## 2 Numerical Methodology
### Chemistry Reduction with Flamelet/Progress Variable Approach (FPV)
#### 2.3.1 Transported Filtered Joint Probability Density Function  
According to [PERSON] in [53] and [PERSON] in [31; 32; 54], a transported equation of a density weighted sub-grid probability density function (PDF) for different scalar quantities required to describe the reaction (FPV controlling variables) and for given instance \(t\) and point \(x_{i}\), can be derived from Equation (1) as follows:  
Figure 1: Distribution of production rate of heat release in the mixture fraction space colored with source term of the progress variable (PV): (**a**) Flame-D table, (**b**) Oxyflame B3 table. \[ \begin{split}\bar{\rho}\frac{\partial\bar{F}_{ggs}\left(\psi\right)}{ \partial t}&+\bar{\rho}\bar{u}_{j}\frac{\partial\bar{F}_{ggs} \left(\psi\right)}{\partial x_{j}}-\sum_{\alpha=1}^{N_{\alpha}}\frac{ \partial\left(\bar{\rho}\omega_{\alpha}\bar{P}_{ggs}\left(\psi\right)\right) }{\partial\psi_{\alpha}}\\ &=-\frac{\partial}{\partial x_{j}}\left[\left(\left(\bar{\rho} \bar{u}_{j}-\bar{\rho}\bar{u}_{j}\right)\left|\underline{\Phi}=\underline{ \psi}\right)\bar{F}_{ggs}\left(\underline{\psi}\right)\right]\\ &-\sum_{\alpha=1}^{N_{\alpha}}\sum_{\beta=1}^{N_{\beta}}\frac{ \partial^{2}}{\partial\psi_{\alpha}\psi_{\beta}}\left[\left(\frac{\mu}{ \sigma}\frac{\partial\psi_{\alpha}}{\partial x_{j}}\frac{\partial\psi_{ \beta}}{\partial x_{j}}\left|\underline{\Phi}=\underline{\psi}\right)\bar{F}_ {ggs}\left(\underline{\psi}\right)\right]\end{split}  \tag{11}\]  
where the assumption of equal diffusivity has been applied [55]. In Equation (11),  
\[\bar{P}_{ggs}\left(\underline{\psi};\left(x_{i},t\right)\right)=\int_{\psi} \frac{\rho\left(x_{i}-x_{i}^{\prime}\right)}{\bar{\rho}}F\left(\underline{\psi };x_{i}^{\prime},t\right)g\left(x_{i}-x_{i}^{\prime},\Delta(x_{i})\right)dx_{ i}^{\prime} \tag{12}\]  
with \(F\left(\underline{\psi};x_{i}^{\prime},t\right)=\prod_{\alpha=1}^{N_{\alpha}} \delta(\psi_{\alpha}-F_{\alpha}(x_{i},t))\) being the joint PDF. The quantity \(\underline{\psi}\) stands for sample space for the variable \(F_{\alpha}\) and \(\delta\) represents the Dirac delta function. The density weighted sub-grid PDF, \(\bar{P}_{ggs}\), then represents the probability to capture the variable \(F_{\alpha}\) between two succeeding phase spaces \(\psi\) and \(\psi+d\psi\), for different realizations of the controlling variable within the filter volume described in Equation (1). On the left hand side of Equation (11), the first term \(\blacklozenge\) represents the accumulation part describing the rate of change in the physical space, the second term \(\blacklozenge\) is the convection term due to the filtered velocity, and the last term \(\blacklozenge\) stands for the closed chemical production source term in the phase space. Unlike the three terms defined already which are in closed form, the terms in the right hand side of Equation (11) require modelling as they are unclosed. The fourth term \(\blacklozenge\) describes the transport at the sub-grid level of the FDF. The last term \(\blacklozenge\) represents the influence of the molecular mixing at scales smaller than the size of the filter width. The transport of the SGS FDF is postulated by a simple gradient ansatz in accordance to the Smagorinsky model used above. To close the micro-mixing term, the linear mean-square estimation model approach (known as interaction by exchange of mean, IEM) [56] is applied in this work for its simplicity. Correspondingly, the modeled expression for the filtered joint probability density function \(\bar{P}_{ggs}\) can be written as  
\[