# Numerical Simulation of Non-Isothermal Mixing Flow Characteristics with ELES Method
## 3 Simulations with the Numerical Method ELES  
### Meshing and Independence Validations for Grid  
The different mesh requirements used in simulation may influence the computational accuracy and the computational time. Therefore, a grid independence check was performed  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Treatment near the Wall** & **Treatment in the Free Shear Layer** & **Criteria** \\ \hline Case-C & \(N_{y}=60\), \(\Delta y_{1}=0.02\), \(y^{*}\sim 1\) & \(\Delta<2.5\) & More relaxed requirements put forward by N) [PERSON] et al. [28] and Equation (21) \\ \hline Case-D & \(N_{y}=80\), \(\Delta y_{1}=0.02\), \(y^{*}\sim 1\) & \(\Delta<1.5\) & Strict requirements put forward by NJ [PERSON] et al. [28] and Equation (21) \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Numerical Settings**} \\ \hline \multirow{4}{*}{Spatial discretization} & Momentum & Bounded central differencing in LES; \\  & & 2 nd order upwind in RANS \\ \cline{2-3}  & K \& \(\Omega\) & 2 nd order upwind in RANS \\ \cline{2-3}  & Gradients & 1 st order upwind \\ \cline{2-3}  & Pressure & 2 nd order upwind \\ \hline \hline \multirow{4}{*}{Time Discretization} & Transient formulation & Bounded second order implicit \\ \cline{2-3}  & Time advancement & Iterative time advancement with \\ \cline{1-1} \cline{2-3}  & Time step interval & 0.001 (s) \\ \cline{1-1} \cline{2-3}  & Iteration every 1 time-step & 20 iterations \\ \hline \multirow{2}{*}{Models} & RANS portion & K-omega SST \\ \cline{1-1} \cline{2-3}  & LES portion & WMLES \\ \hline Precursor simulations & RANS domain & K-omega SST \\ \hline \hline \end{tabular}
\end{table}
Table 3: A summary of the current numerical simulation.  
before the numerical computation to establish an appropriate grid structure in order to maintain a balance between computational accuracy and time. Four cases using the different mesh requirements listed in Table 2 were selected for comparison under the same computational conditions.  
Additionally, two statistical variables, which were mean axial velocity \(V_{Z}\) and axial velocity fluctuation intensity at position \(Z=-0.53\)\(D_{m}\) in the W-ref case, were taken as a comparison. Both the mean velocity and the intensity were normalized by the bulk velocity in the main pipe, \(V_{m}\). Definitions are as below:  
\[\overline{V}_{z}^{*}=\frac{\overline{V}_{z}}{V_{m}} \tag{23}\]  
\[V_{z-sd}^{*}=\frac{1}{V_{m}}\sqrt{\frac{\sum_{i}\left(V_{z}^{i}-\overline{V}_{ z}\right)^{2}}{N}} \tag{24}\]  
where \(N\) is the total number of sampling data and \(V_{z}^{i}\) is an instantaneous value at a certain position along the vertical line.  
Considering that the frequency of the particle image velocimetry (PIV) used in reference [5] was 15 Hz (0.066 s) and approximately 17 s for the measurement, just for comparison, the sampling interval was set as 0.066 s (66 time-steps) and a period of approximately 10 s was taken for statistics after calculation convergence in the current work.  
The simulation results of four grids are shown in Figure 4. It shows that the results reproduce the tendency of the distribution pretty well. At \(-0.53\)\(D_{m}\) upstream of the tee, there is no big difference between the mean axial velocity among the grids. Although the results of the four cases vary to some degree, the calculated mean axial velocity is in good agreement with the measured data. However, the results of the axial velocity fluctuation intensity in Case-C and Case-D show better than that of Case-A and Case-B. Case-A underestimated the experimental data, while Case-B performed worse predictions near the wall than others. Considering the computational cost, Case-C seems the best choice among the four grids since it not only has high enough accuracy but also saves computational resources. Therefore, the grid structure of Case-C was selected in the present research.