The behaviour of shock waves at large distances from the point of explosion

Abstract

This paper is concerned with the investigation of the behaviour of shock waves at large distances from the point of explosion. The work was done in 1957 and is a continuation of [1]. The numerical calculation [1] of the propagation of a strong spherical shock wave in an ideal gas at rest having the density ρ 0 under pressure P 0 was carried out up to a pressure drop of 1.008 at the shock wave front. As a result the formation of two phases of hydrodynamic parameters were discovered—compression and rarefaction, spreading directly behind the shock wave. The moment of their formation corresponds to a pressure drop of about 1–14 at the front. In the remaining region around the centre the values of the pressure, density and velocity of the particles are close to those in the unperturbed medium. The computing net comprised the entire gradually expanding region from the centre to the shock wave front. Because of this the number of computing points pertaining to the actually perturbed region consisting of two phases gradually went down and the calculation could not be continued far enough. A new calculation was undertaken to explain the behaviour of shock waves far from the centre of explosion. The distribution of hydrodynamic parameters corresponding to a pressure drop of 1.1079 at the shock wave front was taken as the initial data. The computing net embraced only the region of perturbed motion directly behind the shock wave front. In the remainder of the space behind the shock wave the gas was taken to be at rest. In view of the fact that at a late stage of development of the explosion the hydrodynamic quantities have almost the same values as in an unperturbed medium, new variables were introduced which are corrections to some principal parts of the values of the pressure, density and velocity of the particles. Similarly the linear acoustic part was separated while calculating the coordinates of the shock wave front. The number of computing points was 48. With this set-up calculation was continued up to a pressure drop of 1.0018 at the front. S. A. Khristianovich's method [2] was used to continue the calculation. Shock waves at a considerable distance from the point of explosion were considered in [2] on the assumption that the difference between the pressures at the front and in the unperturbed medium is small. A finite relation was obtained from which the distribution of hydrodynamic parameters in the perturbed region of the spherical shock wave at a moment of time t can be calculated, when the distribution of these parameters at some other moment t 0 is known. Taking the last distribution of quantities in the perturbed region calculated by the numerical method as the initial distribution, it is possible to obtain with sufficient accuracy the distribution for larger values of time by Khristianovich's method. The procedure for numerical calculation of the given problem is given below and it is shown how Khristianovich's method can be used to continue the calculation. The results given indicate the formation of a secondary shock wave in the negative phase region. The authors had the advantage of valuable consultations with V. V. Rusanov in the course of the work.