## 4 Discussion and Conclusions  
A new differential subgrid stress model (DSM) for large eddy simulations of turbulent flows was developed, a priori term-by-term calibrated and validated. For this purpose, an open DNS database [24,25] was used. The least squares method was found to be inappropriate due to its tendency to underestimate the modeled field norm. To avoid this problem, a new calibration method was proposed which maximizes the correlation coefficient of a modeled field while maintaining its correct norm. Even with this method, it was found that the calibrated model may appear to be unstable, so an appropriate choice of closure functional form is important in addition to determining the coefficient values. In the present study, only the inclusion of terms which were nonlinear in strain and rotation rates allowed us to stabilize the pressure-strain model. To our knowledge, all well-known pressure-strain models such as SSG [37] are linear in their mean (resolved) velocity gradients, which follows from the exact form of \(\overline{p^{\prime}S^{\prime}_{ij}}\) in homogeneous turbulence. Thus, the new model can be considered as an extension of classical models to the inhomogeneous turbulence conditions found in the inertial range. Another result which we considered to be unexpected was the importance of the resolved velocity gradients in the triple-velocity correlation model (8). Although it is not a common practice to include such terms, this allowed us to increase the correlation coefficient from 0.3 to 0.85. The idea was inspired by [22], in which the importance of the velocity gradients in \(\overline{u^{\prime}_{i}u^{\prime}_{j}u^{\prime}_{k}}\) closure was highlighted. The observations made in the current paper may be important for the development of future advanced subgrid stress models. The model assessment was carried out in an LES of non-premixed turbulent combustion. The impact of the new subgrid stress model on the scalar fields was shown by a comparison of the DSM with the Smagorinsky, dynamic Smagorinsky and WALE models. There appeared to be no significant differences in the resolved velocity and mixture fraction fields, which may indicate that the backward influence of the subgrid scales on the resolved ones was relatively small in the considered problem. However, the new model was able to simulate backscatter in contrast to the Smagorinsky and WALE models, an effect observed in DNS. In the DSM, the backscatter relates to the misalignment between the principal axes of \(\overline{u^{\prime}_{i}u^{\prime}_{j}}\) and \(S_{ij}\). This misalignment also leads to the relative inefficiency of the turbulent energy cascade [3], which manifests itself as an increased subgrid stress level of DSM compared with the Smagorinsky and WALE models, while the energy transfer rate is approximately the same. The ability to capture such flow features may play a major role in more complex flows where backscatter is a flow-determining effect. Note that the backscatter is also captured by the dynamic Smagorinsky model, but in this case, it is caused by the presence of negative \(v_{\text{sgs}}\) values. It was shown that the probability density function of energy flux obtained with the dynamic Smagorinsky model is too wide compared with the filtered DNS data. Concerning the mixture fraction statistics, while their resolved parts behave similarly independent of a subgrid model, their subgrid parts significantly depend on the stress field. In the conducted simulations, the subgrid scalar variance was found to be two times lower with the DSM than with the Smagorinsky and WALE models. This was due to a smaller production term and a significantly larger scalar dissipation term. We believe that these differences may be important when advanced turbulence-combustion interaction models are employed. Conceptualization, V.S. and V.V. ; methodology, A.T. and R.B. ; programming, R.B., L.U. and A.T.; data analysis and interpretation, A.T., R.B. and V.S. All authors have read and agreed to the published version of the manuscript.