## 2 Subgrid Stress Tensor Equation Closure and Calibration
### Calibration Results  
At the first step, we calibrated the dissipation rate model of the form (4) and (5). In Table 1, the calibrated coefficients are presented for different filter sizes together with their correlation coefficients. The model correlates well with an exact \(\varepsilon\) field, yielding Corr of about 0.71-0.86. After averaging the coefficient values for different filter sizes and keeping two decimal digits, we propose the following dissipation model for use:  
\[ \begin{split}\varepsilon&=\max\Bigl{\{}C_{E}^{(0)}+ C_{E}^{(1)}A_{2}+C_{E}^{(2)}A_{3},\ C_{E}^{lim}\Bigr{\}}\tfrac{\xi^{3/2}}{ \Delta},\\ C_{E}^{(0)}&=0.8,\ \ C_{E}^{(1)}=-1.4,\ \ C_{E}^{(2)}=2.1,\ \ C_{E}^{lim}=0.21.\end{split}  \tag{12}\] \begin{table}
\begin{tabular}{c c c c c} \hline Filter Size, \(\Delta/h\) & \(C_{E}^{(0)}\) & \(C_{E}^{(1)}\) & \(C_{E}^{(2)}\) & Correlation Coefficient \\ \hline
64 & 0.823 & –1.308 & 1.848 & 0.861 \\
128 & 0.804 & –1.380 & 2.103 & 0.808 \\
256 & 0.783 & –1.421 & 2.305 & 0.717 \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients in the dissipation rate model (4) and (5). Figure 2: Iteration process scheme: correlation coefficient contour and \(\parallel\widetilde{\mathbf{b}}\parallel_{1}=\parallel\mathbf{b}\parallel_{1}\) isoline are shown. Note that the inclusion of \(A_{2}\) and \(A_{3}\) is justified because the correlation coefficient of a constant coefficient model (with only \(C_{E}^{(0)}\) non-zero) is approximately 0.11 lower. The maximum function is introduced to avoid possible undershoots of the dissipation rate in strongly anisotropic flow regions. We selected the value of \(C_{E}^{lim}\) by referring to the probability density function of \"exact\" \(C_{E}=\varepsilon\Delta/k^{3/2}\) distribution, as calculated from DNS data (see Figure 3). \(C_{E}\) values lower than \(C_{E}^{lim}=0.21\) are observed, with less than 1% probability. At the second step, we investigated the constant coefficient model (7). The results are shown in Table 2.  
Averaged coefficient values are \(C_{\phi 1}=1.8\), \(C_{\phi 2}=1.3\), \(C_{\phi 3}=0.39\) and \(C_{\phi 4}=0.69\). Unfortunately, the preliminary simulations using this model showed that it is unstable due to the value of \(C_{\phi 2}\) being too high. Our attempt to introduce the dependence of \(C_{\phi 1},\ldots,C_{\phi 4}\) on \(A_{2}\) and \(A_{3}\) did not help. The model remained unstable, while its correlation coefficient increased only by 0.2%. After a series of attempts, we found a workable solution, which was to replace the \(C_{\phi 2}k\mathbf{S}\) term in the model with the following:  
\[ \begin{array}{c}C_{\phi 2}^{(0)}k\mathbf{S}+C_{\phi 2}^{(1)}\sqrt{k} \Delta(\mathbf{S}\mathbf{\Omega}-\mathbf{\Omega}\mathbf{S})+C_{\phi 2}^{(2)} \sqrt{k}\Delta\Big{(}\mathbf{\Omega}^{2}-\frac{1}{3}\mathrm{tr}(\mathbf{ \Omega}^{2})\mathbf{I}\Big{)}\\ +C_{\phi 2}^{(3)}\Delta^{2}\big{(}\mathbf{S}\mathbf{\Omega}\mathbf{\Omega}+ \mathbf{\Omega}\mathbf{\Omega}\mathbf{S}-\frac{2}{3}\mathrm{tr}(\mathbf{S} \mathbf{\Omega}\mathbf{\Omega})\mathbf{I}-\mathrm{tr}\big{(}\mathbf{\Omega}^{ 2})\mathbf{S}\big{)}.\end{array}  \tag{13}\]  
The introduced nonlinear combinations of \(\mathbf{S}\) and \(\mathbf{\Omega}\) are widely used in the framework of EARSM [26]. The values of model coefficients obtained during the calibration are collected in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Filter Size, \(\Delta\)/\(h\)** & \(C_{\phi 1}\) & \(C_{\phi 2}\) & \(C_{\phi 3}\) & \(C_{\phi 4}\) & **Correlation Coefficient** \\ \hline
64 & 2.020 & 1.364 & 0.449 & 0.791 & 0.590 \\
128 & 1.850 & 1.283 & 0.384 & 0.681 & 0.664 \\
256 & 1.630 & 1.222 & 0.324 & 0.587 & 0.747 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients in the constant coefficient model (7). Figure 3: \(C_{E}\) probability density function for \(\Delta=256h\) The region on the left corresponding to 1% of total probability is shown in red.