## 2 Numerical Methodology
### Chemistry Reduction with Flamelet/Progress Variable Approach (FPV)  
In the classical FPV approach, the problem of turbulent combustion model and simulation consists of determining the mean chemical source term, \(\overline{\omega}\gamma_{k}\), using the appropriate chemical reaction model or mechanism to solve Equation (8). In the present paper, a tabulated chemistry method following the flamelet/progress variable (FPV) approach [7] was employed. Accordingly, a steady-state flamelet library was generated based on two controlling variables, namely the mixture fraction and a reaction progress variable. All relevant information (e.g., the chemical rates of formation/destruction, chemical composition, temperature, density, viscosity, etc.) was therefore stored in the tables as a function of just the two variables. In this way, other flame thermochemical states (i.e., fully burning state, transient solution between burned and unburned states and the unburned state) can be parameterized with the advantage that the reaction progress variable allows description of local extinction and reignition phenomena [15,16,18,49]. In order to build such FPV tables, a laminar counter-flow flame was first solved using the flamelet generator FlameMaster from [50], where the chemical mechanism in use consists of 325 reactions and 53 species available in GRI-MECH 3.0 [51]. The results obtained from 1D flamelets were then mapped into the mixture fraction and progress variable space. Finally, each instantaneous thermochemical state was retrieved by means of the two controlling variables. Given that different definitions of the reaction progress variable (PV) could be used [52], the following progress variables were employed in the current study for the Sandia flame D and the oxy-flame B3, respectively, in accordance with [4,25]. \[PV_{Flame\_D}=[Y_{\text{H2O}}]+[Y_{\text{CO2}}]+[Y_{\text{CO}}] \tag{9}\]\[PV_{Oxyfuel\_B3}=[\mathrm{Y_{H2O}}]. \tag{10}\]  
where \([-]\) denotes the specie mass fraction divided by the specie molar mass. Figure 1 presents some features from the FPV tables created according to the progress variable for both Flame D and Flame B3 (see Equations (9) and (10)). Thereby the distribution of the heat release production rate is shown colored by the reaction rate of the progress variable in the mixture fraction space. Considering the high amount of CO\({}_{2}\) diluted in the oxidizer in Oxyflame B\({}_{3}\) instead of N\({}_{2}\) as it is in Flame D and since the heat transfer properties of CO\({}_{2}\) are different from those of N\({}_{2}\), one can expect a significant variation of the production rate of the heat release. In Figure 1a referring to Flame D with 79% of N\({}_{2}\) in the oxidizer, the maximum value of heat release locally reaches 4.10\({}^{9}\) J/m\({}^{3}\)s. It is in the oxy-flame B\({}_{3}\) radically increased to 7.10\({}^{9}\) J/m\({}^{3}\)s where an amount of 68% of CO\({}_{2}\) is injected as diluent instead of N\({}_{2}\) (Figure 1b). Turbulence Chemistry Interaction, Transported Filtered Joint Probability Density Function and Eulerian Stochastic Field Approach  
To achieve complete statistical modelling of turbulent combustion it is essential to consider both sub-grid scale turbulent motions and chemical reactions along with their interaction in a reliable way. Approaches based on solving the transported probability density function offer compelling advantages in providing an effective resolution to the closure problems that arise from averaging or filtering of the highly nonlinear chemical source terms, from modeling the effects of convection and body forces. In [4] the predictive capability of the hybrid ESF/FPV method in RANS was very well established. This is due to the inherent property of the transported PDF method to include the complete statistical information about its variables. Instead of solving Equation (8) as in the classical FPV approach, for which a presumed PDF is usually adopted, the present paper adapts the RANS-based hybrid ESF/FPV approach to LES.