Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

  • Kh. F. Valiyev
  • A. N. KraikoEmail author


In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.


shock wave reflection method of characteristics center of symmetry unbounded speed of sound different adiabatic exponents sonic shock wave of finite intensity new self-similar solutions with two reflected shock waves 


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  1. 1.
    Kh. F. Valiyev and A. N. Kraiko, “Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “Reflected” shock wave,” J. Appl. Math. Mech. 75, 675–690 (2011).MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959; Nauka, Moscow, 1987).zbMATHGoogle Scholar
  3. 3.
    G. Guderley, “Starke kugelige und zylindrische Verdichtungsstöße in der Nähe des Kugelmittelpunktes bzw. der zylinderachse,” Luftfartforschung 19, 302–312 (1942).MathSciNetGoogle Scholar
  4. 4.
    G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics (Consultants Bureau, New York, 1979; Gidrometeoizdat, Leningrad, 1982).zbMATHCrossRefGoogle Scholar
  5. 5.
    G. L. Grodzovskii, “Self-similar gas motion induced by a strong explosion,” Dokl. Akad. Nauk SSSR 111, 969–971 (1956).Google Scholar
  6. 6.
    Kh. F. Valiyev, “The reflection of a shock wave from a centre or axis of symmetry at adiabatic exponents from 1.2 to 3,” J. Appl. Math. Mech. 73, 281–289 (2009).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kh. F. Valiyev and A. N. Kraiko, “Cylindrically and spherically symmetrical rapid intense compression of an ideal perfect Gas with adiabatic exponents from 1.001 to 3,” J. Appl. Math. Mech. 75, 218–226 (2011).CrossRefGoogle Scholar
  8. 8.
    L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).Google Scholar
  9. 9.
    G. G. Chernyi, Gas Dynamics (Nauka, Moscow, 1988; CRC, Boca Raton, 1994).Google Scholar
  10. 10.
    A. N. Kraiko, Theoretical Gas Dynamics: Classics and the Present Day (Torus, Moscow, 2010) [in Russian].Google Scholar
  11. 11.
    G. G. Chernyi, Hypersonic Gas Flows (Fizmatgiz, Moscow, 1959) [in Russian].Google Scholar
  12. 12.
    S. S. Kvashnina and G. G. Chernyi, “Steady detonating gas flow past a cone,” Prikl. Mat. Mekh. 23(1), 182–186 (1959).MathSciNetGoogle Scholar
  13. 13.
    G. G. Chernyi, “Self-similar problems of combustible gas mixture flow past bodies,” Fluid Dyn. 1(6), 5–13 (1966).CrossRefGoogle Scholar
  14. 14.
    A. N. Kraiko and N. I. Tillyaeva, “Combustible gas mixture flow past a cone with Chapman-Jouguet detonation wave,” Prikl. Mat. Mekh. 77(1), 3–14 (2013).Google Scholar
  15. 15.
    V. A. Levin and G. G. Chernyi, “Asymptotic laws of the behavior of detonation waves,” Prikl. Mat. Mekh. 31, 393–405 (1967).Google Scholar
  16. 16.
    E. A. Afanas’eva and V. A. Levin, “Multifront detonation combustion of matter,” Fluid Dyn. 17, 268–272 (1982).CrossRefGoogle Scholar
  17. 17.
    B. Wendroff, “The Riemann problem for materials with nonconvex equations of state. I: Isentropic flow,” J. Math. Anal. Appl. 38, 454–456 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    B. Wendroff, “The Riemann problem for materials with nonconvex equations of state. II: General flow,” J. Math. Anal. Appl. 38, 640–658 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Liu Tai-Ping, “The Riemann problem for general systems of conservation laws,” J. Differ. Equations 18, 218–234 (1975).zbMATHCrossRefGoogle Scholar
  20. 20.
    Liu Tai-Ping, “Existence and uniqueness theorems for Riemann problems,” Trans. Am. Math. Soc. 212, 375–382 (1975).zbMATHCrossRefGoogle Scholar
  21. 21.
    B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).Google Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Baranov Central Institute of Aviation Motors (CIAM)MoscowRussia

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