## 1 Introduction
It is worth mentioning that using such a tabulated chemistry technique calls for assumptions regarding the flame structure along with the statistical turbulence-chemistry interaction (TCI) phenomena. For that purpose, a probability density function (PDF) of the controlling parameters of the table is usually introduced. To determine this PDF, two main methodologies are suggested in the literature [4; 5; 7; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. The first method consists in assuming a presumed shape for the PDF as reported in the most contributions which apply the FPV approach [8; 10; 16; 17; 18]. Such a shape assumption for the PDF for all scalar distributions calls usually for considering statistical independence between single PDF, which is not the case in reality. Very few considerations relax this assumption, as in [41]. It is noteworthy that the chemical source term within this first method is expressed in an unclosed form. It is usually modeled from the FPV table as function of the reaction progress variable along with the presumed PDF. According to previous LES studies in oxy-flames [17; 22], the application of a presumed shape of PDF with Beta-function experienced shortcomings regarding the general prediction capability of the temperature distribution and minor species evolution. As pointed out in [17], the results obtained within the LES-FPV framework are affected by the pre-calculated flame structures through the tabulated transport properties and source terms from the manifolds, and all data are additionally PDF-integrated. Similar limitations were reported within the RANS context on oxy-fuel configurations in [4]. The second method to determine the PDF is therefore adopted in this study. It is a more efficient approach that describes the PDF by solving a transported probability density function (T-PDF) equation, as reported, for example, in [4; 27; 28; 29; 30; 31; 32; 33; 34; 38; 39; 40]. Two major formulations of the T-PDF method have been proposed in the literature, namely the Lagrangian Monte-Carlo approach as applied in [28; 29] and the Eulerian based approach in [30; 31; 32; 33; 34; 37; 38; 39; 40]. Solving a filtered transported PDF equation means applying a modeled form of the equations of the PDF or filtered density function (FDF) in LES, where the filtered and the sub-grid scalar field can be formulated. One attractive approach to model the FDF equations consists of using the Eulerian stochastic field (ESF) approach [31; 32; 33] and [34]. Following [PERSON] et al. in [35] and [PERSON] al. in [36], multiple stochastic fields for each scalar are utilized to describe the FDF undergoing diffusion, turbulent convection and chemical reaction. The resulting stochastic fields (SFi) are continuous and differentiable in space. Within this method, the chemical source term emerges in a closed form, allowing modeling of the combustion and the involved TCI in an accurate way. Many previous LES investigations have employed the ESF method coupled with different chemical mechanism approaches in both simple and complex scales to simulate various jet flames in an air environment. In particular, the ESF method was applied with reduced mechanisms in order to study premixed and partially-premixed flames in [31; 32] and [37; 42]. From these studies, it was concluded that 8 SFi could well characterize the influence of the sub-grid fluctuations on the combustion properties. [PERSON] et al. in [35; 38; 39] employed the ESF method in both RANS and LES contexts in combination with detailed chemistry. Even though all these previous studies proved the efficiency of the ESF approach in a LES context, the choice of the chemical mechanism technique has an enormous effect in terms of computational requirements. Only very few studies have focused on developing a LES framework for the ESF method in coupling with a tabulated chemistry. In their large eddy simulation (LES) study, [PERSON] et al. [33] reported a coupling of the ESF approach with the FGM-based tabulated approach.