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

Abstract

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 η, 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 η-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.

Keywords:

• laminar counterflow spray flame
• flamelet formulation
• mixture fraction
• effective composition
• bifurcation

Acknowledgements

Helpful discussions with Professor Sirignano on the spray-flamelet formulation are appreciated.

Disclosure statement

Conflict of interest: The authors declare that they have no conflict of interests.

Research involving Human Participants and/or Animals: Not applicable for this paper.

Informed consent: All the authors approve this submission.

Notes

1. Since the definition of mixture fraction is reserved for a conserved quantity, Z_g from Equation (1d(1d) ) does not strictly represent a mixture fraction. However, for reasons of consistency with previous works, we follow this convention.

2. The sign of dη is chosen to be positive in order to derive a monotonically increasing coordinate from the oxidiser side to the fuel side.

3. Despite the fact that this assumption is not exact for variable-density flows, it reduces the computational complexity of the counterflow while retaining the main physics. This approximation is often used as a simplified model for two-phase flame analysis.

4. For L → ∞, the pre-evaporated case is retrieved.

5. The liquid phase does not have a diffusion term, and is therefore characterised by an infinite Lewis number.

6. To take into account the variability of the evaporation time, the vaporisation Stokes number is approximated by where τ_{v, ref} = 0.04 s and d_ref = 40 μm.

7. It is noted that the flame transition from single- to double-reaction and vice versa is sensitive to the numerical procedure that is used to vary the strain rate and droplet diameter.

8. The assumption of constant liquid temperature is not valid for real applications [3], the transient heating time being of primary importance. However, since the main concern about the definition of a composition space is the effect of the vaporisation rate, this assumption has no consequence for the suitability of our methodology when liquid temperature variations are taken into account.

9. It is worth mentioning that this assumption could be relaxed to take into account density effects on the flow structure, by using the Howarth–Dorodnitzyn approximation under the classical boundary layer approximation [35].

Additional information

Funding

The authors gratefully acknowledge financial support through NASA [Award No. NNX14CM43P], [Award No. NNM13AA11G] as well as SAFRAN support for the postdoctoral stay of A. Vié.