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Journal of Applied and Industrial Mathematics

, Volume 1, Issue 3, pp 343–350 | Cite as

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

  • K. V. Kurmaeva
  • S. S. Titov
Article

Abstract

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.

Keywords

Symmetry Axis Logarithmic Singularity Transonic Flow Axisymmetric Problem Thin Body 
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References

  1. 1.
    A. F. Sidorov, Selected Works: Mathematics and Mechanics (Fizmatlit, Moscow, 2001) [in Russian].Google Scholar
  2. 2.
    L. V. Ovsyannikov, “Convergence of the Mayer Series for an Axisymmetric Nozzle,” in Calculation of the Part of the Two-Dimensional and Axisymmetric Nozzles with a Curvilinear Break, Suppl. 1, Ed. by E. Martensen and K. von Sengbusch (Izd. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962), pp. 41–43 [in Russian].Google Scholar
  3. 3.
    Kh. Sh. Tzyan, Ts. Ts. Lin, and E. Reissner, “On Two-Dimensional Unsteady Motion of a Thin Body in an Incompressible Fluid,” in Gas Dynamics (Inostrannaya Literatura, Moscow, 1950), pp. 183–196 [in Russian].Google Scholar
  4. 4.
    V. M. Teshukov, “A Spatial Problem of Propagation of a Contact Discontinuity in an Ideal Gas,” Dinamika Sploshn. Sredy, Issue 32, 82–94 (1977).Google Scholar
  5. 5.
    V. M. Teshukov, “Construction of the Shock Wave Front in a Spatial Piston Problem,” Dinamika Sploshn. Sredy, Issue 33, 114–133 (1978).Google Scholar
  6. 6.
    V. M. Teshukov, “Centered Waves in Three-Dimensional Gas Flows,” Dinamika Sploshn. Sredy, Issue 39, 102–118 (1979).Google Scholar
  7. 7.
    V. M. Teshukov, “Destruction of an Arbitrary Discontinuity on a Curvilinear Surface,” Zh. Prikl. Mekh. i Tekhn. Fiz., No. 2, 126–133 (1980).Google Scholar
  8. 8.
    V. M. Teshukov, “A Spatial Analog of the Riemann and Prandtl—Meyer Centered Waves,” Zh. Prikl. Mekh. i Tekhn. Fiz., No. 4, 98–106 (1982).Google Scholar
  9. 9.
    S. P. Bautin, Analytic Heat Wave (Fizmatlit, Moscow, 2003) [in Russian].Google Scholar
  10. 10.
    S. S. Titov, “On Transonic Gas Flow Around Thin Solids of Revolution,” in Analytical Methods of Continuum Mechanics (Izd. Inst. Mat. Mekh., Sverdlovsk, 1979), Issue 33, pp. 65–72.Google Scholar
  11. 11.
    L. V. Ovsyannikov, Lectures on the Fundamentals of Gas Dynamics (Izd. Inst. Komp’yuternykh Issledovanii, Izhevsk, 2003) [in Russian].Google Scholar
  12. 12.
    S. S. Titov, “Solution of Equations with Singularities in Analytic Scales of Banach Spaces,” Preprint (Ural State Academy of Architecture and Arts, Ekaterinburg, 1999).Google Scholar
  13. 13.
    H. W. Liepmann and A. Roshko, Elements of Gas Dynamics (Wiley, New York, 1957; Inostrannaya Literatura, Moscow, 1960).Google Scholar
  14. 14.
    K. G. Guderley, The Theory of Transonic Flow (Inostrannaya Literatura, Moscow, 1960; Pergamon Press, Oxford, 1962).Google Scholar
  15. 15.
    M. Van Dyke, “Supersonic Flow past Bodies of Revolution,” Aeronaut. Sci. 18 (1951).Google Scholar
  16. 16.
    N. A. Blinkov and I. A. Chernov, “Reduction of an Unsteady Transonic Equation to a Quasisteady Equation,” in Aerodynamics: Two-Dimensional and Axisymmetric Fluid Flows (Izd. Saratov. Gos. Univ., Saratov, 1988), pp. 110–131.Google Scholar
  17. 17.
    N. A. Lar’kin, Smooth Solutions to the Equations of Transonic Gas Dynamics (Nauka, Novosibirsk, 1991) [in Russian].Google Scholar
  18. 18.
    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (Nauka, Moscow, 1983) [in Russian].MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • K. V. Kurmaeva
    • 1
  • S. S. Titov
    • 1
  1. 1.Ural State University of Railway EngineeringYekaterinburgRussia

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