On the generalisation of the mixture fraction to a monotonic mixing-describing variable for the flamelet formulation of spray flames

Spray flames are complex combustion configurations that require the consideration of competing processes between evaporation, mixing and chemical reactions. The classical mixture-fraction formulation, commonly employed for the representation of gaseous diffusion flames, cannot be used for spray flames owing to its non-monotonicity. This is a consequence of the presence of an evaporation source term in the corresponding conservation equation. By addressing this issue, a new mixing-describing variable, called the effective composition variable , is introduced to enable the general analysis of spray-flame structures in composition space. This quantity combines the gaseous mixture fraction Z(g) and the liquid-to-gas mass ratio Z(l), and is defined as . This new expression reduces to the classical mixture-fraction definition for gaseous systems, thereby ensuring consistency. The versatility of this new expression is demonstrated in application to the analysis of counterflow spray flames. Following this analysis, this effective composition variable is employed for the derivation of a spray-flamelet formulation. The consistent representation in both effective composition space and physical space is guaranteed by construction and the feasibility of solving the resulting spray-flamelet equations in this newly defined composition space is demonstrated numerically. A model for the scalar dissipation rate is proposed to close the derived spray-flamelet equations. The laminar one-dimensional counterflow spray-flamelet equations are numerically solved in -space and compared to the physical-space solutions. It is shown that the hysteresis and bifurcation characterising the flame structure response to variations of droplet diameter and strain rate are correctly reproduced by the proposed composition-space formulation.