## 2 Numerical Methodology and DNS Database
The quantities \(\langle Q\rangle_{R}\) and \(\langle Q\rangle_{P}\) refer the averaged/filtered values of \(Q\) conditioned upon reactants and products, respectively. The first term on the right-hand side of Equation (3) accounts for the effects of thermal expansion arising from heat release, whereas the second and third terms arise due to non-reacting turbulence effects and the last term on the right-hand side of Equation (3) originates from the interior of the flame. The last term on the right-hand side of Equation (3) remains small in magnitude for \(Da\gg 1\) under which the PDF of \(c\) can be considered to be bimodal. Whether the PDF of \(c\) is bimodal for the cases considered here is not relevant for the discussion in this paper, and Equation (3) provides important physical insights into the qualitative behaviour of \(\langle\rho u_{i}^{{}^{\prime\prime}}u_{j}^{{}^{\prime\prime}}\rangle\) irrespective of the validity of the bimodality of the \(c\)-PDF. [PERSON] et al. [29] demonstrated that slip velocity \([\langle u_{i}\rangle_{P}-\langle u_{i}\rangle_{R}]\) can be expressed as follows:  
\[[\langle u_{i}\rangle_{P}-\langle u_{i}\rangle_{R}]=-\big{(}\alpha_{E}u^{ \prime}+\tau S_{L}\big{)}M_{i} \tag{4}\]  
where \(M_{i}=-(\tilde{\alpha}\tilde{c}/\partial x_{i})/|\
abla\tilde{c}|\) is \(i^{th}\) component of the normal vector based on the flame brush and \(\alpha_{E}\) is a model parameter of the order of unity. Based on Equation (4), [PERSON] et al. [28] defined a nondimensional parameter known as the Bray number \(NB\sim\tau S_{L}/u^{\prime}\sim\tau/\big{(}Da^{1/2}Ka\big{)}\) to decide if the velocity jump across the flame brush due to heat release is greater (\(NB>1\)) or smaller (\(NB<1\)) than the turbulent velocity fluctuations. For cases A-C, the Bray number is given by \(4.5,\ 0.6\) and \(0.3\), respectively, based on the initial conditions, which shows that the Reynolds stresses in case A are likely to be dominated by effects of heat release, whereas cases B and C are increasingly influenced by turbulent velocity fluctuations. By contrast, the large-scale strain rate \(a_{turb}\) in the context of RANS can be scaled as \(u^{\prime}/l\)[30], whereas the strain rate induced by thermal expansion \(a_{chem}\) can be scaled as \(\tau S_{L}/\delta_{th}\)[31]. Therefore, the ratio of the strain rates induced by thermal expansion due to heat release to the large-scale turbulent strain rate can be scaled as \(a_{chem}/a_{turb}\sim\tau Da\)[31]. The values of \(Da\) in Table 1 reveal that the influence of \(a_{chem}\) is likely to progressively weaken in comparison to the effects of \(a_{turb}\) from case A to case C. Thus, the statistics of \(\varepsilon_{ij}\) in case A are likely to be strongly influenced by heat release, whereas these statistics are expected to be influenced by turbulent straining in case C. The relative strength of thermal expansion in comparison to the background flow turbulence will affect the flow anisotropies as discussed in Section 4. Figure 1: Top: Instantaneous distribution of reaction progress variable with superimposed isocontours corresponding to \(c=0.1,\ 0.5,\ 0.9\). The white line shows the position of the mean flame brush corresponding to \(\overline{z}=0.5\), which demarks the position of the \(y\)-\(z\) plane. Middle and bottom: Instantaneous, normalised velocity fluctuations in direction aligned with \((u^{\prime}/S_{L})\) and normal to \((v^{\prime}/S_{L})\) mean flame propagation direction. Results are shown for case A in the \(x\)-\(y\) (**left**) and \(y\)-\(z\) plane (**right**). The flame propagates from right to left along the negative \(x\)-direction. The field of view represents the computational domain of dimension \(26.1\delta_{th}\times 26.1\delta_{th}\). Figure 2: Top: Instantaneous distribution of reaction progress variable with superimposed isocontours corresponding to \(c=0.1,\ 0.5,\ 0.9\). The white line shows the position of the mean flame brush corresponding to \(\overline{c}=0.5\), which demarks the position of the \(y\)-\(z\) plane.