## 4 Results and Discussion
Data for the Reynolds stresses were shifted in vertical direction by 0.01 in order to be able to distinguish it from the dissipation data. Figure 8 shows that the anisotropies of \(T_{ij}\) and \(\varepsilon_{ij}\) are nearly identical, with only marginally more isotropy for \(\varepsilon_{ij}\) towards the channel center (indicated by the crosses). The same holds true for bubbly channel flow, as reported in [36], and the findings are also consistent with the early experimental data from turbulent boundary layers by [PERSON] et al. [4]: Very close to the wall in the viscous sublayer of the channel flow, the turbulence is essentially two-component, \(v\) being much smaller than \(u\) and \(w\). Anisotropy reaches a peak at a dimensionless wall distance of about \(y^{+}=7\) close to the 1C state and subsequently becomes increasingly isotropic towards the channel center [2]. Apart from the artificial offset, Figure 8 shows nearly the same behaviour for anisotropy based on Reynolds stress and dissipation tensors. The results for the anisotropy of the tensors \(T_{ij}\) and \(\varepsilon_{ij}\) are shown in the Lumley triangle for cases A-C in Figures 9-11. The first row in each plot shows the results for unfiltered data, followed by the data after applying a high pass filter of sizes \(\Delta/\delta_{th}=5.8,\ 1.4,\ 0.4\), such that, for the last row, only the smallest structures remain. Focusing first on the unfiltered results in the first row of Figures 9-11, two things are apparent: (i) there is a pronounced anisotropy of the Reynolds stresses, particularly for case A, where thermal expansion effects dominate the turbulent velocity fluctuations. The turbulent state of the largest part of the statistically planar flame brush can be found on the axisymmetric expansion border which, for the flame propagation direction aligned with the \(x\)-direction, signifies a situation where \(T_{11}>T_{22}=T_{33}\) (refer to Figure 4a). The initial background fluid motion is isotropic, and with increasing turbulence intensity (alternatively increasing \(Ka\) or decreasing \(Da\), see Table 1), the relative contribution of thermal expansion effects weakens, and the flow fields becomes more isotropic. These observations are consistent with earlier findings in [15; 16; 17]: Turbulent premixed flames can cause strong anisotropies such that the turbulent state is located on the axisymmetric expansion border and reaches, for small turbulence intensities (e.g., \(u^{\prime}/S_{L}\approx 1\)) from the isotropic state two-thirds up the way to one component's endpoint. For large turbulence intensities (e.g., \(u^{\prime}/S_{L}>10\)), a nearly isotropic behaviour can be observed: (ii) The dissipation tensor is considerably more isotropic than the Reynolds stress tensor but this difference decreases with decreasing \(\Delta\). For example, the maximum departure from the origin can reach up to \(\xi=0.27\ (0.13)\) and \(\eta=0.27\ (0.13)\) for \(T_{ij}\) in case A (case C), whereas the maximum departure from the origin extends to \(\xi=0.11\ (-0.024)\) and \(\eta=0.11\ (0.024)\) for \(\varepsilon_{ij}\) for the unfiltered condition. This situation changes to \(\xi=-0.13\ (-0.028)\) and \(\eta=0.13\ (0.028)\) for \(T_{ij}\) in case A (case C),  
Figure 8: Anisotropy based on Reynolds stress and dissipation tensor for the channel flow data presented in [36]. The point closest to the wall is marked with a circle, whereas the channel center location is marked with a cross. Results for \(T_{ij}\) are shifted in direction of \(\eta\) by 0.01.  
whereas the maximum departure from the origin extends to \(\xi=-0.064\) (\(-0.03\)) and \(\eta=0.064\) (\(0.03\)) for \(\varepsilon_{ij}\) for \(\Delta/\delta_{th}=0.4\).