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Selfmodelled GAS flow from the surface of a circular cone or wedge with an associated shock wave

USSR Computational Mathematics and Mathematical Physics
Publication Date
DOI: 10.1016/0041-5553(65)90038-8
  • Mathematics


Abstract Gas flow in modern energy transport systems, wind tunnels etc., is in general described by a system of non-stationary equations. Because of the geometrical symmetry of the system, it is usually possible to discard one of the space coordinates in the equations. For selfmodelled flows the number of independent variables is reduced to two. The approach to problems of this latter type is therefore considerably simplified, in spite of the fact that the equations may be of a mixed type. We shall consider the seifmodelled magnetic flow of a gas from the walls of a circular cone or two-dimensional sector as applied to the theory of a pulse discharge in a conical chamber (see [l] - [2]). In the case of an ideally conducting medium (strong skin effect) a cavern, increasing with time and filled by magnetic field, is formed between the chamber walls and the gas. The influence of the magnetic field is similar to the action of a rotationally symmetric piston, compressing the gas towards the axis of symmetry and forcing it from the conical chamber. To determine the shape of the cavern we have to use the condition for the equality of the magnetic pressure and the gas pressure at the cavern boundary. We consider in this paper the gas-dynamical problem of conical compression. The pressure on the piston and other required functions have been found for a given law of expansion of the cavern. For selfmodelled motion this law may be written in the following genral form: r 1 = [ tf( θ 1)] 6, where t is time, and r 1 and θ 1 are the polar coordinates of the piston. Conical compression is obtained in the limit as δ → ∞. in this compression a shock wave is associated with the vertex of the conical chamber. A numerical solution of the problem is obtained by the characteristics method. In this case it is a matter of indifference in principle whether the pressure distribution over the piston, or the piston motion, is given.

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