# 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  
In order to verify and validate the large eddy simulation method for the turbulent mixing layer, the 2D original case computation, the time step study and the grid resolution investigation are carried out in this paper. Then the results are compared with those of experiments and computations in literatures and those of our 3D computation. The parameters of the original case are described in Sections 2.2.1 and 2.2.2, where \(\alpha\) in the inlet disturbance is taken as 0.0045. Figure 1 shows the comparison of the vorticity thickness growth curve of the turbulent mixing layer with the relevant computational results in reference [56] (Figure 1 includes two kinds of inlet cases without forcing and with forcing, single precision and double precision, sp represents single precision, dp represents double precision). Table 1 shows the comparison of Reynolds stress and turbulent mixing layer growth rate with the experimental and computational results in references [56; 57; 59; 60; 61; 62]. It can be seen that these results match well. In order to study the suitability of the time step, a new time step computational case is carried out. The new time step size of \(5\times 10^{-8}\) s is half of the original time step size of \(1\times 10^{-7}\) s. The computational settings except for the time step are consistent with the original calculation. The computational results are shown in Table 2. It can be seen that the results of the new time step are very close to the original one, indicating that the time step of \(1\times 10^{-7}\) s is suitable. Since the cross-wise size of the computational domain of the original case is too large, it can be optimized. The cross-wise range of the optimized computational domain is \(-25\delta_{\omega}(0)\)-\(25\delta_{\omega}(0)\). Firstly, the corresponding optimized grid is generated according to the grid point distribution method of the original grid. Then, the \(x\)-direction is refined by 2 times, \(y\)-direction is refined by 2 times, \(x\), \(y\)-direction is refined by 1.414 times, \(x\) and \(y\)-direction is refined by 2 times at the same time to generate four sets of refined grids. The computational settings of the new grid are consistent with those of the original one. The computational results are shown in Table 3. It can be seen that both \(x\) and \(y\) direction mesh have a great influence on the results. From 1.414-fold to 2-fold, the results converge gradually. Especially, the Reynolds stress in the downstream direction is very close to the previous DNS results. In order to further verify the above 2D computations, 3D studies are also conducted in this paper. The 3D computational domain extends from 0 to \(350\delta_{\omega}(0)\) in the streamwise x-direction, from \(-25\delta_{\omega}(0)\) to \(25\delta_{\omega}(0)\) in the crosswise y-direction and from 0 to \(10\delta_{\omega}(0)\) in the spanwise \(z\)-direction. The grid points in the \(x\), \(y\) and \(z\) directions are 1151, 145 and 51, respectively. The total grid points are of about 8 million (it takes about 50,000 CPU hours to finish the computation of 12 flow-through cycles). In this 3D grid, the minimum grid spacings in the \(x\), \(y\) and \(z\) directions at the mixing layer centerline are \(0.2\delta_{\omega}(0)\), \(0.165\delta_{\omega}(0)\) and \(0.2\delta_{\omega}(0)\), respectively. The 3D grid also has a buffer zone, which is the same as that in the 2D grid. The spanwise boundaries in the 3D grid are treated as periodic. The rest of the boundary conditions and the numerical schemes are consistent with the 2D cases. To generate three-dimensional disturbance, in addition to adding random disturbance in the