Volume 10, no. 4Pages 46 - 55

Focusing of Cylindrically Symmetric Shock in a Gas

V.F. Kuropatenko, F.G. Magazov, E.S. Shestakovskaya
The analytical solution of the problem of converging the shock in the cylindrical vessel with an impermeable wall is constructed for arbitrary self-similar coefficients in Lagrangian coordinates. The negative velocity is set at the cylinder boundary. At the initial time the shock spreads from this point into the center of symmetry. The cylinder boundary moves under the particular law which conforms to the movement of the shock. It moves in Euler coordinates, but the boundary trajectory is a vertical line in Lagrangian coordinates. Generally speaking, all the trajectories of the particles are vertical lines. The value of entropy which appeared on the shock etains along each of these lines. Equations that determine the structure of the gas flow between the shock front and the boundary as a function of time and the Lagrangian coordinate are obtained, as well as the dependence of the entropy on the shock velocity. Thus, the problem is solved for Lagrangian coordinates. It is fundamentally different from previously known formulations of the problem of the self-convergence of the self-similar shock to the center of symmetry and its reflection from the center which were constructed for the infinite area in Euler coordinates for a unique self-similar coefficient corresponding to the unique value of the adiabatic index.
Full text
shock; cylindrical symmetry; ideal gas; analytical solution.
1. Lax P.D., Richtmyer R.D. Survey of Stability of Linear Finite Difference Equations. Communications on Pure and Applied Mathematics, 1956, vol. 9, pp. 267-293.
2. Roach P.J. Computational Fluid Dynamics. Albuquerque, Hermosa Publishers, 1998.
3. Guderley G. Starke kugelige und zylindrische verdichtungsstobe in der nahe des kugelmittelpunktes bzw. der zylinderachse. Luftfartforschung, 1942, vol. 19, no. 9, pp. 302-312. (in German)
4. Sedov L.I. [On the Transient Motion of a Compressible Fluid]. Soviet Mathematics, 1945, vol. 47, no. 2, pp. 94-96. (in Russian)
5. Stanjukovich К.P. [Similar Solutions of the Equations of Fluid Mechanics, Possessing Central Symmetry]. Soviet Mathematics, 1945, vol. 48, no. 5, pp. 331-333. (in Russian)
6. Brushlinskii K.V., Kazhdan Ja.M. On Auto-Models in the Solution of Certain Problems of Gas Dynamics. Russian Mathematical Surveys, 1963, vol. 18, no. 2, pp. 1-22.
7. Sedov L.I. Metody podobija i razmernosti v mehanike [Methods of Similarity and Dimensionality in Mechanics]. Moscow, Gosudarstvennoe izdatel'stvo tehniko-teoreticheskoy literatury, 1954.
8. Sidorov А.F. [Processes Conical Shock-Free Compression and Expansion of Gas]. tJournal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 4, pp. 81-92. (in Russian)
9. Sidorov А.F. [On Optimal Unstressed Compression of the Gas Layers]. Soviet Mathematics, 1990, vol. 313, no. 2, pp. 283-287. (in Russian)
10. Kraiko A.N. Teoreticheskaya gazovaya dinamika: klassika i sovremennost' [Theoretical Gas Dynamics: Classic and Modern]. Moscow, Torus Press, 2010.
11. Kraiko А.N. Rapid Cylindrically and Spherically Symmetric Strong Compression of a Perfect Gas. Journal of Applied Mathematics and Mechanics, 2007, vol. 71, no. 5, pp. 676-689.
12. Kuropatenko V.F. Modeli mehaniki sploshnyh sred [Models of Continuum Mechanics]. Chelyabinsk, Chelyabinsk State University, 2007.
13. Kuropatenko V.F., Shestakovskaya E.S., Yakimova M.N. Dynamic Compression of a Cold Gas Sphere. Doklady Phisics, 2015, vol. 60, no. 4, pp. 180-182.
14. Kuropatenko V.F., Shestakovskaya E.S. Analytical Solution of the Problem of a Shock Wave in the Collapsing Gas in Lagrangian Coordinates. AIP Conference Proceedings, 2016, vol. 1770, p. 030069. DOI: 10.1063/1.4964011