Transport equations for an incompressible reactive flow with two separated phases

This report is devoted to the analysis of the premixed combustion modelling in the approximation leading to the sole use of a progress variable to characterize the mixture state. First, we shortly describe the TFC model to set the problematic. Then we recall the constitutive instantaneous equations and their averaged counterparts. We examine in details the limit case in which the reaction takes place in an infinitely thin sheet. This situation is formaly identical to the modelling of two separated fluids exchanging matter through their interface. In this case, there is an exact dependence of the so-called counter gradient transport term on the source term. The form of the dependence proves the counter gradient nature of the term which was up to now only intuited. It also shows that their is fundamentally only one unclosed term in the averaged progress variable equation. We examine the closure assumption of the source term in this framework and naturally re-derive the TFC model. We propose the basic idea for a slight improvement of the TFC model that takes into account the finite speed of the increasing brush width. The treatment of the limit case is done by use of generalized functions. They are quite delicate to manipulate and there is no strong familiarity with them in the combustion community leading to the effective difficulty in judging the correctness of the results. For this reason, we re-derive the corresponding results in the general case of regular functions.

@TechReport{MB03a,
  author       = {Moreau, V. and Battaglia, V.},
  title        = {Transport equations for an incompressible reactive flow with two separated phases},
  institution  = {CRS4, Center for Advanced Studies, Research and Development in Sardinia},
  number       = {03/42},
  month        = {july},
  year         = {2003},
  address      = {Cagliari, Italy},
  type         = {techreport},
  keywords     = {combustion, modelling, counter-gradient transport, TFC model},
  url          = {https://publications.crs4.it/pubdocs/2003/MB03a},
}