## 4 Results and Discussion
Figure 12 shows that the subgrid scale stresses are considerably more isotropic than their averaged counterparts, and this statement holds for the entire range of filter width. Again, isotropy increases with increasing \(u^{\prime}/S_{L}\), qualitatively similar to the Reynolds/Favre-averaged equivalents. The same physical explanations which were mentioned earlier in the context of RANS to explain the greater extent of isotropy for \(T_{ij}\) and \(\varepsilon_{ij}\) for the cases with high turbulence intensities are also qualitatively valid in the context of subgrid quantities. Figure 9: Anisotropy of the tensors \(T_{ij}\) and \(\varepsilon_{ij}\) shown in the Lumley triangle for case A. A Gaussian high pass filter with (bottom to top) \(\Delta/\delta_{th}=0.4,\ 1.4\) and \(5.8\) is used and compared to the results from the unfiltered data. The results are shown conditional on reaction progress variable \(c\), as indicated by the colorbar. In addition, the first (\(c=0.05\)) and last points (\(c=0.95\)) are marked by a circle and a cross, respectively. Figure 10: Anisotropy of the tensors \(T_{ij}\) and \(\varepsilon_{ij}\) shown in the Lumley triangle for case B. A Gaussian high pass filter with (bottom to top) \(\Delta/\delta_{th}=0.4,\ 1.4\) and \(5.8\) is used and compared to the results from unfiltered data. The results are shown conditional on reaction progress variable \(c\), as indicated by the colorbar. In addition, the first (\(c=0.05\)) and last points (\(c=0.95\)) are marked by a circle and a cross, respectively. Figure 6.0 Anisotropy of the tensors \(T_{ij}\) and \(c_{ij}\) shown in the Lumley triangle for case C. A Gaussian high pass filter with (bottom to top) \(\Delta/\delta_{hi}=0.4,\ 1.4\) and \(5.8\) is used and compared to the results from unfiltered data. The results are shown conditional on reaction progress variable \(c\), as indicated by the colorbar. In addition, the first (\(c=0.05\)) and last points (\(c=0.95\)) are marked by a circle and a cross, respectively. The modification of anisotropy distribution in the Lumley triangle in turbulent pre-mixed flames in comparison to that in the corresponding non-reacting flow has been reported in [15], and these findings have, in the meantime, been confirmed independently by a few other DNS groups [37; 38; 39]: Turbulent premixed flames cause strong anisotropies such that the turbulent state is located on the axisymmetric expansion border and reaches,  
Figure 6.1 Anisotropy of the subgrid scale tensors \(\tau_{ij}\) shown in the Lumley triangle for cases A and B. A Gaussian low pass filter with (bottom to top) \(\Delta/\delta_{th}=0.4,\ 1.4,5.8\) and \(11.6\) is used. The results are shown conditional on reaction progress variable \(c\), as indicated by the colorbar. In addition, the first (\(c=0.05\)) and last points (\(c=0.95\)) are marked by a circle and a cross, respectively. depending on the turbulence intensity (Karlovitz number), from the isotropic state two-thirds up the way to the one component endpoint. Furthermore, early measurements of the enhancement of turbulence anisotropy in large-scale, low-intensity turbulent pre-mixed propane-air flames using two-component measurements have been reported by [PERSON] et al. [40]. It has been shown in the past [15; 16; 41] that the phenomenon of counter-gradient transport or counter-gradient stresses, as theoretically predicted by [PERSON] and co-workers [28], is closely related to the axisymmetric expansion turbulent state. Its existence in the context of LES has been confirmed by very recent and advanced measurement techniques [42] that result in the conclusion that LES models should allow for upscale energy transfer in the vicinity of the flame. In addition, in the context of unsteady RANS or hybrid RANS/LES, the present results suggest the invalidity of the Boussinesq assumption and the need for an anisotropic correction, which will depend on how much of the turbulence kinetic energy can be resolved, as revealed by the multiscale analysis.