## 3 Large Eddy Simulations of Isothermal Combustion  
### Problem Statement and Numerical Setup  
To study the subgrid model's influence on the mixing and combustion process, we consider a three-layer top-hat mixture fraction field, transported by a homogeneous turbulent velocity field. Four simulations were performed using the new DSM, and the Smagorinsky, dynamic Smagorinsky and WALE models. In all cases, the flow corresponds to a non-premixed turbulent combustion regime without the backward effect of heat release. Since we use a compressible solver, we solve a compressible counterpart of (1) and (2), supplemented by an energy equation (molecular transport omitted due to formal \(Re\to\infty\) limit):  
\[\frac{\partial\overline{p}}{\partial t}+\frac{\partial\overline{p}\overline{ u}_{i}}{\partial x_{i}}=0, \tag{22}\]  
\[\frac{\partial\overline{p}\overline{u}_{i}}{\partial t}+\frac{\partial}{ \partial x_{k}}\left[\overline{p}\overline{u}_{i}\overline{u}_{k}+\overline{ p}\delta_{ik}+\overline{p}\overline{u^{\prime}_{i}u^{\prime}_{k}}\right]=0, \tag{23}\]  
\[\frac{\partial\overline{p}\overline{E}}{\partial t}+\frac{\partial}{ \partial x_{k}}\left[\overline{p}\overline{E}\overline{u}_{k}+\overline{p} \overline{u^{\prime}_{i}u^{\prime}_{k}}\overline{u}_{i}+\frac{\overline{p} \overline{u^{\prime}_{i}u^{\prime}_{i}u^{\prime}_{k}}}{2}\right]=0. \tag{24}\]  
This system is closed either by six equations obtained by introducing density into (3), or by algebraic relation (18). The expressions for total energy and state equation are as follows:  
\[\overline{E}=\frac{\overline{u}_{i}\overline{u}_{i}}{2}+\frac{R\overline{T}}{ \kappa-1},\ \ \ \ \ \overline{p}=\overline{p}R\overline{T}, \tag{25}\]  
where \(R\approx 287\,\mathrm{J}/\left(\mathrm{kg}\cdot\mathrm{K}\right)\), \(\kappa=1.4\). To minimize the compressibility effects, we make sure that the resolved field Mach number, \(M=\left(\overline{\left(\overline{u}_{i}\overline{u}_{i}\right)}^{1/2}\right)/ \left(\kappa R(T)\right)^{1/2}\), stays less than \(0.1\) during the simulations (angle brackets denote averaging over the whole computational domain). Accordingly, we neglect the moments involving density fluctuations in (22)-(25). The equation system is solved numerically using multiblock finite-volume in-house code _zFlare_[8] with a purely central second-order kinetic energy-preserving scheme [29] and an explicit three-step Heun time-integration method [30]. The time step, \(\Delta t\), satisfies the CFL condition, \(\left(\overline{u}_{i}+\left(\kappa R(T)\right)^{1/2}\right)\Delta t/H\leq 1\), \(i=1,\ 2,\ 3\), at each cell. The computational domain is a triply periodic box, \(\left[-L/2,\ L/2\right]^{3}\), \(L=2\pi\) m. Initially, we fill the domain with a realization of the velocity field, according to [31]. The velocity field is represented as a sum of transverse harmonic waves with von Karman energy spectrum. The initial integral length scale, as determined by the first zero of the transverse correlation function, is chosen to be \(L_{0}=L/4\), and the initial resolved energy is \(E_{k0}=\overline{(u^{\prime}_{i}u^{\prime}_{j})}/2=1000\) m\({}^{2}\)/s\({}^{2}\), which leads to initial large eddy turnover time scale of \(T_{0}=L_{0}/E_{k0}^{1/2}=0.157\) s. For DSM, the initial uniform isotropic subgrid stress field is specified, \(\overline{u^{\prime}_{i}u^{\prime}_{i}}=2k_{0}\delta_{ij}/3\), where \(k_{0}=150\) m\({}^{2}\)/s\({}^{2}\). At the first stage of the simulation, the initial field is freely decayed, transforming into a developed turbulent field until the resolved energy drops to \(E_{k1}=100\) m\({}^{2}\)/s\({}^{2}\). At the second stage, we impose a three-layer mixture fraction field, as follows:  
\[\overline{z}= \begin{cases}1,&-L/6\leq x\leq L/6,\\ 0,&\text{otherwise},\end{cases} \ \ \ \ \ \ \ \overline{z^{\prime 2}}=10^{-9},\]where \(\overline{z}\) is a spatially filtered mixture fraction and \(\overline{z^{\prime 2}}\) its subgrid-scale variance.