Experimental multi-scalar measurements in laboratory flames have provided important databases for the validation of turbulent combustion closure models. In this work, we present a framework for databased closure in turbulent combustion and establish an a priori validation of this framework. The approach is based on the construction of joint probability density functions (PDFs) and conditional statistics using experimental data based on the parameterization of the composition space using principal component analysis (PCA). The PCA on the data identifies key parameters, principal components (PCs), on which joint scalar PDFs and conditional scalar means can be constructed. To enable a generic implementation for the construction of joint scalar PDFs, we use the multidimensional kernel density estimation (KDE) approach. An a priori validation of the PCA-KDE approach is carried out using the Sandia piloted jet turbulent flames D, E and F. The analysis of the results suggests that a few PCs are adequate to reproduce the statistics, resulting in a low-dimensional representation of the joint scalars PDFs and the scalars' conditional means. A reconstruction of the scalars' means and RMS statistics are in agreement of the corresponding statistics extracted directly from the experimental data. We also propose one strategy to recover missing species and construct conditional means for the reaction rates based on a variation of the pairwise-mixing stirred reactor (PMSR) model. The model is demonstrated using numerical simulations based on the one-dimensional turbulence (ODT) model for the same flames. (T. Echekki). 3 computational cost for complex chemical systems and large computational problems where a large number of notional particles must be retained for an accurate evaluation of the PDFs. Alternative strategies to alleviate the potential cost of a joint PDF transport equation have been proposed, including strategies to parameterize the composition space (e.g. by adopting a flamelet assumption) or via the multi-environment PDF method [10].The two limits of a presumed PDF shape with a limited set of parameters and the solution for a PDF transport equation leave ample room in between for intermediate approaches, which may be based on the construction of PDFs based on simulation [11][12][13][14] or experimental data. For example, Goldin and Menon [11,12], Sankaran et al [13] and Calhoon et al [14] built joint scalar PDFs, which are parameterized by an appropriate set of lower moments using simulations based on the linear-eddy model (LEM) [15].The PDFs are generally trained on simpler canonical flows, such as scalar decay in homogeneous turbulence or co-flowing jet configurations. The LEM is a 1-D model that can accommodate any degree of chemical complexity based on the coupling of reaction and diffusion processes in a deterministic fashion and a stochastic implementation for turbulent stirring. As the model is implemented in physical space, it is capable of generating statistics from which a description of the composition sp...

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In this work, we demonstrate a framework for developing closure models in turbulent combustion using experimental multi-scalar measurements. The framework is based on the construction of conditional means and joint scalar PDFs from experimental data based on the parameterization of composition space using principal component analysis (PCA). The resulting principal components (PCs) act as both conditioning variables and transport variables. Their chemical source terms are constructed starting from instantaneous temperature and species measurements using a variant of the pairwise mixing stirred reactor (PMSR) approach. A multi-dimensional kernel density estimation (KDE) approach is used to construct the joint PDFs in PC space. Convolutions of these joint PDFs with conditional means are used to determine the unconditional means for the closure terms: the mean PCs chemical source terms and the density. These means are parameterized in terms of the mean PCs using artificial neural networks (ANN). The framework is demonstrated a posteriori using the data from the Sandia piloted turbulent jet flames D, E and F by performing RANS calculations. The radial profiles of mean and RMS of temperature and measured species mass fractions agree well with the experimental means for these flames.

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Probability density function (PDF) based turbulent combustion modelling is limited by the need to store multi-dimensional PDF tables that can take up large amounts of memory. A significant saving in storage can be achieved by using various machine-learning techniques that represent the thermo-chemical quantities of a PDF table using mathematical functions. These functions can be computationally more expensive than the existing interpolation methods used for thermo-chemical quantities. More importantly, the training time can amount to a considerable portion of the simulation time. In this work, we address these issues by introducing an adaptive training algorithm that relies on multi-layer perception (MLP) neural networks for regression and self-organizing maps (SOMs) for clustering data to tabulate using different networks. The algorithm is designed to address both the multi-dimensionality of the PDF table as well as the computational efficiency of the proposed algorithm. SOM clustering divides the PDF table into several parts based on similarities in data. Each cluster of data is trained using an MLP algorithm on simple network architectures to generate 'local' functions for thermo-chemical quantities. The algorithm is validated for the socalled DLR-A turbulent jet diffusion flame using both RANS and LES simulations and the results of the PDF tabulation are compared to the standard linear interpolation method. The comparison yields a very good agreement between the two tabulation techniques and establishes the MLP-SOM approach as a viable method for PDF tabulation.