## 3 Large Eddy Simulations of Isothermal Combustion
### Subgrid Model Performance  
To determine the constants in subgrid viscosity models, we conducted a series of simulations (only the first stages) in a computational domain of \(64\times 64\times 64\) cells, varying the constants in the range \(0.1\leq C_{s}\leq 0.2\), \(-0.1\leq C_{dyn}^{min}\leq 0\), \(0.05\leq C_{dyn}^{max}\leq 0.2\) and \(0.3\leq C_{W}\leq 0.6\). We analyzed the longitudinal energy spectrum \(E_{uu}(k_{x})\) at the end of each simulation, as follows:  
\[E_{uu}(k_{x})=\left\langle\left|\int_{-L/2}^{L/2}u(x,\,y,\,z)e^{-ikx}dx\right| \right\rangle^{2}yz,\]  
where \(\left\langle\dots\right\rangle_{yz}\) denotes the averaging over all lines, \(y,\,z=const\). The values corresponding to the Kolmogorov spectrum, \(E_{uu}(k_{x})\propto k_{x}^{-5/3}\), in the range of \(7\leq k_{x}L/2\pi\leq 24\) were selected, which appeared to be equal to \(C_{s}=0.13\), \(C_{dyn}^{min}=-0.03\), \(C_{dyn}^{max}=0.1\) and \(C_{W}=0.4\). The same procedure was repeated for DSM, in which the ratio \(\Delta/H\) was varied (\(H\) is the computational grid spacing first appeared in Equation (19)). The optimum choice was found to be \(\Delta=0.4\)\(H\). In Figure 4, the optimal spectra are illustrated for all models along with the resolved kinetic energy decay. The oscillating character of energy is due to small dilatational motions present in the flow field. After a relaxation period of approximately \(T_{0}/2\), the energy decays according to a power law of \(E_{k}\sim t^{-1.1}\). All the models give similar statistics. However, this does not mean that the resolved fields are statistically equivalent. The energy decay rate and the energy spectrum correspond to the first two statistical moments of the velocity field. This is insufficient to fully describe the velocity field, e.g., its intermittency is left uncharacterized [3]. Interestingly, despite the discussed similarity in the resolved flow statistics, the sub-grid turbulence behaves very differently with these models. At the end of the first stage, the Smagorinsky model predicts the volume-averaged subgrid kinetic energy to be \(0.027E_{k1}\), the WALE model predicts it to be \(0.030E_{k1}\), the dynamic Smagorinsky model gives \(0.10E_{k1}\), while the DSM model gives \(0.055E_{k1}\). The high value obtained with the dynamic Smagorinsky model was surprising to us. This may indicate that the choice of \(v_{ggs}\) clipping in (20) is somewhat arbitrary and should be performed in some other way; another possible explanation is the use of \(k\) clipping in (21), which tends to increase the volume-averaged \(k\) value due to the presence of flow regions, where \(v_{ggs}<0\). The DSM predicts the subgrid kinetic energy to be approximately two times higher than the Smagorinsky and WALE models. Most interestingly, DSM simulates backscatter regions where energy flux is locally directed from subgrid scales toward resolved ones which is consistent with experimental [2] and DNS [3] data. The same holds for the dynamic Smagorinsky model, although it overestimates the frequency of backscatter events. In Figure 5, the probability density functions of scale-to-scale energy flux \(P=-\overline{u_{i}^{\prime}u_{j}^{\prime}}\)\(S_{ij}\) obtained with all models are plotted and compared with filtered DNS data. DSM is qualitatively correct in this distribution, which is encouraging while negative \(P\) values are absent with Smagorinsky and WALE models. The dynamic Smagorinsky model predicts too wide distribution of the energy flux. Note that all models produce approximately the same volume-averaged energy flux which follows from the similarity of lines on the right of Figure 4.  
The fact that the energy fluxes are close to each other while subgrid energies differ by a factor of 2 (not taking into account the dynamic Smagorinsky model) may be interpreted from the perspective of [3]. In [3], the authors analyzed the same DNS dataset as in the current paper and concluded that the turbulent cascade is relatively \"ineffective\" in transporting the energy across the scales.