## 5 Conclusions  
The anisotropy of the Reynolds stress, dissipation, and subgrid scale tensor in turbulent premixed flames with a range of different turbulence intensities has been characterised using the Lumley triangle by performing a multiscale analysis. For this purpose, an existing DNS database of statistically planar turbulent premixed flames has been filtered for a large range of filter widths ranging from dimensions smaller than the thermal flame thickness to structures that are larger than the integral scale. The main findings, which are reported for the first time for turbulent premixed flames, can be summarised as follows:  
1. Although the initial background fluid motion is isotropic, there is a pronounced anisotropy towards the axisymmetric expansion for low turbulence intensities. This effect diminishes with increasing turbulence intensity.
2. In general, this anisotropy decreases with decreasing high pass filter width (i.e., when only smaller and smaller structures remain), but for case A (i.e., the case with the smallest value of \(u^{\prime}/S_{L}\)), the state of the turbulent flow flips towards the axisymmetric contraction side. This trend has been explained by the different size of flow structures associated with velocity components in the direction of mean flame propagation and transverse to it.
3. The dissipation tensor tends to be more isotropic than the Reynolds stress tensor, which is in agreement with the theoretical expectation. The behaviour is different to the turbulent boundary layers, where a much closer coupling can be observed.
4. In agreement with theoretical arguments, subgrid scale stresses are more isotropic than their averaged counterparts for all filter widths. Nevertheless, the axisymmetric expansion state is observed for low turbulence intensities.
5. No remarkable differences were observed when using Reynolds averaging versus Favre averaging procedures and definitions.
6. The application of a 2D (in direction perpendicular to flame propagation) filter results qualitatively in very similar outcomes but the anisotropies remain stronger. While the pronounced anisotropy towards axisymmetric expansion for low turbulence intensities has been observed in earlier work for Reynolds stresses [15; 16; 17], the present analysis confirms this trend for the dissipation tensor and further extends the findings to a multiscale analysis and shows several unexpected (e.g., the flipping of the turbulent state mentioned in (ii)) and interesting aspects (e.g., no difference between Reynolds and Favre-Averaging based definitions (v), or the quantification of 2D vs.  
3D filtering (vi)). However, the analysis is limited to moderate Reynolds numbers, and more analysis will be needed in future to observe if the above trends persist at higher Reynolds numbers.  
Conceptualization, M.K. and N.C.; methodology, M.K.; software, N.U. and T.T.; formal analysis, N.U.; resources, M.K.; writing--original draft preparation, M.K. and N.C.; writing--review and editing, M.K. and N.C.; visualization, N.U. and M.K. supervision, T.T. and M.K. All authors have read and agreed to the published version of the manuscript.  
Funding by dtec.bw--Digitalization and Technology Research Center of the Bundeswehr, Project MORE--is gratefully acknowledged. NC acknowledges Engineering and Physical Sciences Research Council (EP/R029369/1) for financial and computational support.  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.