## 2 Numerical Methodology
### Chemistry Reduction with Flamelet/Progress Variable Approach (FPV)
#### 2.3.2 Numerical Implementation  
The reported simulations have been carried out by means of the open source code OpenFOAM [60], in which the hybrid ESF/FPV was implemented [4] and extended in this study to be coupled with LES turbulence models. The solution procedure is carried out through various numerical steps. According to [4], once the time loop starts, N number of stochastic Fields for the two controlling variables are created, which means \(2^{n}N\) new fields are created for both scalars: N fields for the mixture fraction, \(f\):\(f^{n}\) with \(n=\{1,\,\ldots,\,N\}\) and \(N\) fields for the progress variable, \(PV\):\(PV^{n}\) with \(n=\{1,\,\ldots,\,N\}\). The random Wiener term is computed, at the same stage, separately for each individual stochastic field in all spatial directions: \(dW_{j}^{n}\) with \(j=\{1,\,2,\,3\}\) allowing the random procedure of these created fields for both chemical table controlling variables. At this stage, the momentum and continuity transport equations are calculated where all necessary parameters such as density, viscosity and chemical source term for each stochastic field, are extracted from the chemical table as function of calculated \(f^{n}\) and \(PV^{n}\).  
In the numerical set-up of the studied cases, a stochastic field properties file (in terms of OpenFOAM code) is adopted in the way that the number of stochastic fields can be specified, either \(N\) = 1, 2,..., or 128. Hence, the stochastic differential equations from ESF/FPV approach read:  
\[\begin{split}\overline{\rho}d\big{(}\xi_{PV}^{n}\big{)}& =-\overline{\rho}\widetilde{u}_{j}\frac{\partial\big{(}\xi_{PV}^{n} \big{)}}{\partial x_{j}}dt+\frac{\partial}{\partial x_{j}}\Big{[}\Big{(} \frac{\ddot{\overline{u}}}{\sigma}+\frac{\mu_{\text{eg}}}{\sigma_{\text{eg}}} \Big{)}\frac{\partial\xi_{PV}^{n}}{\partial x_{j}}\Big{]}dt+\overline{\rho} \dot{\omega}_{PV}\big{(}\xi_{PV}^{n}\big{)}dt\\ &\quad-\frac{\mu+\mu_{\text{eg}}}{\Delta^{2}}\Big{(}\xi_{PV}^{n} -\widetilde{F}_{PV}\Big{)}dt+\sqrt{2\overline{\rho}\Big{(}\frac{\mu_{\text{ eg}}}{\sigma_{\text{eg}}}\Big{)}\frac{\partial\xi_{PV}^{n}}{\partial x_{j}}}dW_{j}^{n} \end{split} \tag{20}\]  
\[\begin{split}\overline{\rho}d\Big{(}\xi_{f}^{n}\big{)}& =-\overline{\rho}\widetilde{u}_{j}\frac{\partial\big{(}\xi_{f}^{n} \big{)}}{\partial x_{j}}dt+\frac{\partial}{\partial x_{j}}\Big{[}\Big{(} \frac{\ddot{\overline{u}}}{\sigma}+\frac{\mu_{\text{eg}}}{\sigma_{\text{eg}} }\Big{)}\frac{\partial\xi_{f}^{n}}{\partial x_{j}}\Big{]}dt-\frac{\mu+\mu_{ \text{eg}}}{\Delta^{2}}\Big{(}\xi_{f}^{n}-\widetilde{F}_{f}\Big{)}dt\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +\sqrt{2\overline{\rho}\Big{(}\frac{\mu_{\text{eg}}}{\sigma_{\text{eg}}}\Big{)} \frac{\partial\xi_{f}^{n}}{\partial x_{j}}}dW_{j}^{n}\end{split} \tag{21}\]  
Since only two controlling variables are addressed for the chemical look-up table, Equation (8) is replaced by the SDE equations in the context of ESF/FPV. In Equation (20), the filtered chemical reaction source term is evaluated by averaging the source terms of different stochastic fields, and there with the sub-grid level fluctuations due to stochastic terms are taken into account.  
Following the numerical set-up explained in [4] and keeping the main numerical scheme features, momentum predictor, pressure solver and momentum corrector are applied to solve the compressible Navier-Stokes governing equations (Equations (6) and (7)). A second order central differencing scheme is employed for the convection term of the velocity field, and a second order conservative scheme is applied for the Laplacian terms. A minmod differencing scheme is addressed allowing the solution total variation diminishing (TVD) for scalar flux and the second order backward integration method is used for the time derivative terms for LES calculations. Further details about the discretization procedure and the numerical schemes are explained in the Open-FOAM programmer's guide [60].