ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Filimonova A. М. Dynamics and advection in a vortex parquet. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 4, pp. 71-84. DOI: 10.18500/0869-6632-2019-27-4-71-84

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 120)
Полный текст в формате PDF(En):
(downloads: 87)
Article type: 

Dynamics and advection in a vortex parquet

Filimonova Alexandra Михайловна, Southern Federal University

Issue. The article is devoted to a numerical study of the dynamics and advection in a vortex parquet. A vortex structure, consisting of vortex patches on the entire plane, is considered. The mathematical model is formulated as a system of two partial differential equations in terms of vorticity and stream function. The dynamics of the vortex structures is considered in a rectangular area under the assumption that periodic boundary conditions are imposed on the stream function. Investigation methods. The non-stationary problem is solved by the meshless vortex-in-cell method, based on the vorticity field approximation by its values in liquid particles and stream function expansion in the Fourier series cut. Results. Vortex structure consisting of four patches with different directions is investigated. The results of a numerical study of the dynamics and interaction of the structure are presented. The influence of the patch radius and the relative position of positively and negatively directed patches on the processes of interaction and mixing is studied. The obtained results correspond to the following possible scenarios: the initial configuration does not change over time; the initial configuration forms a new structure, which is maintained for longer times; the initial configuration returns to its initial state after a certain period of time. The processes of mass transfer of vorticity by liquid particles on a plane were calculated and analyzed. The results of a numerical analysis of the particles dynamics and trajectories on the entire plane and the field of local Lyapunov exponents are presented.

  1. Grigoryev Yu.N., Vshivkov V.A. Numerical Methods «Particle-in-Cells». Novosibirsk: Nauka, 2000 (in Russian).
  2. Dynnikova G.Ya. Fast technique for solving the N-body problem in flow simulation by vortex methods. Computational Mathematics and Mathematical Physics, 2009, vol. 49, no. 8, pp. 1389– 1396.
  3. Cottet G.H., Koumoutsakos P.D. Vortex methods. Cambridge University Press, 2000.
  4. Govorukhin V.N. A vortex method for computing two-dimensional inviscid incompressible flows. Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 6, pp. 1061–1073.
  5. Govorukhin V.N. Numerical analysis of the dynamics of distributed vortex configurations. Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 8, pp. 1474–1487.
  6. Revina S.V. Stability of the Kolmogorov flow and its modifications. Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 6, pp. 995–1012.
  7. Fortova S.V., Oparina E.I., Belotserkovskaya M.S. Numerical simulation of the Kolmogorov flow under the influence of the periodic field of the external force //Journal of Physics: Conference Series, 2018. Vol. 1128. 012089.
  8. Dombre T., Frisch U., Greene J.M., He´non M., Mehr A., Soward A.M. Chaotic streamlines in the ABC flows // Journal of Fluid Mechanics. 1986. Vol. 167. Pp. 353–391.
  9. Govorukhin V.N., Morgulis A.B., Yudovich V.I., Zaslavsky G.M. Chaotic advection in compressible helical flow // Physical Review E. 1999. Vol. 60, № 3. Pp. 2788–2798.
  10. Zaslavskii G.M., Sagdeev R.Z., Usikov D.A., Chernikov A.A. Minimal chaos, stochastic webs, and structures of quasicrystal symmetry. Sov. Phys. Usp, 1988, vol. 31, pp. 887–915.
  11. Zaslavky G.M. The Physics of Chaos in Hamiltonian Systems. Second edition. Imperial College Press, 2007.
  12. Govorukhin V.N., Filimonova A.M. Numerical calculation of planar geophysical flows of an inviscid incompressible fluid by a meshfree-spectral method. Computer research and modelling, 2019, vol. 11, no. 3, pp. 413–426 (in Russian).
  13. Govorukhin V.N. On the choice of a method for integrating the equations of motion of a set of fluid particles. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 4, pp. 706–718. 
  14. Aubry A., Chartier P. Pseudo-symplectic Runge–Kutta methods // BIT. 1998. Vol. 38, № 3. Pp. 439–461.
  15. Shadden S., Lekien F., Marsden J. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows // Phys. D: Nonlinear Phenomena. 2005. Vol. 212, № 3–4. Pp. 271–304.
  16. Haller G. Finding finite-time invariant manifolds in two-dimensional velocity fields // Chaos. 2000. Vol. 10, № 1. Pp. 99–108.
Short text (in English):
(downloads: 100)