Skip to main content
SpringerLink
Log in

Qualitative analysis of dynamics in Kolmogorov’s problem on a flow of a viscous incompressible fluid

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We numerically analyze the qualitative behavior of solutions in the two-dimensional problem on a periodic flow of a viscous incompressible fluid with an exciting force (the Kolmogorov problem). To construct an approximate solution, we use the Bubnov–Galerkin method based on spline approximations. We prove the convergence of the method to the solution of the problem in the weak sense. We compare the obtained results with the results of other authors as well as with the Galerkin method based on trigonometric polynomials and with a hybrid finite-volume finite-element method. We carry out a bifurcation analysis of the problem and indicate a scenario of passage from regular solutions to chaotic ones.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I. and Meshalkin, L.D., Mathematical Life in USSR: A. N. Kolmogorov Sem. on Selected Problems of Analysis (1958–1959), Uspekhi Mat. Nauk, 1960, vol. 15, no. 1 (91), pp. 247–250.

    Google Scholar 

  2. Meshalkin, L.D. and Sinai, Ya.G., Investigation of the Stability of a Stationary Solution of a System of Equations for the Plane Movement of an Incompressible Viscous Liquid, Prikl. Mat. Mekh., 1961, vol. 25, no. 6, pp. 1140–1143.

    Google Scholar 

  3. Yudovich, V.I., On Self-Oscillations Arising in the Loss of Stability of Parallel Flows of a Viscous Fluid under Long-Wave Periodic Perturbations, Izv. Akad. Nauk SSSR. Mekh. Zhidk. Gaza, 1973, no. 1, pp. 32–35.

    Google Scholar 

  4. Yudovich, V.I., An Example of the Loss of Stability and the Generation of a Secondary Flow of a Fluid in a Closed Container, Mat. Sb., 1967, vol. 74 (116), no. 4, pp. 565–579.

    MathSciNet  Google Scholar 

  5. Sivashinsky, G., Weak Turbulence in Periodic Flows, Phys. D, 1985, vol. 17, no. 2, pp. 243–255.

    Article  MATH  MathSciNet  Google Scholar 

  6. Okamoto, H. and Shoji, M., Bifurcation Diagrams in Kolmogorov’s Problem of Viscous Incompressible Fluid on 2D Flat Tori, Japan J. Indust. Appl. Math., 1993, vol. 10, no. 2, pp. 191–218.

    Article  MATH  MathSciNet  Google Scholar 

  7. Belotserkovskii, S.O., Mirabel’, A.Ya., and Chusov, M.A., On the Construction of an After-Critical Model for a Plane Periodic Flow, Izv. Akad. Nauk SSSR Fiz. Atmosf. Okeana 1978, vol. 14, no. 1, pp. 11–14.

    Google Scholar 

  8. Zhi-Min Chen, Hopf Bifurcation of the Three-Dimensional Navier–Stokes Equations, J. Math. Anal. Appl., 1999, vol. 237, no. 2, pp. 583–608.

    Article  MATH  MathSciNet  Google Scholar 

  9. Kim, S.C. and Okamoto, H., Bifurcations and Inviscid Limit of Rhombic Navier–Stokes Flows in Tori, IMA J. Appl. Math., 2003, vol. 68, no. 2, pp. 119–134.

    Article  MATH  MathSciNet  Google Scholar 

  10. Belotserkovskii, S.O., Modeling of Flows of a Viscous Incompressible Fluid on the Basis of Navier–Stokes Equations, Cand. Sci. (Phys.–Math.) Dissertation, Moscow, 1979.

    Google Scholar 

  11. Dan Lucas and Rich Kerswell, Spatiotemporal Dynamics in 2D Kolmogorov Flow over Large Domains, arXiv: 1308.3356v1, Received March 12, 2014.

    Google Scholar 

  12. Evstigneev, N.M. and Magnitskii, N.A., FSM Scenarios of Laminar–Turbulent Transition in Incompressible Fluids. Chapter 10, in Nonlinearity, Bifurcation and Chaos–Theory and Applications, INTECH, 2013, pp. 250–280.

    Google Scholar 

  13. Evstigneev, N.M. and Magnitskii, N.A., On Possible Scenarios of Transition to Turbulence in Rayleigh–Bénard Convection, Dokl. Akad. Nauk, 2010, vol. 433, no. 3, pp. 318–322.

    MathSciNet  Google Scholar 

  14. Evstigneev, N.M., Magnitskii, N.A., and Sidorov, S.V., Nonlinear Dynamics of Laminar–Turbulent Transition in Three Dimensional Rayleigh–Bénard Convection, Commun. Nonlinear Sci. Numer. Simul., 2010, vol. 15, no. 10, pp. 2851–2859.

    Article  MATH  MathSciNet  Google Scholar 

  15. Silaev, D.A., Amiliyushenko, A.V., Luk’janov, A.I., and Korotaev, D.O., Semilocal Smoothing Spline of Class C 1, J. Math. Sci., 2007, vol. 143, no. 4, pp. 3401–3414.

    Article  MathSciNet  Google Scholar 

  16. Silaev, D.A., Twice Continuously Differentiable S-Splines, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2007, no. 2, pp. 12–17.

    MathSciNet  Google Scholar 

  17. Silaev, D.A., Twice Continuously Differentiable Semilocal Smoothing Spline, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2009, no. 5, pp. 187–194.

    MathSciNet  Google Scholar 

  18. Silaev, D.A. and Korotaev, D.O., Solution of Boundary Value Problems with the Use of S-Spline, Komp’yut. Issled. Model., 2009, vol. 1, no. 2, pp. 161–171.

    Google Scholar 

  19. Zav’yalov, Yu.S., Kvasov, B.I., and Miroshnichenko, V.L., Metody splain-funktsii (Methods of Spline Functions), Moscow: Nauka, 1980.

    Google Scholar 

  20. Temam, R., Navier–Stokes Equations. Theory and Numerical Analysis, Amsterdam: North-Holland, 1977. Translated under the title Uravneniya Nav’e–Stoksa. Teoriya i chislennyi analiz, Moscow: Mir, 1981.

    MATH  Google Scholar 

  21. Ming-Jun Lai1, Chun Liu, and Paul Wenston, Bivariate Spline Method for Numerical Solution of Time Evolution Navier–Stokes Equations over Polygons in Stream Function Formulation, Numer. Methods Partial Differential Equations, 2003, vol. 19, no. 6, pp. 776–827.

    Article  MATH  MathSciNet  Google Scholar 

  22. Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor, Strong Stability-Preserving High-Order Time Discretization Methods, SIAM Rev., 2001, vol. 43, no. 1, pp. 89–112.

    Article  MATH  MathSciNet  Google Scholar 

  23. Evstigneev, N.M., Magnitskii, N.A., and Sidorov, S.V., On the Nature of Turbulence in Rayleigh–Bénard Convection, Differ. Uravn., 2009, vol. 45, no. 1, pp. 69–73.

    MathSciNet  Google Scholar 

  24. Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press., 1995. Translated under the title Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Gorodetskii, A.S., Ed., Moscow: MTsNMO, 2005.

    Book  MATH  Google Scholar 

  25. Magnitskii, N.A., Teoriya dinamicheskogo khaosa (Theory of Dynamical Chaos), Moscow, 2011.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute Systems Analysis, Russian Academy of Sciences, Moscow, Russia

    N. M. Evstigneev, N. A. Magnitskii & D. A. Silaev

  2. Lomonosov Moscow State University, Moscow, Russia

    N. M. Evstigneev, N. A. Magnitskii & D. A. Silaev

Authors
  1. N. M. Evstigneev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Magnitskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. A. Silaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. M. Evstigneev.

Additional information

Original Russian Text © N.M. Evstigneev, N.A. Magnitskii, D.A. Silaev, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 10, pp. 1302–1314.

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. & Silaev, D.A. Qualitative analysis of dynamics in Kolmogorov’s problem on a flow of a viscous incompressible fluid. Diff Equat 51, 1292–1305 (2015). https://doi.org/10.1134/S0012266115100055

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266115100055

Keywords

Search

Navigation