Modeling of sub-grid conditional mixing statistics in turbulent sprays using machine learning methods

Deep artificial neural networks (ANNs) are used for modeling sub-grid scale mixing quantities such as the filtered density function (FDF) of the mixture fraction and the conditional scalar dissipation rate. A deep ANN with four hidden layers is trained with carrier-phase direct numerical simulations (CP-DNS) of turbulent spray combustion. A priori validation corroborates that ANN predictions of the mixture fraction FDF and the conditional scalar dissipation rate are in very good agreement with CP-DNS data. ANN modeled solutions show much better performance with a mean error of around 1%, which is one order of magnitude smaller than that of standard modeling approaches such as the β-FDF and its modified version. The predicted conditional scalar dissipation rate agrees very well with CP-DNS data over the entire mixture fraction space, whereas conventional models derived for pure gas phase combustion fail to describe ⟨N|ξ = η⟩ in regions with higher mixture fraction and low probability. In the second part of this paper, uncertainties associated with ANN predictions are analyzed. It is shown that a suitable selection of training sets can reduce the size of the necessary test database by ∼50% without compromising the accuracy. Feature importance analysis is used to analyze the importance of different combustion model parameters. While the droplet evaporating rate, the droplet number density, and the mixture fraction remain the dominant features, the influence of turbulence related parameters only becomes important if turbulence levels are sufficiently high.