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Vortices and invariant surfaces generated by symmetries for the 3D Navier–Stokes equations

Author

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  • Grassi, V.
  • Leo, R.A.
  • Soliani, G.
  • Tempesta, P.

Abstract

We show that certain infinitesimal operators of the Lie-point symmetries of the incompressible 3D Navier–Stokes equations give rise to vortex solutions with different characteristics. This approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity ω and the vortex stretching vector Sω, where S is the strain matrix. The symmetry algebra associated with the Navier–Stokes equations turns out to be infinite-dimensional. New vortical structures, generalizing in some cases well-known configurations such as, for example, the Burgers and Lundgren solutions, are obtained and discussed in relation to the value of the dynamic angle φ=arctan|ω→∧Sω→|/ω→·Sω→. A systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed, and the corresponding invariant surfaces are recognized.

Suggested Citation

  • Grassi, V. & Leo, R.A. & Soliani, G. & Tempesta, P., 2000. "Vortices and invariant surfaces generated by symmetries for the 3D Navier–Stokes equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 286(1), pages 79-108.
    • Handle: RePEc:eee:phsmap:v:286:y:2000:i:1:p:79-108
    DOI: 10.1016/S0378-4371(00)00223-5
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    Cited by:

    1. Razafindralandy, D. & Hamdouni, A. & Al Sayed, N., 2012. "Lie-symmetry group and modeling in non-isothermal fluid mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4624-4636.
    2. Grassi, V. & Leo, R.A. & Soliani, G. & Tempesta, P., 2001. "A group analysis of the 2D Navier–Stokes–Fourier equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(3), pages 421-434.

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    Keywords

    Lie groups; Fluid dynamics;

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