## 1 Introduction
To overcome this disagreeability, novel techniques were recently proposed, like the so-called hybrid transported-tabulated chemistry (HTTC) approach, combining the transport of the main species in the flow with the tabulation of the radical species [14]. The tables resulting from these techniques are coupled either to RANS- or to LES-based Computational Fluid Dynamics (CFD). Focusing especially on the flamelet/progress variable (FPV) method as suggested by [PERSON] in [8], the mixture fraction and the progress variable, tracing the local reaction progress, are the controlling variables for the multidimensional chemistry. Within the FPV-based combustion model, they are solved by their own transport equations in addition to the continuity, momentum, energy/enthalpy equations in single phase reacting flow cases (e.g., [4; 8; 15; 16; 17]). This method has been widely used, especially for CH4/Airjet flame-D by various authors (e.g., [4; 15; 18]) and for predicting extinction in Sandia piloted jet flames (E and F) (e.g., [16]). Dealing with oxy-combustion modelling, several advanced approaches have been evaluated to study the turbulence-flame interaction phenomena. In the numerical work of [PERSON] et al. in [19], both the laminar steady flamelet model (SFM) and the eddy dissipation concept model (EDC) were used to study the influence of additional oxy-fuel jet burners in gas phase on the productivity of the investigated furnace. Also, [PERSON] et al. in [20], in their oxy-combustion study, compared the same methods in a RANS context with the eddy dissipation model (EDM) for various oxygen enrichments. Both studies employed different chemical mechanisms. However, [PERSON] et al. in [21] discussed the usability and limits of the SFM approach in oxy-fuel combustions. [PERSON] et al. in [22] applied in the LES framework an alternative approach based on the joint PDF method coupled to FGM technique based on the mixture fraction and the reaction progress variable to investigate the behavior of two reacting separated oxy-fuel jets. Thereby the sub-grid PDF shape is described by the presumed beta-PDF assumption. However, according to [23], not all transported scalar properties could be appropriately captured using presumed-PDF methods. Nevertheless, the conditional moment closure (CMC) combustion model has been used for different turbulent studies, including the characteristics of turbulent combustion of natural gas flame in oxy-fuel combustion environments by [PERSON] et al. in [24] and the oxy-fuel coaxial jets by [PERSON] et al. in [25]. However, very few contributions dealt with oxy-combustion using the FPV approach. One of the early turbulent simulations incorporating the FPV approach in the LES framework has been reported in [17], where it was shown that the LES-FPV approach possesses in turbulent regimes potential promise in addressing the issue of differential diffusion in such a flame. Similar to FPV, the FGM method has been used in [22] with Le = 1 to tackle oxy-fuel combustion process in twin-jets. It is worth noting that both FPV and FGM parameterize all species and temperature by a mixture fraction and a reaction progress variable or reaction progress parameter. However, the two models treat the flamelet manifold between equilibrium and flame extinction in a different manner (see in [26]). As pointed out in [26], the FPV model solves flamelet equations in the (unstable) middle and lower branches of the flamelet S-curve, while the FGM model solves an unsteady extinguishing flamelet from the last stable burned steady solution before extinction. Nevertheless, the generated tables show similar behavior in the mixture fraction space while exhibiting different behaviors on the progress variable space. In the current study, the FPV approach is especially adopted due to its better features in both lean and rich flame conditions. It is then used at the first time in coupling with LES to investigate non-premixed oxy-flames.