## 3 Large Eddy Simulations of Isothermal Combustion
### Problem Statement and Numerical Setup
We continue the simulation solving two additional equations, as follows:  
\[\frac{\partial\rho\overline{z}}{\partial t}+\frac{\partial}{\partial x_{k}} \Big{[}\rho\overline{z}\,\overline{u}_{k}+\rho\overline{z^{\prime 2}u^{\prime}_{k}} \Big{]}=0,\]  
\[\frac{\partial\rho\overline{z^{\prime 2}}}{\partial t}+\frac{\partial}{ \partial x_{k}}\Big{[}\rho\overline{z^{\prime 2}}\,\overline{u}_{k}+\rho \overline{z^{\prime 2}u^{\prime}_{k}}\Big{]}=\rho\overline{P}_{z}-\rho \overline{\chi}.\]  
To ensure that the subgrid stress model comparison is straightforward, we use the same closures of these equations for all considered models. Subgrid transport terms are modeled using a simple gradient hypothesis, as follows:  
\[\overline{z^{\prime}u^{\prime}_{k}}=-\sigma_{z1}\
u_{sss}\frac{\partial \overline{z}}{\partial x_{k}},\qquad\overline{z^{\prime 2}u^{\prime}_{k}}=- \sigma_{z2}\
u_{sss}\frac{\partial\overline{z^{\prime 2}}}{\partial x_{k}},\]  
where \(\
u_{sss}\) is determined as described in Section 2.6, and \(\sigma_{z1}=\sigma_{z2}=1\) for simplicity. Accordingly, the production term \(\overline{P}_{z}=-2\overline{z^{\prime}u^{\prime}_{k}}\partial\overline{z}/ \partial x_{k}\) is approximated as  
\[\overline{P}_{z}=2\sigma_{z1}\
u_{sss}\frac{\partial\overline{z}}{\partial x_ {k}}\frac{\partial\overline{z}}{\partial x_{k}}. \tag{26}\]  
Finally, the scalar dissipation term \(\overline{\chi}=-2\
u\ \overline{\partial z^{\prime}/\partial x_{k}\cdot \partial z^{\prime}/\partial x_{k}}\) is represented as  
\[\overline{\chi}=C_{\chi}\frac{\xi}{\overline{k}}\overline{z^{\prime 2}}=C_{ \chi}C_{E0}\frac{\chi}{\Delta}\overline{z^{\prime 2}},\ \ C_{\chi}=1.5,\qquad C_{E0}=0.47, \tag{27}\]  
where the \(C_{\chi}\) value lies in the range proposed in the literature [32] and \(C_{E0}\) is determined from filtered DNS by calibrating a constant coefficient dissipation rate model (4). We intentionally use simple models of \(\overline{z}\) and \(\overline{z^{\prime 2}}\) which can be directly coupled to subgrid viscosity models and DSM. A drawback of this approach is missing backscatter in mixture fraction field (i.e., regions where \(\overline{P}_{z}<0\)). An advanced model with differential equations for \(\overline{z^{\prime}u^{\prime}_{k}}\), which allows the turbulent transport of a mixture fraction to be misaligned with \(-\partial\overline{z}/\partial x_{k}\), will be investigated in the future. Spatially filtered values of mass fractions can be found using a subgrid-scale probability density function (PDF) of the mixture fraction \(z\), as follows:  
\[\overline{Y_{j}}=\int_{0}^{1}Y_{j}(z)PDF(z)dz, \tag{28}\]  
where \(Y_{j}(z)\) corresponds to the exact solution of the stationary laminar flamelet equations [18] in the Burke-Schumann approximation [17] (see Table 6). In Table 6, \(Y_{F}\), \(Y_{O}\) and \(Y_{P}\) are the mass fractions of the fuel, the oxidizer and the product of a single unidirectional reaction \(F+O\to P\), respectively, and \(z_{st}\) is the value of the mixture fraction at the stoichiometric surface, where the flame front is located. Let us denote the mass of the oxidizer required for the combustion of unit of mass of the fuel as \(L_{0}\). Since the model problem with arbitrary substances is considered, we take \(L_{0}=1\), which leads to flame front coordinate \(z_{st}=1/(1+L_{0})=0.5\). \begin{table}
\begin{tabular}{c c} \hline \hline
**z>=\(z_{st}\)** & **z<=\(z_{st}\)** \\ \hline \(Y_{F}=\frac{z-z_{s}}{1-z_{s}}\) & \(Y_{F}=0\) \\ \(Y_{O}=0\) & \(Y_{O}=1-\frac{z}{z_{st}}\) \\ \(Y_{P}=1-Y_{F}\) & \(Y_{P}=1-Y_{O}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Burke–Schumann solution.