## 3 Mathematical Background  
The turbulent state and its anisotropy can be analysed in the so-called Lumley triangle [18]. The boundaries of the triangle are defined in terms of the second \(II_{a}\) and third \(III_{a}\) invariant of the tensor \(a_{ij}\) (and by analogy for \(e_{ij}\)) after introducing variables \(\eta\) and \(\xi\) in the following manner. \[II_{a}=\frac{trace(a)^{2}-trace\big{(}a^{2}\big{)}}{2},\;III_{a}=\det(a),\;\eta =\left(-\frac{1}{3}II_{a}\right)^{1/2},\;\xi=\left(\frac{1}{2}III_{a}\right)^{ 1/3} \tag{5}\]  
Two borders are given by the straight lines connecting the origin (0,0) with the points (\(-1/6,1/6\)) and (\(1/3,1/3\)), respectively, while \(\eta=\left(1/27+2\varsigma^{3}\right)^{1/2}\) represents the third, curved border. Any physically realizable state of the anisotropy tensors has to lie within this triangle and the borders, sketched in Figure 4a, represent an axisymmetric contraction (left border), axisymmetric expansion (right border) and the two-component state, respectively, while the origin demarks the isotropic state [2]. Coordinates \(\xi\) and \(\eta\) are nondimensional quantities; thus, the Lumley triangle provides a general framework for the parameterization of anisotropy. For the purpose of this multiscale analysis, DNS data have been explicitly filtered by using a Gaussian filter kernel \(G(r)\) such that the filtered values of a quantity \(Q\) can be expressed as follows. \[\overline{Q(x)}=\int Q(x-r)G(r)dr,\;G(r)=\left(6/\pi\Delta^{2}\right)^{3/2}\exp \left(-6\;r\cdot r/\Delta^{2}\right) \tag{6}\]  
The application of this low pass filter removes the high wavenumber content of the fluctuating velocity signal (refer to Figure 4b) and an associated high pass filter (which removes the low wavenumber content) can be defined by \(\underline{Q}=Q-\overline{Q}\). In this paper, results will be presented from \(\Delta/\delta_{th}=0.4\), where the flame is partially resolved, up to \(\Delta/\delta_{th}=5.8\), where the flame becomes fully unresolved and \(\Delta\) becomes larger than the integral length scale. While the Reynolds stress tensor is usually defined as \(T_{ij}=\langle u^{\prime}_{i}u^{\prime}_{j}\rangle\), Favre filtering is often applied in variable-density flows. The Favre-averaged Reynolds stress is defined as \(T^{F}_{ij}=\langle\rho u^{\prime}_{i}\,u^{\prime}_{j}\rangle/\langle\rho\rangle\). Similarly, the Reynolds-averaged dissipation tensor can be defined as \(\varepsilon_{ij}=2\langle\
u\;\partial u^{\prime}_{i}/\partial x_{k}\;\partial u ^{\prime}_{j}/\partial x_{k}\rangle\) or in the context of Favre averaging as \(\varepsilon^{F}_{ij}=\langle t_{ij}s^{\prime\prime}_{ij}\rangle/\langle\rho\rangle\), where \(s^{\prime\prime}_{ij}\) is the fluctuating strain tensor and \(t_{ij}=2\rho vs^{\prime\prime}_{ij}-(2\rho v/3)\delta_{ij}s^{\prime\prime}_{ kk}\) is the usual constitutive  
Figure 4: (**a**) Sketch of the Lumley triangle with its borders; (**b**) sketch of turbulence kinetic energy spectrum of velocity before (solid line), after applying a low pass filter which results in \(\overline{u}\) (dash-dotted line) and after applying a high pass filter which results in \(\underline{u}\) (dashed line). The axisymmetric contraction border (\(*\)) signifies a situation where \(\langle u^{\prime}_{1}u^{\prime}_{1}\rangle<\langle u^{\prime}_{2}u^{\prime}_{2 }\rangle=\langle u^{\prime}_{3}u^{\prime}_{3}\rangle\), while the axisymmetric expansion (\(**\)) signifies a situation where \(\langle u^{\prime}_{1}u^{\prime}_{1}\rangle>\langle u^{\prime}_{2}u^{\prime}_{2 }\rangle=\langle u^{\prime}_{3}u^{\prime}_{3}\rangle\). stress-strain relation [32]. Figure 5 shows that there is nearly no difference in the Lumley triangles of the Reynolds-averaged versus Favre-averaged Reynolds stress and dissipation tensors. Henceforth, the Favre-averaged version will be shown, and for the sake of simplicity, it will be denoted \(T_{ij}\) and \(\varepsilon_{ij}\), respectively.