## 5 Conclusions  
In this paper, it was demonstrated that the 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, can be predicted and analyzed using a novel hybrid ESF/FPV model designed for that purpose. 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 under consideration of a detailed chemical reaction mechanism. This novel hybrid ESF/FPV approach removes the weaknesses of the presumed-probability density function (P-PDF)-based FPV modeling along with the solving of associated additional modeled transport equations while rendering the T-PDF computationally less affordable. The prediction capability of the LES hybrid ESF/FPV was first appraised successfully on an air- piloted methane jet flame (Sandia Flame D). Then, it was assessed in analyzing the combustion properties of an oxy-flame, the so-called oxy-flame B\({}_{3}\) under investigation and in capturing CO\({}_{2}\) effect on the behavior of the oxy-flame. These configurations were previously investigated by the authors using a RANS-based formulation of this model. Relying on the results achieved, the following conclusions can be listed:  
* Overall, the LES hybrid filtered ESF/FPV approach demonstrated its high capability in capturing the main flame characteristics and flow field variables not only in the Sandia flame D but also in an oxy-flame configuration with CO\({}_{2}\) and H\({}_{2}\) dilution in oxidizer and fuel streams, respectively, using FPV tables based on Le = 1. This indicates that the H\({}_{2}\) induced differential diffusion effect is not important in the investigated area of this oxy-flame B\({}_{3}\). * Compared to our RANS results in our previous work [4], the LES hybrid ESF/FPV model provides more accurate predictions. * A good convergence of the optimal number of stochastic fields (SFi) strongly depends on the complexity of the combustion case. Even though more SFi could help to achieve fully complete convergence in this regard, in search of saving computational costs it turned out that a simulation with at least 8 stochastic fields leads to an accurate prediction in Sandia flame D as in [31] and at least 16 stochastic fields allow achieving better results in accordance with measurements once complex configuration like the oxy-fuel flame-B\({}_{3}\) is investigated. * Regarding the general prediction of the oxy-flame structure, stability and emissions, it turns out that 68% molar percentage of additional CO\({}_{2}\) enrichment in the oxidizer side leads to 0.39% of CO formation near the burner fuel nozzle and 0.62% at 10 d\({}_{\rm fuel}\) above the nozzle. These amounts of CO-formed gases are clearly high compared to ordinary flame cases with air/fuel conditions with 0.35% near the burner nozzle and 0.62% at 10 d\({}_{\rm fuel}\) above the jet tip. The novel model needs to be further evaluated by using FPV tables with a non-unity Lewis number and by considering advanced micro-mixing models coupled to the LES hybrid ESF/FPV approach. All of these tasks are left for future work. Conceptualization, R.M., and A.S.; Methodology, R.M., M.J. and A.S.; T-PDF solver, M.J.; Hybrid-ESF/FPV solver development, R.M., M.J. and A.S.; Settings up, R.M. ; Simulations and Validation, R.M. Formal Analysis, R.M., and A.S.; Investigation, R.M. ; Resources, A.S., B.F. and J.J.; Data Curation, R.M. ; Writing-Original Draft Preparation, R.M. ; Writing-Review & Editing, A.S., M.J., F.R. and B.F.; Visualization, A.S.; Supervision, A.S. and B.F.; Project Administration, A.S. This project was funded by the European Union's Horizon 2020 under the [PERSON] grant agreement 643134 (CLEAN-Gas).