## 2 Numerical Methodology  
### Large Eddy Simulation Transport Equations  
In this section, the LES hybrid ESF/FPV approach is presented. Actually, it combines in the LES framework a transported joint scalar probability density function (T-PDF) following the Eulerian stochastic field methodology (ESF) and the flamelet progress variable (FPV) turbulent combustion model under consideration of detailed chemical reaction mechanism. The filtered governing equations for describing a turbulent reacting Newtonian fluid flow are obtained after a filtering procedure in space from which the large-scale energetic motions are resolved and the unresolved sub-grid scale structures are accounted for by so-called sub-grid scale (SGS) models (e.g., [5; 6; 7]). The filtered transport equations for LES emerge then in the condensed form as  
\[\frac{\partial(\overline{p}\widetilde{\varphi})}{\partial t}+\frac{\partial( \overline{pu_{i}}\widetilde{\varphi})}{\partial x_{i}}=\frac{\partial}{ \partial x_{i}}\{\overline{Q_{i}}+Q_{i}{}^{ggs}\}+\overline{\pi}_{\varphi}+ \overline{S}_{c} \tag{1}\]  
where the quantity \(\widetilde{\varphi}\) is defined as \(\widetilde{\varphi}=\overline{\rho\varphi}/\overline{p}\)[5; 45]. The symbols \(\overline{(. )}\) and \(\widetilde{(. )}\) express spatially filtered and density-weighted (Favre-filtered) quantities, respectively. In Equation (1), \(u_{i}\) is the velocity component (with \(i=\{\)1,2,3\(\}\)) and \(x_{i}\) the position vector in i-direction. The quantity \(\varphi\) may represent unity, the velocity component, thescalar mass fraction, \(Y\), for the equation of mass, momentum, and progress variable transport, respectively. \(Q_{i}\) represents the molecular flux vector quantity and \(Q_{i}{}^{ggs}\) a SGS flux counterpart, while \(\pi\) stands for a supply term (viscous/pressure term, possible radiation) and \(S_{C}\) for the chemical source term. To close Equation (1), models for the unclosed SGS terms \(Q_{i}{}^{ggs}\) and \(S_{C}\) are required. In the momentum and scalar transport equations this term is the SGS stress tensor and scalar flux vector, respectively, expressed as:  
\[\tau_{ij}^{ggs}=\widehat{\rho u_{i}u_{j}}-\rho\widetilde{u_{i}}\widetilde{u _{j}}\text{ and }Q_{i\varphi_{\alpha}}^{ggs}=\widehat{\rho u_{i}\varphi_{\alpha}}- \rho\widetilde{u_{i}}\widetilde{\varphi_{\alpha}} \tag{2}\]  
Various models have been suggested in the literature (see [5; 7; 46]) to close these terms. In the present paper, the SGS stress tensor is postulated following the eddy viscosity approach as:  
\[\tau_{ij}^{ggs}-\frac{1}{3}\tau_{kk}^{ggs}\delta_{ij}=-2\mu_{ggs}\left( \widetilde{S}_{ij}-\frac{1}{3}\widetilde{S}_{kk}\delta_{ij}\right) \tag{3}\]  
where \(\mu_{ggs}=\overline{p}\left(c_{m}\Delta_{grid}\right)^{2}\left(S_{ij}S_{ij} \right)^{1/2}\) and \(\widetilde{S}_{ij}=\frac{1}{2}\left(\frac{\partial\widetilde{u_{j}}}{ \partial x_{i}}+\frac{\partial\widetilde{u_{i}}}{\partial x_{j}}\right)\), according to [PERSON] [6]. Thereby, \(C_{m}\) is a model coefficient and \(\Delta_{grid}=\left(\Delta_{x}\Delta_{y}\Delta_{z}\right)^{\frac{1}{3}}\) the grid filter width. To account for the stream jet'swall surface, the van Driest wall damping function is introduced [45], so that the grid filter width \(\Delta_{grid}\) is replaced by the expression:  
\[\Delta_{VD}=min\bigg{[}\Delta_{grid},\frac{k_{r}}{C_{\Delta}}\Big{(}1-e^{(-r.A^ {+}/r^{+})}\Big{)}\bigg{]} \tag{4}\]  
where \(k_{r}\) is the Karman constant and \(r^{+}\) is the dimensionless wall distance. The SGS scalar flux \(Q_{i\varphi_{\alpha}}^{sgs}\) is modelled as:  
\[Q_{i\varphi_{\alpha}}^{sgs}=-\frac{\mu_{sgs}}{\sigma_{sgs}}\left(\frac{ \partial\widetilde{\varphi}_{\alpha}}{\partial x_{j}}\right) \tag{5}\]  
were \(\alpha\) refers to species and \(\sigma_{sgs}\) expresses the SGS Schmidt number.