Asymptotic Approach to the Problem of Boundary Layer Instability in Transonic Flow

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Abstract

Tollmien–Schlichting waves can be analyzed using the Prandtl equations involving selfinduced pressure. This circumstance was used as a starting point to examine the properties of the dispersion relation and the eigenmode spectrum, which includes modes with amplitudes increasing with time. The fact that the asymptotic equations for a nonclassical boundary layer (near the lower branch of the neutral curve) have unstable fluctuation solutions is well known in the case of subsonic and transonic flows. At the same time, similar solutions for supersonic external flows do not contain unstable modes. The bifurcation pattern of the behavior of dispersion curves in complex domains gives a mathematical explanation of the sharp change in the stability properties occurring in the transonic range.

Keywords

free interaction boundary layer transonic and subsonic flow stability dispersion relation Airy function Tollmien–Schlichting wave spectrum of eigenmodes increment of growth phase velocity wave number neutral curve 

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References

  1. 1.
    O. S. Ryzhov and I. V. Savenkov, “Stability of a boundary layer with transonic external-flow velocities,” J. Appl. Mech. Tech. Phys. 31 (2), 222–227 (1990).CrossRefGoogle Scholar
  2. 2.
    F. T. Smith, “On the nonparallel flow stability of the Blasius boundary layer,” Proc. R. Soc. London Ser. A 366 (1724), 91–109 (1979).CrossRefMATHGoogle Scholar
  3. 3.
    V. I. Zhuk and O. S. Ryzhov, “Free interaction and stability of a boundary layer in incompressible fluid,” Dokl. Akad. Nauk SSSR 253 (6), 1326–1329 (1980).MathSciNetMATHGoogle Scholar
  4. 4.
    O. S. Ryzhov and V. I. Zhuk, “Stability and Separation of Freely Interacting Boundary Layers,” Lect. Notes Phys. 141, 360–366 (1981).CrossRefGoogle Scholar
  5. 5.
    B. J. Bodonyi and F. T. Smith, “The upper branch stability of the Blasius boundary layer, including non-parallel flow effect,” Proc. R. Soc. London Ser. A 375 (1760), 65–92 (1981).CrossRefMATHGoogle Scholar
  6. 6.
    K. V. Guzaeva and V. I. Zhuk, “Asymptotic theory of perturbations inducing a pressure gradient in a transonic flat-plate boundary layer,” Comput. Math. Math. Phys. 48 (1), 121–138 (2008).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    V. I. Zhuk and A. V. Chernyshev, “Dispersion equations in the stability problem for transonic flows and some of their properties,” Comput. Math. Math. Phys. 50 (1), 121–138 (2010).CrossRefMATHGoogle Scholar
  8. 8.
    V. I. Zhuk and O. S. Ryzhov, “On a property of the linearized boundary layer equations with self-induced pressure,” Dokl. Akad. Nauk SSSR 240 (5), 1042–1045 (1978).MATHGoogle Scholar
  9. 9.
    O. S. Ryzhov and V. I. Zhuk, “Internal waves in the boundary layer with the self-induced pressure,” J. Mécanique 19 (2), 561–580 (1980).MathSciNetMATHGoogle Scholar
  10. 10.
    V. I. Zhuk, Tollmien–Schlichting Waves and Solitons (Nauka, Moscow, 2001) [in Russian].Google Scholar
  11. 11.
    O. S. Ryzhov, “Triple-deck instability of supersonic boundary layers,” AIAA J. 50 (8), 1733–1741 (2012).CrossRefGoogle Scholar
  12. 12.
    V. Ya. Neiland, “Asymptotic problems in the theory of supersonic viscous flows,” in Proceedings of Central Institute of Aerohydrodynamics (Moscow, 1974), No. 1529 [in Russian].Google Scholar
  13. 13.
    K. Stewartson, “Multistructured boundary layers on flat plates and related bodies,” Adv. Appl. Mech. 14, 145–239 (1974).CrossRefGoogle Scholar
  14. 14.
    V. V. Sychev, A. I. Ruban, Vik. V. Sychev, and G. L. Korolev, Asymptotic Theory of Separation Flows, Ed. by V.V. Sychev (Nauka, Moscow, 1987) [in Russian].Google Scholar
  15. 15.
    V. I. Zhuk, “Nonlinear perturbations inducing a natural pressure gradient in a boundary layer on a plate in a transonic flow,” J. Appl. Math. Mech. 57 (5), 825–835 (1993).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    C. C. Lin, E. Reissner, and H. S. Tsien, “On two-dimensional non-steady motion of a slender body in a compressible fluid,” J Math. Phys. 27 (3), 220–231 (1948).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    J. D. Cole and L. P. Cook, Transonic Aerodynamics (North-Holland, Amsterdam, 1986).MATHGoogle Scholar
  18. 18.
    W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965).MATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,”Russian Academy of SciencesMoscowRussia

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