Equivalence of Low-frequency Stability Conditions for Multidimensional Detonations in Three Models of Combustion

We use the classical normal mode approach of hydrodynamic stability theory to define stability determinants (Evans functions) for multidimensional strong detonations in three commonly studied models of combustion: the full reactive Navier-Stokes (RNS) model, and the simpler Zeldovich-von Neumann-Döring (ZND) and Chapman-Jouguet (CJ) models. The determinants are functions of frequencies (λ, η), where λ is a complex variable dual to the time variable, and η ∈ Rd−1 is dual to the transverse spatial variables. The zeros of these determinants in <λ > 0 correspond to perturbations that grow exponentially with time. The CJ determinant, ∆CJ (λ, η), turns out to be explicitly computable. The RNS and ZND determinants are impossible to compute explicitly, but we are able to compute their first-order low-frequency expansions with an error term that is uniformly small with respect to all possible (λ, η) directions. Somewhat surprisingly, this computation yields an Equivalence Theorem: the leading coefficient in the expansions of both the RNS and ZND determinants is a constant multiple of ∆CJ ! In this sense the low-frequency stability conditions for strong detonations in all three models are equivalent. By computing ∆CJ we are able to give low-frequency stability criteria valid for all three models in terms of the physical quantities: Mach number, Gruneisen coefficient, compression ratio, and heat release. The Equivalence Theorem and its surrounding analysis is a step toward the rigorous theoretical justification of the CJ and ZND models as approximations to the full RNS model.