## 3 Large Eddy Simulations of Isothermal Combustion
### Mixture Fraction Statistics  
In Figure 6, the fuel-consumption rate, \(W=-dM_{\rm F}/dt\), is shown where \(M_{\rm F}\) is the total fuel mass obtained by integrating the \(\rho Y_{\rm F}\) field over the volume of the computational domain. Consumption rate is normalized by \(W_{0}\), an initial value of \(W\) in DSM simulation. It is seen that the profiles for all subgrid viscosity models are qualitatively similar while the DSM simulation departs from them near the start of the process and after \(t/T_{0}=0.05\). The maximum ratio of fuel-consumption rates obtained with different models is as high as 1.8 at \(t/T_{0}\approx 0.09\). This indicates the importance of accurate description of subgrid processes in the problem considered. We selected four time instances, \(t=\{0.02T_{0},\ 0.05T_{0},\ 0.1T_{0},\ 0.15T_{0}\}\), for comparative analysis. Below (Figures 7-10), we consider the instantaneous \(\overline{z}\) field (black isoline corresponds to \(\overline{z}=0.5\)), \(\langle\overline{z}\rangle_{YZ}(x)\), profile which is obtained by averaging \(\overline{z}\) over \(YZ\)-planes, resolved mixture fractionmixture fraction variance \(\overline{z^{\prime 2}}_{res}(x)=\langle\overline{z}^{2}\rangle_{YZ}-\langle \overline{z}\rangle_{YZ}^{2}\), subgrid-scale mixture fraction variance \(\overline{z^{\prime 2}}_{\rm gsg}(x)=\langle\overline{z^{\prime 2}}\rangle_{YZ}\) and its production rate \(\langle\overline{P}_{z}\rangle_{YZ}(x)\), and scalar dissipation \(\langle\overline{x}\rangle_{YZ}(x)\). The last two statistics are calculated according to (26), (27). Figure 5: Normalized probability density function of scale-to-scale energy flux. Figure 6: Fuel-consumption rate. Figure 7: Time instance \(t=0.02T_{0}\). Instantaneous \(\overline{z}\) field obtained with the DSM (**upper left**), \(\langle\overline{z}\rangle_{YZ}(x)\) (**upper right**), \(\overline{z^{2}}_{res}\) (**middle left**), \(\overline{z^{2}}_{sg}\)(**middle right**), \(\langle\overline{P}_{z}\rangle_{YZ}(x)\) (**lower left**), \(\langle\overline{x}\rangle_{YZ}(x)\) (**lower right**). Figure 8: Time instance \(t=0.05T_{0}\). Designations as in Figure 7.  
Figure 9: Time instance \(t=0.10T_{0}\). Designations as in Figure 7.  
Again, a comparison of the \(\left\langle\overline{z}\right\rangle_{YZ}\) and \(\overline{z^{\prime 2}}_{res}\) profiles shows that the resolved scalar statistics are similar with all the considered models. However, the subgrid-scale characteristics are significantly different. A typical level of mixture-fraction subgrid-scale variance is two times lower for the DSM than for the Smagorinsky model. A possible explanation of the discrepancy is that the \(\left\langle\overline{z}\right\rangle_{YZ}\) and \(\overline{z^{\prime 2}}_{res}\) profiles are similar. Figure 10: Time instance \(t=0.15T_{0}\). Designations as in Figure 7.  
this difference is as follows. Consider a balance between production and dissipation of \(\overline{z^{\prime 2}}\). In local equilibrium, \(\overline{P}_{z}\approx\overline{\chi}\), so  
\[2\sigma_{z1}\cdot v_{\text{sgs}}\cdot\frac{\partial\overline{z}}{\partial x_{k }}\frac{\partial\overline{z}}{\partial x_{k}}\approx C_{\chi}\frac{\sqrt{2.54v _{\text{sgs}}S}}{\Delta}z^{\prime 2}\quad\Rightarrow\quad\overline{z^{\prime 2}} \propto\sqrt{\frac{v_{\text{sgs}}}{S}}\Delta\frac{\partial\overline{z}}{ \partial x_{k}}\frac{\partial\overline{z}}{\partial x_{k}}\propto\frac{\Delta}{H}.\]  
In DSM, \(\Delta/H=0.4\), while in three other models, \(\Delta/H=1\). This leads to \(\left(\overline{z^{\prime 2}}\right)_{DSM}/\left(\overline{z^{\prime 2}} \right)_{other}\approx 0.4\), which is even lower than the observed ratio of 0.5. To improve the estimate, one has to take into account the diffusion terms in \(\overline{z^{\prime 2}}\) equation and the possible differences in the statistics of the \(\partial\overline{z}/\partial x_{k}\) field, as predicted by different models.