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

Abstract

International audienceSpray 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 due 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 effective composition variable η, is defined to enable the general analysis of spray-flame structures in composition space. This quantity combines the gaseous mixture fraction Zg and the liquid-to-gas mass ratio Z. 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, the effective composition variable η 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 equation 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 the η-space and compared to the physical-space solutions. It is shown that the hysteresis and bifurcation characterizing the flame structure response to variations of droplet diameter and strain rate are correctly reproduced by the proposed composition-space formulation

A generalised spray-flamelet formulation by means of a monotonic variable

“…[2,3,4,5,6]. Unlike gaseous diffusion flames, that are governed by the competition between scalar mixing and chemistry, spray flames are also influenced by evaporation and mass transport of the liquid-fuel into the reaction zone [7], making it a more complex problem.…”

“…Due to the importance of spray flames, several numerical and theoretical investigations have been performed to understand the main physical and chemical processes that govern spray combustion as well as their flame structure in different spatial and temporal scales [2,7]. Regarding numerical investigations, two different approaches can be highlighted: (i) Eulerian Interface Capturing (EIC), and (ii) Particle Tracking (PT) methods.…”

“…Theoretical investigations of the spray-flame structure can give important physical insights into the behaviour of spray flames in simple configurations. Such insights can then be used as a basis to understand more complex combustion systems [7]. Traditionally, the flame structure of laminar gaseous diffusion flames is studied in terms of the gaseous mixture fraction Z [10], a passive scalar that is an appropriate variable to analyse the mixing of the reactants (the dominant physical process in these type of flames).…”

“…Extending this formulation to spray-flames would in principle enable the analysis tools developed for gaseous flames to be applied. However, a direct extrapolation of the classical mixture fraction to spray-flames is not possible because Z becomes non-monotonic due to the presence of vaporisation sources [7,12,13,14]; the constraint of monotonicity is required to guarantee that the solution is single-valued. In addition to Z, other composition spaces have been proposed and analysed in previous studies such as the total mixture fraction [15,16,17], and the conserved mixture fraction [7].…”

“…However, a direct extrapolation of the classical mixture fraction to spray-flames is not possible because Z becomes non-monotonic due to the presence of vaporisation sources [7,12,13,14]; the constraint of monotonicity is required to guarantee that the solution is single-valued. In addition to Z, other composition spaces have been proposed and analysed in previous studies such as the total mixture fraction [15,16,17], and the conserved mixture fraction [7]. The aforementioned alternatives do not have their monotonicity guaranteed mainly due to differential diffusion and the relative velocity that exists between the liquid and gaseous phases.…”