A framework for data-based turbulent combustion closure: A priori validation

Abstract

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...

A framework for data-based turbulent combustion closure: A posteriori validation

“…The use of KDE in combustion is relatively new. However, it was demonstrated by our group in two recent studies [12,13] to construct marginal uni-variate PDFs and joint PDFs. The PDFs, of course, have to be determined using multiple measurements at one point.…”

“…During the preprocessing step, a priori validation may be carried out as presented in Ref. [13]. This validation is an important step to evaluate the data adequacy in terms of size and ability to span the composition space and to determine the number of PCs that need to be retained from the PCA analysis.…”

“…Although, a similar procedure can be implemented for LES based on multi-scalar line measurement data and the construction of filtered probability density functions. As described earlier, PCs are a representation of the thermochemical scalars in a lower dimensional space [13,[22][23][24][25][26][27][28][29][30]. The vectors of PCs and their chemical source terms are linearly related to the vectors of measured thermo-chemical scalars and their chemical source terms, respectively:…”

“…Note, that the expression for the PDF also can include unconditional means for variances or higher moments. However, in our present analysis based on a priori [13] and a posteriori validation (presented here), the unconditional means of the PCs, along with the instantaneous values of the PCS, are sufficient to tabulate the PDFs. Figure 1 summarizes the preprocessing steps for developing the closure elements of the proposed framework starting with multi-scalar measurements.…”

“…In Ref. [13], we proposed a variant of the pairwise mixing stirred reactor (PMSR) [17][18], that mixes and reacts states that belong to the same region of the composition space. These states are initialized with the experimental measurements.…”