Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T05:49:54.638Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixed Lagrangian–Eulerian description of vortical flows for ideal and viscous fluids

Published online by Cambridge University Press:  26 March 2008

E. A. KUZNETSOV
E. A. KUZNETSOV*
Affiliation:
P. N. Lebedev Physical Institute, 53 Leninsky Ave., 119991 Moscow, Russia and L. D. Landau Institute for Theoretical Physics, 2 Kosygin Str., 119334 Moscow, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid is equivalent to the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. The velocity of new auxiliary fluid coincides with the velocity component normal to the vorticity line for the primitive equations. Therefore this new hydrodynamics represents hydrodynamics of vortex lines. Their compressibility reveals a new mechanism for three-dimensional incompressible vortical flows connected with breaking (or overturning) of vortex lines which can be considered as one of the variants of collapses. Transition to the Lagrangian description in the new hydrodynamics corresponds, for the original Euler equations, to a mixed Lagrangian–Eulerian description – the vortex line representation (VLR). The Jacobian of this mapping defines the density of vortex lines. It is shown also that application of VLR to the Navier–Stokes equations results in an equation of diffusive type for the Cauchy invariant. The diffusion tensor for this equation is defined by the VLR metric.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 600 , 10 April 2008 , pp. 167 - 180
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Arnold, V. I. 1986 Theory of Catastrophe, 2nd rev. edn. Springer.CrossRefGoogle Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 172–1746.CrossRefGoogle Scholar
Kerr, R. M. 2005 Vorticity and scaling of collapsing Euler vortices. Phys. Fluids 17, 075103.CrossRefGoogle Scholar
Kuzmin, G. A. 1983 Ideal incompressible hydrodynamics in terms of the vortex momentum density. Phys. Lett. A 96, 88–90.Google Scholar
Kuznetsov, E. A. 2002 Vortex line representation for flows of ideal and viscous fluids. Pis'ma v ZhETF 76, 406–410 (JETP Lett. 76, 346–350 (2002)).Google Scholar
Kuznetsov, E. A. & Mikhailov, A. V. 1980 On the topological meaning of canonical Clebsch variables. Phys. Lett. A 77, 37–38.CrossRefGoogle Scholar
Kuznetsov, E. A., Podvigina, O. N. & Zheligovsky, V. A. 2003 Numerical evidence of breaking of vortex lines in an ideal fluid. In Tubes, Sheets and Singularities in Fluid Dynamics (ed. Bajer, K. & Moffatt, H. K.), pp. 305–316. Kluwer.Google Scholar
Kuznetsov, E. A. & Ruban, V. P. 1998 Hamiltonian dynamics of vortex lines for systems of the hydrodynamic type. Pis'ma v ZhETF 67, 1015 (JETP Lett. 67, 1076–1081 (1998)).Google Scholar
Kuznetsov, E. A. & Ruban, V. P. 2000 a Collapse of vortex lines in hydrodynamics. ZhETF 118, 893 (JETP 91, 776 (2000)).Google Scholar
Kuznetsov, E. A. & Ruban, V. P. 2000 b Hamiltonian dynamics of vortex and magnetic lines in the hydrodynamic type models. Phys. Rev. E 61, 831–841.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1980 The Classical Theory of Fields, Fourth Edn. Butterworth-Heinemann.Google Scholar
Majda, A. 1986 Vorticity and the mathematical theory of incompressible flow. Commun. Pure Appl. Maths 39, 46 pages.CrossRefGoogle Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117.CrossRefGoogle Scholar
Ricca, R. L. 1991 Rediscovery of Da Rios equations. Nature 352, 561.CrossRefGoogle Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225.CrossRefGoogle Scholar
Volovik, G. E. & Mineev, V. P. 1977 Investigation of singularities in superfluid 3He and in liquid crystals by homotopic topology method. Zh. Eksp. Teor. Fiz. 72, 2256 (Sov. Phys. JETP 45, 1186–1196 (1977)).Google Scholar
Yakubovich, E. I. & Zenkovich, D. A. 2001 Matrix fluid dynamics. Proc. Intl Conf. “Progress in Nonlinear Science”, July 2001, N.Novgorod, Russia, vol. II “Frontiers of Nonlinear Physics” (ed. Litvak, A. G.), Nizhny Novgorod, 2002, pp. 282–287; physics/0110004.Google Scholar
Zakharov, V. E. & Kuznetsov, E. A. 1997 Hamiltonian formalism for nonlinear waves. UFN 167, 1137–1168 (Physics-Uspekhi 40, 1087–1116 (1997)).Google Scholar
Zheligovsky, V. A., Kuznetsov, E. A. & Podvigina, O. M. 2001 Numerical modeling of collapse in ideal incompressible hydrodynamics. Pis'ma v ZhETF 74, 402 (JETP Lett. 74, 367 (2001)).Google Scholar