Stability of the Kolmogorov flow and its modifications

  • S. V. Revina


Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x 0, then the expansion coefficients of the velocity perturbations are even functions about x 0 for even powers of the wave number and odd functions about for x 0 odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.


stability of two-dimensional viscous flows Kolmogorov flow long wavelength asymptotics 


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  1. 1.
    V. I. Yudovich, “Example of the generation of a secondary steady or periodic flow when there is loss of a stability of the laminar flow of a viscous incompressible fluid,” Prikl. Mat. Mekh. 29 (3), 455–467 (1965).MathSciNetMATHGoogle Scholar
  2. 2.
    O. Reynolds, “An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels,” Phil. Trans. R. Soc. London 174, 935–982 (1883).CrossRefMATHGoogle Scholar
  3. 3.
    V. I. Arnol’d and L. D. Meshalkin, “A.N. Kolmogorov’s seminar on selected problems in analysis (1958–1959),” Usp. Mat. Nauk 15 (1), 247–250 (1960).Google Scholar
  4. 4.
    L. D. Meshalkin and Ya. G. Sinai, “Stability analysis of the stationary solution to a system of equations governing viscous incompressible planar flows,” Prikl. Mat. Mekh. 25 (6), 1140–1143 (1961).Google Scholar
  5. 5.
    O. V. Troshkin, “Algebraic structure of the two-dimensional stationary Navier–Stokes equations and nonlocal uniqueness theorems,” Dokl. Akad. Nauk SSSR 298 (6), 1372–1376 (1988).MathSciNetGoogle Scholar
  6. 6.
    E. I. Oparina and O. V. Troshkin, “Stability of Kolmogorov flow in a channel with rigid walls,” Dokl. Phys. 49, 583–587 (2004).MathSciNetCrossRefGoogle Scholar
  7. 7.
    O. V. Troshkin, “A dissipative top in a weakly compact Lie algebra and stability of basic flows in a plane channel,” Dokl. Phys. 57, 36–41 (2012).MathSciNetCrossRefGoogle Scholar
  8. 8.
    O. V. Troshkin, “Nonlinear stability of Couette, Poiseuille, and Kolmogorov plane channel flows,” Dokl. Math. 85, 181–185 (2012).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    S. O. Belotserkovskii, A. P. Mirabel’, and M. A. Chusov, “Construction of a postcritical regime for plane periodic flow,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 14 (1), 11–20 (1978).Google Scholar
  10. 10.
    A. M. Obukhov, “Kolmogorov flow and laboratory simulation of it,” Russ. Math. Surv. 35 (4), 113–126 (1983).CrossRefGoogle Scholar
  11. 11.
    S. O. Belotserkovskii and A. M. Oparin, Numerical Experiment: From Order to Chaos (Nauka, Moscow, 2000) [in Russian].MATHGoogle Scholar
  12. 12.
    S. V. Fortova, “Numerical simulation of the three-dimensional Kolmogorov flow in a shear layer,” Comput. Math. Math. Phys. 53 (3), 311–319 (2013).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    O. M. Belotserkovskii, A. V. Konyukhov, A. M. Oparin, O. V. Troshkin, and S. V. Fortova, “Structurization of chaos,” Comput. Math. Math. Phys. 51 (2), 222–234 (2011).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. V. Fortova, “Comparative analysis of eddy cascade formation in various turbulent problems,” Comput. Math. Math. Phys. 55 (2), 298–304 (2015).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    G. Chandler and R. Kerswell, “Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow,” J. Fluid Mech. 722, 554–595 (2013).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. Lucas and R. Kerswell, “Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow,” arXiv: 1406.1820v2 [physics.flu-dyn]. Apr. 1, 2015.MATHGoogle Scholar
  17. 17.
    V. I. Yudovich, “Instability of viscous incompressible parallel flows with respect to spatially periodic perturbations,” in Numerical Methods for Problems in Mathematical Physics (Nauka, Moscow, 1966), pp. 242–249 [in Russian].Google Scholar
  18. 18.
    V. I. Yudovich, “Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under longwavelength periodic disturbances,” Fluid Dyn. 8, 26–29 (1973).CrossRefGoogle Scholar
  19. 19.
    V. I. Yudovich, “Instability of long-wave viscous flows,” Fluid Dyn. 25, 516–521 (1990).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    S. V. Revina and V. I. Yudovich, “Initiation of self-oscillations at loss of stability of spatially-periodic threedimensional viscous flows with respect to long-wave perturbations,” Fluid Dyn. 36, 192–203 (2001).CrossRefMATHGoogle Scholar
  21. 21.
    A. P. Melekhov and S. V. Revina, “Onset of self-oscillations upon the loss of stability of spatially periodic twodimensional viscous fluid flows relative to long-wave perturbations,” Fluid Dyn. 43, 203–216 (2008).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    S. V. Revina, “Long wavelength asymptotics for the problem of stability of time periodic flows,” Proceedings of the 15th International Conference on Modern Problems in Continuum Mechanics (Southern Federal Univ., Rostov- on-Don, 2011), Vol. 1, pp. 204–208.Google Scholar
  23. 23.
    S. V. Revina, “Recurrence formulas for long wavelength asymptotics in the problem of shear flow stability,” Comput. Math. Math. Phys. 53 (8), 1207–1220 (2013).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    S. V. Revina, “Stability of spatially periodic and nearly periodic viscous flows,” Mat. Forum (Itogi Nauki Yug Ross.) 8 (Part 2), 130–141 (2014).MathSciNetGoogle Scholar
  25. 25.
    S. V. Revina, Stability of the Kolmogorov Flow and Its Modifications, Available from VINITI, No. 8-B2016 (Moscow, 2016).Google Scholar
  26. 26.
    M. M. Vainberg and V. A. Trenogin, “The methods of Lyapunov and Schmidt in the theory of non-linear equations and their further development,” Russ. Math. Surv. 17 (2), 1–60 (1962).CrossRefMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Southern Federal UniversityRostov-on-DonRussia
  2. 2.Southern Mathematical Institute, Vladikavkaz Research CenterRussian Academy of Sciences and the Government of the Republic of North Ossetia-AlaniaVladikavkaz, Republic of North Ossetia-AlaniaRussia

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