## 2 Subgrid Stress Tensor Equation Closure and Calibration
### Calibration Results
Compared with the constant coefficient model, the correlation coefficient increased, on average, by 1%; however, the extended model turned out to produce stable solutions. After averaging the data in Table 3, we formulated the following model:  
\[ \begin{split}\mathbf{\Phi}=-C_{\Phi 1}\mathbf{a}e+C_{\Phi 2}^{(0)}k\mathbf{S}+C_{ \Phi 2}^{(1)}\sqrt{k}\Delta(\mathbf{S}\mathbf{\Omega}-\mathbf{\Omega}\mathbf{S})+C_{\Phi 2}^{(2)} \sqrt{k}\Delta\Big{(}\mathbf{\Omega}^{2}-\frac{1}{3}\mathbf{\mathrm{tr}}\big{(}\mathbf{ \Omega}^{2}\big{)}\mathbf{\mathrm{I}}\Big{)}\\ +C_{\Phi 2}^{(3)}\Delta^{2}\big{(}\mathbf{S}\mathbf{\Omega}\mathbf{\Omega}+\mathbf{ \Omega}\mathbf{\Omega}\mathbf{S}-\frac{2}{3}\mathbf{\mathrm{tr}}(\mathbf{S}\mathbf{\Omega}\mathbf{ \Omega})\mathbf{\mathrm{I}}-\mathbf{\mathrm{tr}}\big{(}\mathbf{\Omega}^{2}\big{)}\mathbf{S} \big{)}+C_{\Phi 3}k\big{(}\mathbf{aS}+\mathbf{Sa}-\frac{2}{3}\mathbf{\mathrm{tr}}(\mathbf{aS})\mathbf{ \mathrm{I}}\big{)}+C_{\Phi 4}k(\mathbf{\Omega a}-\mathbf{a}\mathbf{\Omega}),\\ C_{\Phi 1}=3.0,\ \ \ C_{\Phi 2}^{(0)}=1.04,\ \ \ C_{\Phi 2}^{(1)}=-0.136,\ \ \ C_{ \Phi 2}^{(2)}=-0.058,\ \ \ C_{\Phi 2}^{(3)}=-0.23,\ \ \ C_{\Phi 3}=0.34,\ \ \ C_{\Phi 4}=1.2. \end{split}  \tag{14}\]  
Consider now a triple-velocity correlation model (8). Again, starting with constant coefficients, \(C_{T1},\dots,C_{T3}\), we obtained the results collected in Table 4. Interestingly, the corresponding term norms relate as follows:  
\[\parallel C_{T1}k^{1/2}\Delta\sum_{\{i\to j\to k\}}\frac{\partial u_{i}^{ \prime}u_{j}^{\prime}}{\partial x_{k}}\parallel_{2}:\parallel C_{T2}k^{1/2} \Delta\sum_{\{i\to j\to k\}}a_{kl}\frac{\partial\overline{u_{i}^{\prime}u_{ j}^{\prime}}}{\partial x_{l}}\parallel_{2}:\parallel C_{T3}\Delta^{2}\sum_{\{i\to j \to k\}}\frac{\partial\overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{l} }\parallel_{2}=6:1:53. \tag{15}\]  
The second term in the model is clearly negligible, so we omitted it. The recalibration of the remaining coefficients is presented in Table 5. Fortunately, the high correlation coefficient was not affected by the simplification. An attempt to make transport coefficients dependent on \(A_{2}\) and \(A_{3}\) was unsuccessful, giving no significant improvements. The final model reads as follows:  
\[ \begin{split}\overline{u_{i}^{\prime}u_{j}^{\prime}u_{k}^{\prime }}=\sum_{\{i\to j\to k\}}\Big{[}-C_{T1}k^{1/2}\Delta\delta_{kl}+C_{T3}\Delta^{2 }\frac{\partial\overline{u_{i}}}{\partial x_{l}}\Big{]}\frac{\partial \overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{l}},\\ C_{T1}=0.019,\ \ \ C_{T3}=0.064.\end{split}  \tag{16}\]  
Making the model coefficients dependent on \(A_{2}\) and \(A_{3}\) allowed us to increase Corr only by 0.2-0.3%, which is impractical. Note that, according to (15), the leading term in the model (16) is the one containing the resolved velocity gradients. More traditional models, which do not contain explicit dependence on velocity gradients, perform significantly worse. Our tests show that a correlation coefficient for a model with only \(C_{T1}\) non-zero has a Corr in the range 0.27-0.33, depending on \(\Delta\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Filter Size, \(\Delta/\mathbf{h}\)** & \(C_{T1}\) & \(C_{T3}\) & **Correlation Coefficient** \\ \hline
64 & 0.0178 & 0.0084 & 0.0615 & 0.869 \\
128 & 0.0176 & 0.0042 & 0.0617 & 0.868 \\
256 & 0.0207 & 0.0094 & 0.0665 & 0.844 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients in the constant coefficient model (8). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Filter Size, \(\Delta/\mathbf{h}\)** & \(C_{\Phi 1}\) & \(C_{\Phi 2}^{(0)}\) & \(C_{\Phi 2}^{(1)}\) & \(C_{\Phi 2}^{(2)}\) & \(C_{\Phi 2}^{(3)}\) & \(C_{\Phi 3}\) & \(C_{\Phi 4}\) & **Correlation Coefficient** \\ \hline
64 & 3.305 & 1.060 & –0.134 & –0.044 & –0.269 & 0.356 & 1.335 & 0.604 \\
128 & 3.046 & 1.030 & –0.137 & –0.061 & –0.233 & 0.359 & 1.202 & 0.684 \\
256 & 2.740 & 1.029 & –0.137 & –0.069 & –0.191 & 0.313 & 1.014 & 0.760 \\ \hline \hline \end{tabular}
\end{table}