On the structure of self-similar detonation waves in TNT charges

A phase-plane method is proposed to model flow fields bounded by constant-velocity detonation waves propagating in TNT charges. Similarity transformations are used to formulate the problem in the phase plane of non-dimensional sound speed $Z$ versus non-dimensional velocity $F$. The formulation results in two coupled ordinary differential equations that are solved simultaneously. The solution corresponds to an integral curve $Z(F)$ in the phase plane, starting at the Chapman–Jouguet (CJ) point and terminating at the singularity $A$, which is the sonic point within the wave. The system is closed by computing thermodynamic variables along the expansion isentrope passing through the CJ point, forming, in effect, the complete equation of state of the thermodynamic system. The CJ condition and isentropic states are computed by the Cheetah thermodynamic code. Solutions are developed for planar, cylindrical, and spherical detonations. Species profiles are also computed; carbon graphite is found to be the predominant component ($\approx$ 10 mol/kg). The similarity solution is used to initialize a 1D gas-dynamic simulation that predicts the initial expansion of the detonation products and the formation of a blast wave in air. Such simulations provide an insight into the thermodynamic states and species concentrations that create the initial optical emissions from TNT fireballs.