The modeling of turbulent reactive flows is a subject of contemporary research. Current turbulent-reaction models cannot account for realistic complexities such as distinct species mass-diffusion coefficients. Under the assumption of a single, constant, mass-diffusion coefficient, a conserved-scalar equation is typically derived in turbulent reactive flows by taking the difference between chemical-species conservation equations having opposite reaction rates (in the sense that the reactant has an opposite reaction rate to the product), thereby creating an equation devoid of reaction terms. Assuming the reaction regions are very thin and are merely contorted by turbulence, chemistry and turbulence can be decoupled, and the evolving statistics of the conserved scalar describe the reaction progression. No such equation has yet been derived for distinct mass-diffusion coefficient cases where the single coefficient is now replaced by a full matrix. Considering that mass diffusion is responsible for reactants approaching at the molecular level and for reaction initiation, this lack of mathematical framework is very disturbing.

This study inquired whether one can derive conserved-scalar equations in this general case, and whether these equations would indeed be useful for flamelet modeling. A model for turbulent reactive flow called “flamelet theory” is indiscriminately used in numerical simulations without checking whether the assumptions on which it is based are consistent with other aspects of the problem.

A derivation was presented, and it has been found that the flamelet theory assumptions are not generally consistent with the general equations. A methodology has also been proposed on how to check whether assumptions could be made for the general equations that would render them compatible with flamelet theory. Simulations of liquid rocket engine combustion for astronautic applications, and of gas turbine engines for aeronautic applications, all rely on these concepts.

For species mixing, under the general situation of a full multi-component species mass diffusion flux containing not only the mass diffusion terms but also the Soret and/or the pressure gradient terms, a set of conserved scalars cannot be obtained akin to those used in the context of flamelet theory. Even when the Soret and/or the pressure gradient terms are negligible — due to the non-symmetric property of the diffusion matrix — it cannot be assumed that, if diagonalizable, its eigenvalues are all real, which precludes replacing the (N − 1) set of species by a (N − 1) set of conserved scalars of the form utilizable in flamelet theory.

For reacting species, a set of conserved scalars can be defined — these being the elemental mass fractions — but even they would not be amenable to a transformation to a set necessary for the flamelet approach. Thus, simulations of turbulent reactive flows for many applications such as liquid rocket propulsion (oxygen/hydrogen) or gas turbine engines (hydrocarbon/air combustion at high pressure where the Soret effect has been demonstrated to be important) show that the current flamelet methodology for describing turbulent combustion is not applicable.

As a constructive approach, it was proposed that in practical applications, one should write the diffusion matrix as a sum of symmetric and antisymmetric matrices, and evaluate at a multitude of points in the thermodynamic space the importance of the latter matrix with respect to the former. If it turns out that the antisymmetric matrix is negligible with respect to the symmetric one, only the symmetric one can be retained, and conserved scalars for flamelet theory applications can be constructed.

*This software is available for license through the Jet Propulsion Laboratory, and you may request a license at: https://download.jpl.nasa.gov/ops/request/request_introduction.cfm . NPO-48090*