Two-Gamma Jump Relations for Gaseous Detonation Waves

Abstract

Accurate jump relations for Chapman-Jouguet (CJ) and overdriven gaseous detonation waves are derived. The difficulty in accurately approximating the energy equation in a perfect gas two-gamma detonation jump formulation is resolved by adjusting the authentic combustion heat release in terms of linearly approximated up- and down-stream sensible enthalpies at the CJ condition where it is exactly satisfied for CJ and well approximated for overdriven waves. The basis of the success of the derived two-gamma jump relations is explained. Explicit thermodynamic jump relations across a normal detonation wave are obtained. These approximate jump relations depend on the following four parameters: γ upstream isentropic exponent (or the gamma giving pertinent upstream sound speed for defining M_j); γ_J, CJ isentropic exponent (or the gamma giving pertinent sound speed for a CJ state); M_j upstream Mach number; and M_J, CJ Mach number. γ_J and M_j can be obtained numerically, or experimentally with the derived jump relations. Comparisons of exact numerical and the present approximate calculation for jump conditions of CJ and overdriven detonations for methane- and hydrogen-oxygen systems show excellent agreement over a wide range of upstream Mach number, temperature, pressure, and mixture conditions.