Closure of the Scalar Dissipation Rate in the Spray Flamelet Equations
Hernan Olguin  1@  , Arne Scholtissek  2@  , Sebastian Gonzalez  1@  , Matthias Ihme  3@  , Christian Hasse  2@  , Eva Gutheil  4@  
1 : Universidad Tecnica Federico Santa Maria [Valparaiso]
Avenida España 1680, Valparaíso -  Chile
2 : Institute for Simulation of Reactive Thermo-Fluid Systems, TU Darmstadt
3 : Department of Mechanical Engineering, Stanford University
4 : Interdisciplinary Center for Scientific Computing, Heidelberg University

Closure of the Scalar Dissipation Rate (SDR) in the Spray Flamelet Equations (SFE) is addressed in this paper. For this, the gradient $g_{\xi}$ of mixture fraction ${\xi}$ is used instead of the SDR itself. First, a transport equation for this variable is derived and transformed into mixture fraction space. Moreover, the spray flamelet equations of the species mass fractions and of gas temperature are re-derived in terms of $g_{\xi}$ for consistency, considering differential diffusion effects. In the resulting set of equations, two different kind of unclosed quantities appear: 1) Sources of mass and energy due to evaporation and 2) the spatial gradient of the product of gas velocity and gas density, $\hat{a}$. Numerical simulations of different counterflow ethanol/air flames in physical space are carried out 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. The validation confirms that the new set of SFE perfectly reproduces the counterflow structure when the right profiles of the unclosed quantities are available. Moreover, the appropriateness of imposing a constant value for the closure of $\hat{a}$ is tested in terms of the ability of the SFE of properly predicting both $g_{\xi}$ and the spray flamelet structure of the reference counterflow flames. Two different alternatives are considered for the constant value of $\hat{a}$: its value at the stoichiometric point and its value at the air side of the counterflow configuration. The budgets of the individual terms in the SFE are analyzed, giving special emphasis to the effects of evaporation on the quality of the assumptions. It is found that the main physical phenomena taking place in the counterflow flames studied can be properly recovered when the value of $\hat{a}$ at stoichiometry is employed. The proposed approach provides advancement towards the development of a comprehensive and self-contained spray flamelet theory.

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