Matched Asymptotic Expansions and the Closure Problem for Combustion Waves

Abstract

The closure problem for combustion waves arises when applying the method of matched asymptotic expansions for large activation energy to many nonsteady combustion problems. The exponential nature of the dependence of the reaction rate on temperature and the large coefficient (activation energy) in the exponent lead to the first order correction for temperature appearing in the equations at leading order. Equations describing the first order correction involve the second order correction, and so on. These terms can be scaled away for steady solutions, but when considering nonsteady propagation, they remain for any constant scaling of temperature. The closure problem refers to the fact that the equations must be solved at all orders before the leading order solution can be determined. One traditional approach to alleviate the problem is to truncate the series. While sacrificing the distinction between scales of temperature variation ahead of and behind the flame, these methods allow the replacement of the distributed reaction with a delta function reaction rate model. We present an alternative procedure which uses generalized matched asymptotic expansions and an associated truncation-like restriction to allow formulation of a model which retains the scales of temperature variation at the expense of a simple delta function interpretation of the model. The procedure is demonstrated through the derivation of models for gasless combustion as well as the diffusional-thermal (constant density approximation) theory for gaseous combustion. Comparison with results using standard truncation methods reveals quantitative differences in linear stability results. Stability of planar uniformly propagating wave solutions for the diffusional-thermal model is shown to depend on both Lewis number (ratioof thermal and species diffusivities) and Zeldovich number (ratio of diffusive to reactive length scales). This procedure should be of interest to those exploring weakly nonlinear analysis of steady solutions to gasless and diffusional-thermal combustion problems and to those developing models for other combustion configurations such as porous media combustion.