# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 2 Large Eddy Simulation of Turbulent Mixing Layer and Dynamic Mode Decomposition
### Large Eddy Simulation Method
#### 2.2.4 Verification and Validation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(Re_{\delta_{\omega}(0)}\) & \(\frac{\sigma_{\omega}}{\Delta U}\) & \(\frac{\sigma_{\omega}}{\Delta U}\) & \(\frac{\sigma_{\omega}}{\Delta U}\) & \(\frac{1}{\eta}\frac{d\delta_{\omega}(0)}{d\chi}\) & **Reference** \\ \hline - & 0.031 & 0.019 & 0.0225 & 0.009 & 0.19 & [PERSON] and [PERSON] [59] \\ - & 0.036 & 0.014 & 0.0225 & 0.013 & 0.16 & [PERSON] and [PERSON] [60] \\
1800 & 0.032 & 0.020 & 0.022 & 0.010 & 0.163 & [PERSON] and [PERSON] [61] \\
720 & 0.040 & 0.084 & - & 0.023 & 0.15 & 2-D DNS: [PERSON] and [PERSON] [62] \\
720 & 0.048 & 0.078 & - & 0.012 & 0.15 & 2-D DNS: [PERSON] [57] \\
720 & 0.044 & 0.080 & - & 0.015 & 0.17 & 2-D LES: [PERSON] et al. [56] \\
720 & 0.044 & 0.080 & - & 0.022 & 0.164 & Present 2-D LES single precision \\
720 & 0.053 & 0.076 & - & 0.024 & 0.173 & Present 2-D LES double precision \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the results of the original case with the results of previous experiments and computations. Figure 1: Vorticity thickness growth rate of turbulent mixing layer in the original case. y-direction, we also set the velocity fluctuation algorithm of the spectral synthesizer [58] at the inlet. Table 4 demonstrates the 3D and 2D computational results. In this study, the 3D results have been found to be close to the 2D ones, which is consistent with the observations in [PERSON] et al. [56]. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Original (2-D LES)** & **NewTS (2-D LES)** & **Uzun [57] (2-D DNS)** \\ \hline \(\Delta t(\mathrm{s})\) & \(1\times 10^{-7}\) & \(5\times 10^{-8}\) & \(6\times 10^{-8}\) \\ Sampling time (FTCs) & 8 & 8 & 6 \\ Sampling frequency(\(\Delta t\)) & 1 & 1 & 1 \\ Number of samples & 28,000 & 56,000 & 35,000 \\ Growth rate, \(\delta\delta_{\omega}(x)/dx\) & 0.055 & 0.055 & 0.050 \\ \(\sigma_{xx}/\Delta U^{2}\) at \(x=708\omega_{\omega}(x)\) & 0.002 & 0.002 & 0.013 \\ \(\sigma_{xx}/\Delta U^{2}\) at \(x=100\omega_{\omega}(x)\) & 0.010 & 0.008 & 0.034 \\ \(\sigma_{xx}/\Delta U^{2}\) at \(x=130\omega_{\omega}(x)\) & 0.023 & 0.020 & 0.042 \\ \(\sigma_{xx}/\Delta U^{2}\) at \(x=160\omega_{\omega}(x)\) & 0.043 & 0.042 & 0.0490 \\ \(\sigma_{xx}/\Delta U^{2}\) at \(x=190\omega_{\omega}(x)\) & 0.049 & 0.042 & 0.046 \\ \(\sigma_{yy}/\Delta U^{2}\) at \(x=70\delta_{\omega}(x)\) & 0.002 & 0.001 & 0.027 \\ \(\sigma_{yy}/\Delta U^{2}\) at \(x=100\delta_{\omega}(x)\) & 0.032 & 0.023 & 0.051 \\ \(\sigma_{yy}/\Delta U^{2}\) at \(x=130\delta_{\omega}(x)\) & 0.047 & 0.047 & 0.071 \\ \(\sigma_{yy}/\Delta U^{2}\) at \(x=160\delta_{\omega}(x)\) & 0.057 & 0.053 & 0.078 \\ \(\sigma_{yy}/\Delta U^{2}\) at \(x=190\delta_{\omega}(x)\) & 0.069 & 0.076 & 0.080 \\ \(|\sigma_{xy}|/\Delta U^{2}\) at \(x=70\delta_{\omega}(x)\) & 0.001 & 0.001 & 0.008 \\ \(|\sigma_{xy}|/\Delta U^{2}\) at \(x=100\omega_{\omega}(x)\) & 0.009 & 0.009 & 0.015 \\ \(|\sigma_{xy}|/\Delta U^{2}\) at \(x=130\delta_{\omega}(x)\) & 0.007 & 0.006 & 0.014 \\ \(|\sigma_{xy}|/\Delta U^{2}\) at \(x=160\delta_{\omega}(x)\) & 0.020 & 0.021 & 0.009 \\ \(|\sigma_{xy}|/\Delta U^{2}\) at \(x=190\delta_{\omega}(x)\) & 0.012 & 0.015 & 0.011 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of different time step computational results with previous DNS results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Case** & \ \begin{tabular}{c} **Optimized:** \\ \(\triangle\mathbf{x}\), \(\triangle\mathbf{y}\) \\ \end{tabular}  & \(\triangle\mathbf{x}\), \(\triangle\mathbf{y}\) & \(\triangle\mathbf{x}\), \(\triangle\mathbf{y}\)/2 & \(\triangle\mathbf{x}\)/2, \(\triangle\mathbf{y}\) & \(\triangle\mathbf{y}\)/1.414 & \(\triangle\mathbf{x}\)/2, \(\triangle\mathbf{y}\)/2 & \begin{tabular}{c} **Uzun [57]** \\ **(2-D DNS)** \\ \end{tabular}