Radially convergent shock waves using a Mie-Grüneisen equation of state*

In 1942, Guderley first investigated the problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave that converges in an ideal, inviscid gas. Since then, Guderley type problems have been studied in various disciplines of physics and engineering (i.e., laser fusion and verification activities). Van Dyke and Guttmann provided the seminal solution to a theorized set of initial conditions for the Guderley problem, the 1D radial piston problem. Potential applications of research concerning the 1D radial piston problem have since been limited by Van Dyke and Guttmann’s choice of the ideal gas EOS.

The purpose of this work is to expand the current knowledge base to include a more complex and broadly relevant constitutive law - the Mie-Grüneisen (MG) EOS. The MG EOS models solid materials which resist compression more than their ideal gaseous counterparts. Further, the MG EOS can be reduced to reproduce the results from the Van Dyke and Guttmann investigation. This work uses the solution to the 1D planar piston problem as the first term in a quasi-self-similar series expansion to estimate the solution of the 1D radial piston problem for a material with a non-ideal EOS. Through this analysis, the state-variables in the shocked region of a material are calculated.