Closure of the scalar dissipation rate in the spray flamelet equations through a transport equation for the gradient of the mixture fraction - Olguín Astudillo, Hernán Andrés; Gutheil, Eva (Combustion and flame, 2019)
In this paper, the closure of the Scalar Dissipation Rate (SDR) in the Spray Flamelet Equations (SFE) is addressed. For this purpose, the gradient gξ of the mixture fraction ξ is used instead of the SDR itself. A transport equation for this variable is derived and transformed from physical into mixture fraction space. Moreover, the spray flamelet equations of the species mass fractions and of gas temperature are re-derived in terms of gξ for consistency, considering differential diffusion effects. Numerical simulations of different axi-symmetric counterflow ethanol/air flames are carried out in physical space using a well established model and the results are employed for the validation and analysis of the newly proposed set of SFE. In particular, a non-premixed gas flame is established as a base case and then perturbed by means of different mono-disperse sprays injected from the air side of the configuration. In the newly proposed SFE, two different kind of unclosed quantities appear: The spatial gradient of the product of gas velocity and gas density, a^, and sources of mass and energy due to evaporation. In the present work, different alternatives for the closure of a^ are presented and analyzed, where the following approaches are proposed: 1) Introducing a stream-like function and using the global mass and axial momentum balance equations to derive an expression for a^ and 2) Assuming a constant value for this variable. Moreover, two different constant values of a^ are considered: its value at the stoichiometric mixture fraction and its value at the air side of the counterflow configuration. The evaporation-related source terms are closed through projections of the numerical results from physical into mixture fraction space. The suitability of the proposed approaches is tested in terms of the ability of the SFE of properly predicting both gξ and the spray flamelet structure of the reference counterflow flames. Additionally, the contributions of the individual terms in the SFE are analyzed, with special focus on the effects of evaporation. The validation confirms that the new set of SFE accurately reproduces the counterflow profiles when the stream-like function is employed, whereas employing the value of a^ at stoichiometry still adequately describes the relative importance of the different physical and chemical phenomena taking place in the flames studied (the contribution of the different terms in the SFE), even when the flame structure itself shows non-negligible deviations. Finally, using the value of a^ at the left inflow boundary leads to poor predictions in most situations. The present results represent a significant advance towards the development of a comprehensive and self-contained spray flamelet theory.