Transverse ionizing MHD detonation waves. Part 1. Structure, Hugoniot curves and Chapman–Jouguet limit

Abstract

The structure and jump relations across a one-dimensional ionizing magnetohydrodynamic detonation wave with transverse electromagnetic fields are studied. It is shown that the downstream point in a three-dimensional phase space, constructed from the gas velocity, magnetic field and burned fraction of gas, is of a singular nature and leads to a unique solution. In the zero viscosity, infinite transitional electrical conductivity limit, the ‘fast wave’ degenerates into a pre-compression ionizing shock followed by a perfectly-conducting deflagration zone, and the ‘slow wave’ degenerates into an ordinary gas dynamic pre-compression shock followed by a deflagration zone in which the transition to perfect conductivity occurs. In both cases, a relationship is derived connecting the electric field (which cannot be arbitrarily imposed) and both the upstream variables and parameters describing the ionization transition point. This relationship must be appended to the Rankine–Hugoniot jump relations and a Chapman-Jouguet condition (or its equivalent) to provide a unique relationship between the assumed unknown shock-speed and the downstream state as a function of the assumed known upstream state.The ionization temperature is found to affect the jump and determines the electric field developed upstream. It is found that the Hugoniot consists of two branches: a ‘normal’ one which retains the character of an ordinary detonation Hugoniot, and an ‘anomalous’ one which does not. Although both branches represent detonations endowed with a transition structure, it is suggested that ‘anomalous’ detonations are either unstable or non-evolutionary. The evolution of the topology of the Hugoniot, as a function of the upstream magnetic field and ionizing temperature, is studied and correlated with Chapman-Jouguet propagations. Results are presented for the Chapman-Jouguet jump values and detonation speeds, as functions of dimensionless parameters that represent the upstream magnetic and electric field. Relationships between this upstream electric field and ionization temperature is derived and found to differ according as ionization precedes or follows ignition.