Volume 9, no. 1Pages 5 - 19
Shock Waves in Gas SphereV.F. Kuropatenko, E.S. Shestakovskaya, M.N. Yakimova
Mathematical modelling is widely applied for researches in all natural sciences, industries, economy, biology and other areas. Already existing or new created models and numerical methods are used for the solution of specific problems. The most reliable way to check the adequacy of the differential scheme is to compare the numerical solution with the precise solution of the problem where it is possible. As an example of such 'reference' solution we construct a precise solution for the problem of a convergent shock wave and dynamic gas compression in a spherical vessel with an impermeable wall. Initially, the external border of the gas begins to move stepwise with a negative velocity, and the shock wave begins to propagate from border to gas. Acceleration of the border and sphericity determine the motion of the shock wave and the structure of the gas flow between the shock front and border. The considered problem formulation is fundamentally different from previously known statements of the problem of self-similar shock wave convergence to the center of symmetry and its reflection from the center with no boundary of gas. Full text
- shock wave; analytical solution; ideal gas; spherical symmetry.
- 1. Guderley G. Starke kugelige und zylindrische Verdichtungsstobe in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfartforschung, 1942, vol. 19, no. 9, pp. 302-312.
2. Sedov L.I. [On the Transient Motion of a Compressible Fluid]. Doklady Akademii Nauk SSSR, 1945, vol. 47, no. 2, pp. 94-96. (in Russian)
3. Stanjukovich К.P. [Similar Solutions of the Equations of Fluid Mechanics, Possessing Central Symmetry]. Doklady Akademii Nauk SSSR, 1945, vol. 48, no. 5, pp. 331-333. (in Russian)
4. 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.
5. Sedov, L.I. Metody podobiya i razmernosti v mehanike [Methods of Similarity and Dimensionality in Mechanics]. Moscow, Teh. teor. lit., 1954. 326 p.
6. Sidorov А.F. [Processes Conical Shock-free Compression and Expansion of Gas]. Journal of Applied Mathematics and Mechanics, 1994, vol 58, no. 4, pp. 81-92. (in Russian)
7. 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.
8. Kuropatenko V.F. Modeli Mehaniki Sploshnyh Sred [Models of Continuum Mechanics]. Chelyabinsk, CSU, 2007. 302 p.
9. Kuropatenko V.F., Kuznecova V.I., Mihajlova G.N., Kovalenko G.V., Sapozhnikova G.N. [The Complex WAVE and Heterogeneous Software Difference Method for Calculating Unsteady Motion of Compressible Continua]. Issues of Atomic Science and Physics Simulation Tehniki. Series: Mathematical Modeling of Physical Processes, 1989, no. 2, pp. 9-25. (in Russian)
10. Kuropatenko V.F., Yakimova M.N. A Method for Shock Calculation. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 2, pp. 60-70.