## 1 Introduction
To the knowledge of the authors, coupling the ESF method to the FPV tabulated chemistry strategy is new in the LES context. In an oxy-fuel environment, [PERSON] et al. [4] were the first to develop a hybrid ESF/FPV approach within the RANS framework and to validate it in three oxy-fuel jet flames (A1, A3 and B3) as experimentally investigated in [43]. They compared the results with those obtained by means of presumed PDF and with experimental data. In contrast, the oxy-fuel jet flames (A1 and A3) have been investigated in LES, including a Smagorinsky model and a conditional moment closure (CMC) combustion model by [PERSON] and [PERSON] [25]. Since a unity Lewis number diffusion model was used, the model suffers from real limitations in predicting the main flame characteristics and major reacting species as well as the temperature evolution at different positions. Relying on the abovementioned limitations of the LES-FPV approach and on the attractive features of the hybrid ESF/FPV approach, it is promising to extend the hybrid ESF/FPV approach in which the TCI is well accounted for to LES. This task is attempted in the present paper, where the resulting LES hybrid ESF/FPV approach includes in a first step FPV table generated with Le = 1 in order to evaluate its effect on the prediction of the oxy-flame properties. The current paper reports on the predictions and analysis of combustion characteristics of a turbulent oxy-methane non-premixed flame operating under highly diluted conditions of CO\({}_{2}\) and H\({}_{2}\) in oxidizer and fuel streams, respectively, by rather applying a novel, well-designed numerical combustion model. The latter consists of a LES extension of a recently suggested hybrid ESF/FPV approach in RANS context by the authors [4]. It combines a transported joint scalar probability density function (T-PDF) following the Eulerian stochastic field methodology (ESF) and a flamelet progress variable (FPV) turbulent combustion model which includes detailed chemical reaction mechanism. This hybrid ESF/FPV approach overcomes the limitations of the presumed-probability density function (P-PDF) by accurately characterizing the influence of the sub-grid fluctuations on the flame structure and on combustion properties, along with the TCI. The prediction capability of the LES hybrid ESF/FPV was first appraised on the well-known air-piloted methane jet flame (Sandia Flame D). Then, it was evaluated in analyzing the combustion properties of a non-premixed oxy-flame and in capturing the CO\({}_{2}\) dilution effect on the oxy-fuel flame behavior. To this end, the turbulent non-premixed methane oxy-flame, the so-called oxy-flame B3, as experimentally studied in [43] and already numerically investigated in RANS context in [4], was analyzed. Finally, the impact of generating the FPV chemistry table under condition of unity Lewis number even with CO\({}_{2}\) dilution was examined on the general prediction of the oxy-fuel flame structure, stability and emissions. In this configuration, a significant CO\({}_{2}\) dilution level was considered in the oxidizer and a high percentage of H\({}_{2}\) enrichment in the fuel stream. As pointed out in [44], the CO\({}_{2}\) diluted oxy-fuel flame evolves differently compared to a classical air-fuel flame and more stabilization mechanisms are required. These changes in the combustion environment of the oxy-fuel case make it more attractive and challenging for sub grid scale and combustion modeling. The remainder of the paper is structured as follows. Section 2 outlines the numerical methodology. Section 3 presents the experimental configurations with corresponding numerical set-up details. Section 4 outlines and discusses the obtained results. Concluding remarks are summarized in the last section.