motion of fluid, part of which is moving rotationally and part irrotationally. Phil. Trans. R. Soc. Lond. A 175, 363–410.Google Scholar
Hopfinger, E. J. & Heijst van, G. J. F. 1993 Vortices in rotating fluids. Ann. Rev. Fluid Mech. 25, 241–289.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Kirchhoff, G. 1976 Vorlesungen über mathematische Physik: Mechanik. Teubner.Google Scholar
Kloosterziel, R. C. & Heijst van, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.Google Scholar
Lamb, H. 1895 Hydrodynamics (2nd edn). Cambridge University Press.Google Scholar
Lamb, H. 1906 Hydrodynamics (3rd edn). Cambridge University Press.Google Scholar
Loitsyanskii, L. G. 1966 Mechanics of Liquids and Gases. Pergamon.Google Scholar
Love, A. E. H. 1887 On recent English researches in vortex motion. Math. Annalen 30. 326–344.CrossRefGoogle Scholar
Love, A. E. H. 1893 On the stability of certain vortex motions. Proc. London Math. Soc. 25, 18–34.Google Scholar
Love, A. E. H. 1901 Hydrodynamik: Theoretische Ausführungen. In Encyklopädie der Mathematischen Wissenschaften (ed. Klein, F. & Müller, C.), vol. 4, pp. 84–147. Teubner.CrossRefGoogle Scholar
Maxwell, J. C. 1855 a Manuscript on the steady motion of an incompressible fluid. In The Scientific Letters and Papers of James Clerk Maxwell (ed. Harman, P. M.), 1990, vol. 1, pp. 295–299. Cambridge University Press.Google Scholar
Maxwell, J.C. 1855b Letter to William Thomson. In The Scientific Letters and Papers of James Clerk Maxwell (ed. Harman, P. M.), 1990, vol. 1, pp. 309–313. Cambridge University Press.Google Scholar
Maxwell, J. C. 1961 a On physical lines of force. Part I. The theory of molecular vortices applied to magnetic phenomena. Phil. Mag. (4) 21, 161–175.Google Scholar
Maxwell, J. C. 1961 b On physical lines of force. Part II. The theory of molecular vortices applied to electric currents. Phil.Mag. (4) 21, 281–291, 338–348.Google Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. Olsen, J. H., Goldburg, A. & Rogers, M.), pp. 339–354. Plenum.CrossRefGoogle Scholar
Nguyen Duc, J.-M. & Sommeria, J. 1988 Experimental characterization of steady two-dimensional vortex couples. J. Fluid Mech. 192, 175–192.Google Scholar
Nycander, J. & Isichenko, M. B. 1990 Motion of dipole vortices in a weakly inhomogeneous medium and related convective transport. Phys. Fluids B 2, 2042–2047.Google Scholar
Polvani, L. & Wisdom, J. 1990 Chaotic Lagrangian trajectories around an elliptical patch embedded in a constant and uniform background shear flow. Phys. Fluids A 2, 123–126.Google Scholar
Rankine, W.J.M. 1858 A Manual of Applied Mechanics. Griffin.Google Scholar
Ricca, R. L. 1991 Rediscovery of Da Rios equations. Nature 352, 561–562.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Satkevich, A. A. 1923 Aerodynamics as the Theoretical Basis of Aerial Flight. Petrograd Institute of Ways of Communications Press.Google Scholar
Slezkin, N. G. 1988 Generalization of the Chaplygin problem of a cylindrical vortex. Sov. Phys. Dokl. 33, 170–171.Google Scholar
Stokes, G.G. 1842 On the steady motion of incompressible fluids. Trans. Camb. Phil. Soc. 7, 439–453.Google Scholar
Voropayev, S. I. Afanasyev, Y. D. & Filippov, I. A. 1991 Horizontal jets and vortex dipoles in a stratified fluid. J. Fluid Mech. 227, 543–566.Google Scholar
Wien, W. 1900 Lehrbuch der Hydrodynamik. Hirzel.Google Scholar
Yarmitskii, A. G. 1992 A whirlwindlike Chaplygin vortex. Fluid Dyn. 27, 489–494.Google Scholar