## 4 Results and Discussion
The differences in anisotropy between \(T_{ij}\) and \(\varepsilon_{ij}\) tensors are in contrast to the observations from shear flows, as illustrated in Figure 8. Nevertheless, with increasing \(u^{\prime}/S_{L}\) (which is equivalent to increasing \(Ka\) or decreasing \(Da\) for a given value of \(l/\delta_{th}\)), isotropy increases (i.e., from case A to case C). This shows that linear relationships between the anisotropy of \(T_{ij}\) and \(\varepsilon_{ij}\) that might have been developed for shear flows are unlikely to work well for turbulent premixed flames. The effects of anisotropy on \(T_{ij}\) is dictated by the relative strengths of turbulent velocity fluctuation \(u^{\prime}\) and the velocity jump due to thermal expansion, which can be characterised in terms of \(NB\sim\tau S_{L}/u^{\prime}\sim\tau/\left(Da^{1/2}Ka\right)\), whereas the anisotropy of \(\varepsilon_{ij}\) is dictated by the relative strengths of \(a_{turb}\sim u^{\prime}/l\) and \(a_{chem}\sim\tau S_{L}/\delta_{th}\), which is dictated by \(\tau Da\). This implies that a simple linear relationship between the anisotropy of \(T_{ij}\) and \(\varepsilon_{ij}\) is unlikely to be valid for premixed turbulent flames. For a high pass filter of width \(\Delta\), flow structures smaller than \(\Delta\) are retained such that the second, third and fourth row in Figures 9-11 represent increasingly small flow structures. For case C, the following behaviour is as expected: Both tensors \(T_{ij}\) and \(\varepsilon_{ij}\) tend to become more isotropic with decreasing filter width \(\Delta\). Isotropy, in particular, increases once the filter becomes smaller than the integral scale (i.e., \(\Delta/\delta_{th}=4.58\) according to Table 1). By contrast, case A shows an unexpected behaviour and the turbulent state switches from axisymmetric expansion (lower right border) to the axisymmetric contraction side (lower left border) with a decrease in \(\Delta\), but considerably anisotropy can be observed even for the smallest filter width considered here. This unexpected behaviour warrants an explanation, which is provided in the following manner. The axisymmetric contraction signifies a situation where \(T_{11}<T_{22}=T_{33}\) in the present scenario. This means that, for the smallest flow scales, the strong dilatation effects, responsible for the opposite scenario (i.e., \(T_{11}>T_{22}=T_{33}\)), are no longer present. This can be explained as follows. Figure 1 shows that flow structures corresponding to the fluctuating \(u\) velocity component are considerably larger compared to the \(v\) (or \(w\), not shown) structures. By filtering the flow field with a filter width smaller than these, \(u\) velocity structures diminish their strength considerably until they finally vanish and only the smaller velocity fluctuations of the transverse components are left. Ultimately, this indicates that not only the fluctuation strength is anisotropic for case A but also the size of the structures associated with different velocity components is different. The behaviour of Case B is somewhere in between case A and case C. It is worth remarking that the value of \(k\) decreased by roughly 50% at the time statistics were taken. As a result, Bray number \(NB\sim\tau S_{L}/u^{\prime}\) for case B increases from 0.6 to a value close to unity. This also implies that the Damkohler (Karlovitz) number \(Da\) (\(Ka\)) increases (decreases) roughly by a factor of 1.414 (0.6) in comparison to the initial values in case B. Finally, the anisotropy of the subgrid scale stress tensor is analysed in Figure 12 for filter widths \(\Delta/\delta_{th}=0.4,\ 1.4,\ 5.8\) and 11.6. In contrast to the previous results, a low pass filter is used in this context and the unfiltered result (which would result in a vanishing subgrid scale contribution) has been replaced with an even larger filter width of \(\Delta/\delta_{th}=11.6\). Only cases A and B are shown in Figure 12 because the subgrid scale stresses for case C are nearly isotropic even for the largest filter width.