Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks
- First Online:
- 63 Downloads
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.
Keywordsshock 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
Unable to display preview. Download preview PDF.
- 5.G. L. Grodzovskii, “Self-similar gas motion induced by a strong explosion,” Dokl. Akad. Nauk SSSR 111, 969–971 (1956).Google Scholar
- 8.L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).Google Scholar
- 9.G. G. Chernyi, Gas Dynamics (Nauka, Moscow, 1988; CRC, Boca Raton, 1994).Google Scholar
- 10.A. N. Kraiko, Theoretical Gas Dynamics: Classics and the Present Day (Torus, Moscow, 2010) [in Russian].Google Scholar
- 11.G. G. Chernyi, Hypersonic Gas Flows (Fizmatgiz, Moscow, 1959) [in Russian].Google Scholar
- 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.V. A. Levin and G. G. Chernyi, “Asymptotic laws of the behavior of detonation waves,” Prikl. Mat. Mekh. 31, 393–405 (1967).Google Scholar
- 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