Combustion Science and Technology, Vol.188, No.11-12, 2149-2177, 2016
Statistical Analysis of the Reaction Progress Variable and Mixture Fraction Gradients in Flames Propagating into Droplet Mist: A Direct Numerical Simulation Analysis
The statistics of reaction progress variable, c, and mixture fraction, xi, and their gradients (i. e., del c and del xi) in flames propagating in droplet mist, where the fuel was supplied in the form ofmonodisperse droplets, have been analyzed for different values of turbulent velocity fluctuations (u'), droplet equivalence ratios (phi(d)), and droplet diameters (a(d)) 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 fuellean mode, even for phi(d) >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 xi and c (i.e., P(xi, c)), cannot be approximated in terms of discrete delta functions throughout the flame brush for the cases considered here. Furthermore, the magnitude of P(xi, c) annot be adequately approximated by the product of marginal PDFs of xi, and variable, c (i. e., P(xi), P(c)). The statistical properties of the Favre probability density functions (Favre-PDFs) of the mixture fraction, xi, and oxidizer-based reaction progress variable, c, have been analyzed at several locations across the flame brush and a beta-function distribution has been found to capture the Favre-PDFs of xi 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, vertical bar del xi vertical bar and vertical bar del c vertical bar, respectively, but discrepancies between the log-normal distribution and the DNS data were observed at the tails of PDFs. In addition, the interrelation between del xi and del c was examined in terms of the PDFs of the cosine of the angle between them (i.e., cosd(theta)) and it was observed that most droplet cases exhibited much greater likelihood of positive values of cosd(theta)) than negative values. Finally, the joint PDF of vertical bar del xi vertical bar amd vertical bar del c vertical bar, P (vertical bar del xi vertical bar, vertical bar del c vertical bar) has been compared with that of P(vertical bar del xi vertical bar). P(vertical bar del c vertical bar) (i.e., assuming statistical independence of vertical bar del xi vertical bar and vertical bar del c vertical bar a good level of agreement has been obtained. The bivariate log-normal distribution has been considered both assuming correlation betweenj vertical bar del xi vertical bar and vertical bar del c vertical bar assuming no correlation for the purpose of modeling P(vertical bar del xi vertical bar, vertical bar del c vertical bar) and the variant with no correlation has been found to be more successful in capturing qualitative behavior of P(vertical bar del xi vertical bar, vertical bar del xi vertical bar) although quantitative discrepancies have been observed due to inaccuracies involved in parameterizing P(vertical bar del xi vertical bar) and P (vertical bar del c vertical bar)by log-normal distributions.