Focusing of cylindrically symmetric shock in a gas

V. F. Kuropatenko ; F. G. Magazov ; E. S. Shestakovskaya

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 4, pp. 46-55

Résumé

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.

Détail

Keywords: shock; cylindrical symmetry; ideal gas; analytical solution.