## 2 Numerical Methodology and DNS Database  
Three turbulent, statistically planar premixed flames with global Lewis number \(Le=1.0\), representing stoichiometric methane-air flames preheated to 415 K, have been chosen from a larger database that has been described in [19; 20]. The compressible Navier-Stokes equations have been solved in nondimensional form (see, e.g., [21]) using the well-known SENGA code [22]. As this work focuses on the fluid dynamical aspects of reacting flows, a generic single step Arrhenius type irreversible chemistry has been employed, which provides the same qualitative and very similar quantitative behaviour of flame turbulence interaction [23] compared to a detailed chemical mechanism, in particular, with respect to flow anisotropy [15; 16]. The turbulence Reynolds number \(Re_{t}\), normalised turbulent root-mean-square (rms) velocity fluctuation \(u^{\prime}/S_{L}\), integral length scale to thermal flame thickness ratio \(l/\delta_{th}\), Damkohler number \(Da\) and Karlovitz number \(Ka\) for cases A-C are shown in Table 1. The definitions of these quantities are given as follows. \[Da=\frac{IS_{L}}{\delta_{th}u^{\prime}}Ka=\bigg{(}\frac{u^{\prime}}{S_{L}} \bigg{)}^{\frac{3}{2}}\bigg{(}\frac{l}{\delta_{th}}\bigg{)}^{-\frac{1}{2}}\, \delta_{th}=\frac{T_{ad}-T_{0}}{\text{max}|\
abla T|_{L}}\,\tau=\frac{T_{ad}-T _{0}}{T_{0}}\,\,\beta=\frac{T_{ac}(T_{ad}-T_{0})}{T_{ad}^{2}} \tag{2}\]  
Here, \(\delta_{th}\) is the thermal flame thickness, and \(S_{L}\) (with the subscript \(L\) referring to unstrained laminar flame quantities) is the laminar burning velocity. The heat release parameter \(\tau\) (with the adiabatic and fresh gas temperatures \(T_{ad},T_{0}\)) and the Zel'dovich number \(\beta\) (\(T_{ac}\) is the activation temperature) are 4.5 and 6.0, respectively, for the stoichiometric methane-air flames preheated to 415 K. Standard values of Prandtl number (\(Pr=0.7\)) and ratio of specific heats (\(\gamma=1.4\)) have been used, which are consistent with unity Lewis number assumption. Flame-turbulence interaction takes place under decaying turbulence, and the values reported in Table 1 have to be understood as initial values. The simulation time is taken to be the chemical time scale \(t_{chem}=\delta_{th}/S_{L}\), which is in all cases larger than the eddy turnover time \(t_{EDT}=l/u^{\prime}\). Advantages and disadvantages of this particular setup have been discussed in detail in [24], and it has been checked that the results are qualitatively similar to those obtained from a database with unburned gas forcing [25], which are not explicitly shown here for the sake of brevity. While the decaying turbulence setup potentially has a history effect, it is not important for the qualitative nature of the results presented in this work. This approach ensures that flame development is mostly natural and possible artificial effects due to the forcing term in the Navier-Stokes equations can be avoided. While the reacting flow and species fields are initialised by a steady planar unstrained premixed laminar flame solution, turbulent velocity fluctuations are initialised using a homogeneous isotropic incompressible velocity field in conjunction with a model spectrum suggested by [PERSON] [2]. The simulation domain is taken to be a cube with side length \(26.1\delta_{th}\), which is discretised using a uniform Cartesian grid of dimension \(512^{3}\). This ensures sufficient resolution of the flame structure (11 grid points are kept within \(\delta_{th}\)) and the smallest scales in the turbulent flow. It has been found that coarsening the mesh by factor of 2.0 did not make any significant influence on the values of \(S_{L}\) and \(\delta_{th}\) (<1% change); thus, the grid spacing considered here deemed appropriate for the current analysis. The timestep has been determined by the acoustic CFL criterion and it remained at about 0.1 for all computations reported in this paper.