Skip to main content
Log in

Numerical Analysis of Laminar–Turbulent Transition by Methods of Chaotic Dynamics

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

This paper summarizes the results of studies of the laminar–turbulent transition in some fluid and gas dynamics problems obtained by applying numerical methods and methods of chaotic dynamics. The following problems are analyzed: 2D and 3D Kolmogorov problems in a periodic domain, 3D Rayleigh–Benard convection in rectangular domains, 3D backward-facing step flow, and development of 3D Rayleigh–Taylor and Kelvin–Helmholtz instabilities in viscous compressible flows. An analysis confirms that instabilities develop via cascades of subcritical or supercritical bifurcations. In all systems, a universal scenario of the transition to chaos (Feigenbaum–Sharkovskii–Magnitskii scenario) is found along with other scenarios of chaotization of dynamical systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

Similar content being viewed by others

Notes

  1.  The projection of the phase space, the Poincaré section, and the corresponding physical solution are shown in Figs. 1 and 2.

REFERENCES

  1. V. I. Yudovich, Math. Motes 49 (5), 540–545 (1991).

    Google Scholar 

  2. K. V. Demyanko and Yu. M. Nechepurenko, Dokl. Phys. 56 (10), 531–533 (2011).

    Article  Google Scholar 

  3. C. K. Mamun and L. S. Tuckerman, Phys. Fluids 7, 80 (1995). https://doi.org/10.1063/1.868730

    Article  MathSciNet  Google Scholar 

  4. N. V. Nikitin and V. O. Pimanov, Uchen. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki 157 (3) 111–116 (2015).

    Google Scholar 

  5. N. M. Evstigneev, N. A. Magnitskii, and D. A. Silaev, Differ. Equations 51 (10) 1292–1305 (2015).

    Article  MathSciNet  Google Scholar 

  6. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 (4) 299–303 (1941).

    Google Scholar 

  7. A. V. Fursikov, Dokl. Akad. Nauk SSSR 319 (1) 83–87 (1991).

    Google Scholar 

  8. A. L. Afendikov and K. I. Babenko, Mat. Model. 1 (8) 45–74 (1989).

    MathSciNet  Google Scholar 

  9. O. A. Ladyzhenskaya, Zap. Nauchn. Sem. LOMI 27, 91–115 (1972).

    Google Scholar 

  10. N. A. Magnitskii, Theory of Dynamical Chaos (Lenand, Moscow, 2011) [in Russian].

    MATH  Google Scholar 

  11. N. Evstigneev and N. Magnitskii, J. Appl. Nonlinear Dyn. 6, 345–353 (2017). https://doi.org/10.5890/JAND.2017.09.003

    Article  MathSciNet  Google Scholar 

  12. N. M. Evstigneev and N. A. Magnitskii, in Nonlinearity, Bifurcation, and Chaos: Theory and Applications (INTECH, 2013), pp. 250–280. https://doi.org/10.5772/48811

  13. N. M. Evstigneev, Open J. Fluid Dyn. 6, 496–539 (2016). https://doi.org/10.4236/ojfd.2016.64035

    Article  Google Scholar 

  14. N. M. Evstigneev and N. A. Magnitskii, in Turbulence Modeling Approaches: Current State, Development Prospects, Applications (INTECH, 2017), pp. 29–60. https://doi.org/10.5772/67918

  15. N. M. Evstigneev and N. A. Magnitskii, Tr. Inst. Sist. Anal. Ross. Akad. Nauk 62 (4), 85–102 (2012).

    Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research, grant nos. 18-29-10008 mk and 20-07-00066.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to N. M. Evstigneev or N. A. Magnitskii.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Evstigneev, N.M., Magnitskii, N.A. Numerical Analysis of Laminar–Turbulent Transition by Methods of Chaotic Dynamics. Dokl. Math. 101, 110–114 (2020). https://doi.org/10.1134/S1064562420020118

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562420020118

Keywords

Navigation