Statistical Analysis of the Reaction Progress Variable and Mixture Fraction Gradients in Flames Propagating into Droplet Mist: A Direct Numerical Simulation Analysis

Abstract

The statistics of reaction progress variable, c, and mixture fraction, Ԑ, and their gradients (i.e., (Formula presented.) and (Formula presented.)) in flames propagating in droplet mist, where the fuel was supplied in the form of monodisperse droplets, have been analyzed for different values of turbulent velocity fluctuations (uʹ), droplet equivalence ratios (ϕd) , and droplet diameters (ad) based on three-dimensional direct numerical simulations (DNS) in a canonical configuration under decaying turbulence. The combustion process in the gaseous phase has been found to take place predominantly in fuel-lean mode, even for ϕd>1. The probability of finding fuel-lean mixture increases with increasing initial droplet diameter due to slower evaporation of larger droplets. It has been shown that the joint probability density function (i.e., joint PDF) of Ԑ, and c (i.e., PԐ,c), cannot be approximated in terms of discrete delta functions throughout the flame brush for the cases considered here. Furthermore, the magnitude of P(Ԑ, c) cannot be adequately approximated by the product of marginal PDFs of, Ԑ and variable, c(i.e., P(Ԑ).p(c)). The statistical properties of the Favre probability density functions (Favre-PDFs) of the mixture fraction, Ԑ, and oxidizer-based reaction progress variable, c, have been analyzed at several locations across the flame brush and a β-function distribution has been found to capture the Favre-PDFs of Ԑ and c obtained from the DNS data. Furthermore, a log-normal distribution has been shown to capture the qualitative behaviors of the PDFs of the gradient of the mixture fraction and the gradient of the reaction progress variable, (Formula presented.) and (Formula presented.) , respectively, but discrepancies between the log-normal distribution and the DNS data were observed at the tails of PDFs. In addition, the interrelation between (Formula presented.) and (Formula presented.) was examined in terms of the PDFs of the cosine of the angle between them (i.e., (Formula presented.) and it was observed that most droplet cases exhibited much greater likelihood of positive values of (Formula presented.) than negative values. Finally, the joint PDF of (Formula presented.) and (Formula presented.) , (Formula presented.) , has been compared with that of P((Formula presented.) (i.e., assuming statistical independence of (Formula presented.) and (Formula presented.)) and a good level of agreement has been obtained. The bivariate log-normal distribution has been considered both assuming correlation between (Formula presented.) and (Formula presented.) and assuming no correlation for the purpose of modeling (Formula presented.) , and the variant with no correlation has been found to be more successful in capturing qualitative behavior of (Formula presented.) although quantitative discrepancies have been observed due to inaccuracies involved in parameterizing P((Formula presented.) and P((Formula presented.) by log-normal distributions.