_applied sciences_  
_Article_  
**Multiscale Analysis of Anisotropy of Reynolds Stresses, Subgrid Stresses and Dissipation in Statistically Planar Turbulent Premixed Flames**  
**[PERSON] 1\({}^{\copyright}\)**, **[PERSON] 1\({}^{\copyright}\)**, **[PERSON] 1 and [PERSON] 2\({}^{,}\)**\({}^{\copyright}\)**  
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\[a_{ij}=\frac{\langle u^{\prime}_{i}u^{\prime}_{j}\rangle}{2k}-\frac{1}{3}\delta_{ ij}\quad k=\frac{1}{2}\langle u^{\prime}_{i}u^{\prime}_{j}\rangle \tag{1}\]  
where \(k\) is the turbulence kinetic energy. It is noted that the anisotropy of the dissipation tensor \(e_{ij}\) can be defined by the same analogy. Instead of an isotropic relation \(\varepsilon_{ij}=\varepsilon\delta_{ij}/3\), a linear relation between the dissipation tensor and the Reynolds stress anisotropy has been suggested by [PERSON] and [PERSON] [3]. [PERSON] et al. [4] suggested a linear relationship between the dissipation tensor and the anisotropy tensor with a Reynolds number dependent constant of proportionality. By contrast, [PERSON] and [PERSON] [5] proposed an anisotropic model, which approximates anisotropy tensor \(a_{ij}\) by a normalised turbulent dissipation tensor. It becomes clear from the foregoing discussion that the relationship between Reynolds stress tensor and dissipation tensor plays a fundamental role in RANS based turbulence modelling, a topic reviewed in the 1990s by [PERSON] [6] and [PERSON] [7] or more recently in a variety of textbooks [2; 8; 9; 10].  
Apart from the modelling aspect, the physics of small-scale turbulence, which is strongly associated with the mechanism of dissipation, has been an active area of research for several decades (e.g., [PERSON] and [PERSON] [11]). [PERSON] and [PERSON] [12] reported, for turbulent shear flows up to a Taylor scale Reynolds number of \(Re_{\lambda}=1000\), that the postulate of local isotropy is untenable, both at dissipation and inertial scales, and they suggested that it is unlikely to be so even at higher Reynolds numbers. The multiscale behaviour of anisotropy occurring in turbulent boundary layers has been analysed by [PERSON] and [PERSON] [5], who reported that anisotropy does not decay as scales decrease.  
In turbulent premixed combustion, the situation can become even more complicated due to the anisotropic nature of heat release. This is closely related to the phenomenon of counter-gradient transport, which has been theoretically explained by [PERSON] and [PERSON] [13] and [PERSON] [14] and has been observed in many experimental and numerical studies, as reviewed in [PERSON] et al. [15; 16] and [PERSON] et al. [17], where combustion-induced flow anisotropy has been characterised with the help of the so-called Lumley triangle [18]. However, the relation between Reynolds stress tensor and dissipation tensor anisotropies has, to the best knowledge of the authors, not been explored in the context of turbulent premixed combustion. As the landscape of Computational Fluid Dynamics (CFD) methods is full of different modeling approaches ranging from classical RANS models to hybrid RANS-LES to LES methods, it will be of particular interest to perform a multiscale analysis of turbulence anisotropies which reflects the nature of the associated different unclosed terms. In this regard, the main objectives of this study are the following: (i) perform a multiscale analysis of the anisotropies of Reynolds stress, dissipation and subgrid scale tensors for a large range of different filter sizes, (ii) to compare their level of anisotropy for different turbulence intensities and (iii) to provide detailed physical explanations for the observed behaviour.  
The rest of the paper is organised as follows: Section 2 introduces the database and the numerical methods and Section 3 provides the mathematical framework for the subsequent analysis. Results will be presented in Section 4 and the main findings will be summarised in Conclusions.