## 4 Results and Discussion  
The distributions of \(k\) and \(\varepsilon\) in a \(x\)-\(y\) planes are shown in Figure 6. and it is obvious that their behaviour changes significantly from case A to case C. Turbulence kinetic energy and its dissipation can be scaled as \(k\sim 1.5u^{\prime\,2}\) and \(\varepsilon\sim k^{1.5}/l\), respectively [2]. According to Table 1 this suggests that, globally, \(k\) increases by a factor of about 56 and 225 from case A to B and case A to C, while dissipation increases by a factor of 422 and 3375, respectively. This explains the largely different scales in Figure 6. While case A shows indications of augmentations of \(k\) within the flame because of thermal expansion, turbulent velocity fluctuations decay from the unburned to the burned gas side for cases B and C due to the rise of kinematic viscosity. Furthermore, for case A, highest values of dissipation are obtained within the flame, but for cases B and C, the dissipation magnitude decreases from unburned gas to the burned gas side by roughly a factor \(1/5\). Figure 5: Anisotropies of the tensors \(T_{ij}\), \(T_{ij}^{F}\), \(\varepsilon_{ij}\) and \(\varepsilon_{ij}^{F}\) in the Lumley triangle shown exemplarily for case B. The left (right) column shows the turbulent state of the Reynolds stress (dissipation) tensor, while the first (second) row is related to Reynolds (Favre) averaging, respectively. 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. It can further be observed from Figure 6. that the dissipative structures are clearly smaller than the structures carrying turbulence kinetic energy, which is in agreement with the idea of scale separation between the scales injecting and dissipating energy (i.e., \(l\gg\eta\) consistent with the scaling \(l/\eta\sim Re_{t}^{3/4}\), see Table 1, where \(\eta\) denotes the Kolmogorov scale). Figure 6: Instantaneous distribution of normalised turbulence kinetic energy (left) and normalised dissipation in the \(x\)-\(y\) plane for top-bottom cases A–C. The flame propagates from right to left along the negative \(x\)-direction. The field of view represents the computational domain of dimension \(26.1\delta_{hh}\times 26.1\delta_{hh}\). Multiscale [33; 34] or spectral [35] analysis of turbulent premixed flames is frequently conducted in the homogeneous directions, because heat release does not allow a Fourier transform in the direction of mean flame propagation. It might be, therefore, of interest to study the effects of a 2D (i.e., in the \(y\)-\(z\) plane) versus a 3D Gaussian filter on flow anisotropies. The anisotropy of the tensor \(T_{ij}\) is shown in Figure 7 for cases A-C after the application of a high pass filter with \(\Delta/\delta_{th}=0.4\). By comparing the left and right columns, it can clearly be observed that results are qualitatively similar for both filters, but the anisotropies remain stronger in the case of 2D filtering because large structures in \(x\)-direction are not affected by filtering in the \(y\)-\(z\) plane. In the following, only results for the 3D filter will be shown. This does not impose any problems because the flames are sufficiently far away from the \(x\)-boundaries to allow for the application of the 3D filter. In order to be able to correctly classify the following results for turbulent premixed flames, the anisotropies of the Reynolds stress and dissipation tensor from a channel flow simulation are exemplarily shown in Figure 8, where the data were taken from results  
Figure 7: Anisotropy of the tensor \(T_{ij}\) in the Lumley triangle for cases A–C when applying a 2D versus a 3D Gaussian high pass filter with \(\Delta/\delta_{th}=0.4\). The results are shown to be 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. presented in [36].