Asymptotic solutions in non-equilibrium nozzle flow

Abstract

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.