Stability of Quasi-two-dimensional Shear Flows with Arbitrary Velocity Profiles

  • Felix V. Dolzhansky
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 103)

Abstract

Thus one can conclude the following: the linear stability theory of the strictly two-dimensional Kolmogorov flow, i.e., constructed without taking into account external friction, is structurally unstable with respect to the inclusion of the latter and, conversely, the linear theory of the quasi-two-dimensional flow, i.e., constructed by taking into account external friction, qualitatively is not sensitive to the inclusion or exclusion of internal viscosity. Moreover, it is easy to see that for \(\lambda_{0}=\widehat{\lambda}/\widehat{\nu}=R_{\nu}/R_{\lambda} \gg 1\), the results in the quasi-two-dimensional linear theory are almost self-similar in R ν . What is most important is that this conclusion holds for the nonlinear stability theory, as we shall see below.

Is the situation described above an exclusive feature of the Kolmogorov flow, or is it typical for shear flows with arbitrary profiles? To answer this seemingly very difficult question, we will give a new interpretation of the results in the preceding chapter, which will allow us to draw certain conclusions regarding the stability of quasi-two-dimensional shear flows, using the well-developed stability theory of strictly two-dimensional flows.

Keywords

Shear Flow Dispersion Curve Potential Vorticity Neutral Stability Critical Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. R. Betchov and V. Kriminale, Questions of Hydrodynamic Stability, Mir, Moscow, 1971. Google Scholar
  2. F.V. Dolzhansky, V.A. Krymov, and D.Yu. Manin, Stability and quasi-two-dimensional vortex structures of shear flows, UFN, Vol. 160, No. 7, 1990. Google Scholar
  3. P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge Univ. Press, Cambridge, 1981. MATHGoogle Scholar
  4. E.B. Gledzer, F.V. Dolzhansky, and A.M. Obukhov, Systems of Hydrodynamic Type and Their Applications, Nauka, Moscow, 1981. Google Scholar
  5. R. Hide and P.T. Mason, Sloping convection in rotating fluid, Adv. Phys., Vol. 24, No. 1, 1975. Google Scholar
  6. C.C. Lin, The Theory of Hydrodynamic Stability, Cambridge Univ. Press, Cambridge, 1966. Google Scholar
  7. D.Yu. Manin, Stability and supercritical regimes of quasi-two-dimensional shear flows in the presence of external friction (theory), Izv. USSR, Fluid Dyn., Vol. 2, 1989. Google Scholar
  8. N.N. Romanova and S.Yu. Annenkov, Three-wave resonant interactions in unstable media, J. Fluid Mech., Vol. 539, No. 57, 2005. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix V. Dolzhansky
    • 1
  1. 1.

Personalised recommendations