Rapid cylindrically and spherically symmetric strong compression of a perfect gas

Abstract

The problem of the rapid cylindrically and spherically symmetric strong compression of a perfect (non-viscous and non-heat-conducting) gas is solved. The term “rapid” denotes that the compression time is much less than the run time of a sound wave across the initial cylindrical or spherical volume, while the term “strong” in this case means the simultaneous attainment of as large a density and temperature as desired. By definition, rapid compression must begin in a strong shock wave, which propagates to the axis or centre of symmetry. When the shock wave approaches the centre of symmetry this flow is described by the self-similar Guderley equation with an unbounded rise in temperature, pressure and velocity and a finite increase in the density at the centre of symmetry both behind the arriving and behind the reflected shock waves. To obtain as high an increase in the density as desired one must add on a centred compression wave with focus at the centre of symmetry to the overtaking shock wave at the instant it arrives at the centre of symmetry C −-characteristic. Outside a small neighbourhood of the focus one can calculate, by the method of characteristics, the centred wave and the trajectory of the piston which produces it. As for any centred wave, this calculation must be carried out from the centre of symmetry. Since some of the parameters at the focus (certainly the pressure, temperature and velocity of the gas) are unbounded, it is necessary to preface the calculation by the method of characteristics by constructing an analytic solution which holds in a small neighbourhood of the centre of symmetry. Below, after constructing the required solution, the centred waves corresponding to it and the trajectories of the piston producing them are calculated.