## 2 Numerical Methodology
### Chemistry Reduction with Flamelet/Progress Variable Approach (FPV)
#### 2.3.1 Transported Filtered Joint Probability Density Function
\begin{split}\bar{\rho}\frac{\partial\bar{F}_{ggs}\left(\psi \right)}{\partial t}+\bar{\rho}\bar{u}_{j}\frac{\partial\left(\bar{P}_{ggs} \left(\psi\right)\right)}{\partial x_{j}}-\sum_{\alpha=1}^{N_{\alpha}}\frac{ \partial\left(\bar{\rho}\omega_{\alpha}\bar{P}_{ggs}\left(\psi\right)\right) }{\partial\psi_{\alpha}}=\frac{\partial}{\partial x_{j}}\left[\left(\frac{\bar {\mu}}{\sigma}+\frac{\mu_{ggs}}{\sigma_{ggs}}\right)\frac{\partial\bar{P}_{ggs }\left(\psi\right)}{\partial x_{j}}\right]\\ -\frac{\bar{\rho}}{\tau_{ggs}}\sum_{\alpha=1}^{N_{\alpha}}\left[ \left(\psi_{\alpha}-\overline{F}_{\alpha}\right)\bar{P}_{ggs}\psi\right]\end{split}  \tag{13}\]  
In Equation (13) the sub-grid mixing time scale \(\tau_{ggs}\) is modeled by using a mixing time model referring to [PERSON] et al. [54] as:  
\[\frac{1}{\tau_{ggs}}=C_{F}\frac{\mu+\mu_{ggs}}{\bar{\rho}\Delta^{2}}\text{ with }C_{F}=2. \tag{14}\]  
It is worth noting that various approaches for the micro-mixing modeling exist. Recent reviews can be provided in [57; 58]. It turns out that the coefficient \(C_{F}\) can take different values, for more details, see [57; 58]. In order to solve the closed form of the SGS density weighted transported PDF equation (Equation (12)) the Eulerian stochastic field formulation according to [PERSON] [35], [PERSON] [54], and [PERSON] et al. [4] is applied. In conjunction with the FPV chemistry reduction technique, the temporal evolution of the SGS joint PDF is represented by an ensemble of Ns Eulerianstochastic fields \(\xi_{\alpha}^{n}(x_{i},t)\) defined in the entire domain for each controlling variable \(\alpha=\{PV,f\}\) with 1 \(\leq n\leq\) Ns and \(1\leq\alpha\leq N_{\alpha}\). These fields evolve following the equation:  
\[ \begin{array}{c}\overline{\rho}d(\xi_{\alpha}^{n})=-\overline{\rho}\dot{u}_{j }\frac{\partial(\xi_{\alpha}^{n})}{\partial x_{j}}dt+\frac{\partial}{\partial x _{j}}\left[\left(\frac{\widetilde{u}}{\overline{\sigma}}+\frac{u_{\alpha\alpha }}{\sigma_{\alpha\beta}}\right)\frac{\partial x_{\alpha}^{n}}{\partial x_{j}} \right]dt+\overline{\rho}\dot{\omega}_{\alpha}(\xi_{\alpha}^{n})dt-\frac{\mu+ \mu_{\alpha\alpha}}{\Delta^{2}}\left(\xi_{\alpha}^{n}-\widetilde{F}_{\alpha} \right)dt\\ +\sqrt{2\overline{\rho}\left(\frac{u_{\alpha\alpha}}{\sigma_{\alpha\beta}} \right)\frac{\partial x_{\alpha}^{n}}{\partial x_{j}}}dW_{j}^{n}\text{ for }n=[1,\ldots,N]\end{array}  \tag{15}\]  
which has to be solved in addition to the governing equations of mass and momentum (Equations (6) and (7)) within the LES hybrid ESF/FPV approach. Thereby, the chemical source term \(\dot{\omega}_{\alpha}\) is determined based on the evolution of the controlling variable \(\xi_{\alpha}^{n}\), and the last term, \(dW_{j}^{n}\), which stands for the increments of a vectorial Gaussian process, is referred to as the vector Wiener process, which is spatially uniform, varying in time and different for each stochastic field [59]. Hence, for the different stochastic fields Ns and by employing the approximation used in [35], the Wiener term is determined by the discrete time-step \(\Delta_{t}\) between the temporal sample \(t^{n}\) and \(t^{n+1}\), multiplied by the dichotomic vector following (see also in [4]):  
\[dW_{j}^{n}=N(0,1)\sqrt{\Delta_{t}}\text{ with }N(0,1)\approx[-1,+1] \tag{16}\]  
The Wiener process as a random walk is normally distributed for Ns stochastic fields with zero mean and variance of the time increment \(\Delta_{t}\) reading:  
\[\frac{1}{N}\sum_{n=1}^{N}dW_{j}^{n}=0\text{ and }\frac{1}{N}\sum_{n=1}^{N}dW_{j}^ {n}dW_{j}^{n}=\Delta_{t}\text{ for }j\in[1,2,3] \tag{17}\]  
It is important to note that the different stochastic fields are continuous in time due to the Wiener process. The fields are assumed to be smooth with the grid control volume and then discretized at the length scale of the grid size. This assumption is highly relevant, since sub-grid scales are absent in the stochastic fields and stochastic differential equations (SDE) are fully resolved on the grid size level.