Deflagrations and Detonations for Small Heat Release

Abstract : Technical Reports 117 and 118 develop the solutions (using activation-energy asymptotics) of one-dimensional steady combustion waves, deflagrations and detonations respectively, when the Mach number is not small. Even with the great simplifications afforded by the limit of large activation-energy some numerical calculations are necessary. However a completely analytical description of these solutions is possible whenever the heat released during reaction is small. In this paper, we give these explicit analytical solutions for the fast deflagration wave and a simple expression for its speed of propagation. As the speed of propagation approaches the lower Chapman-Jouget wave speed (slightly less than sonic velocity) we show that the velocity structure in front of the flame adjusts to a classical Taylor shock. We also give an explicit analytical solution for detonations traveling at speeds greater than the upper chapman-Jouget velocity (slightly greater than sonic velocity); in particular, such strong detonations are characterized by Taylor-like velocity adjustments both in front of and behind the flame. (For detonations the speed is not determined.) This work serves as the basis for a completely analytical treatment of the transition form deflagration to detonation.