## 2 Governing Equations and Subgrid-Scale Stress Modeling of LES
### The Selection of the DSLM and WALE Models
The non-dimensional time step (\(\triangle tU_{\text{b}}/D\)) is set to 1.024 \(\times\) 10\({}^{-3}\) for keeping the Courant number less than 1. After the flow reaches a statistically steady state, the statistics are calculated for 100 non-dimensional time units. The volume-averaged resolved kinetic energy is monitored and reaches a stationary state in approximately 60-70 non-dimensional time units, and the flow is computed for 100 non-dimensional time units before starting the data sampling for statistics. Figure 2 shows wall-normal distribution of the mean radial velocity and turbulent energy at \(r/D\)=2. As depicted in Figure 2, the velocity is similar for the two turbulence models, while the turbulent kinetic energy at \(y/D\leq 0.16\) is different. The change trend of turbulent kinetic energy near the wall for the WALEM is almost closed to that of the velocity in the wall boundary layer, which indicates that the WALEM can capture the fluctuating velocity near the wall given by the turbulent kinetic energy. According to the criteria presented by [PERSON] et al. [27], when the index s\(\equiv\frac{1}{1+\
icefrac{{\
u}}{{\
u}}}\) is equal to 1, it is not sensitive to grid resolution since it is anticipated that \(\langle\
u_{\text{t}}\rangle\gg\
u\) in most LES applications. That is, the LES is better resolved as the value of \(\langle\
u_{\text{t}}\rangle/\
u\) is smaller. [PERSON] et al. [27] suggested that LES is considered to be good if the index of quality (LESIQ\({}_{\text{V}}=\frac{1}{1+0.05\left(1+\langle\
u_{\text{t}}\rangle/\
u\right)^{0 35}}\)) is greater than 0.8. Figure 3 shows the wall-normal distribution for the ratio of the sub-grid viscosity to the molecular viscosity at \(r/D=0\) and \(r/D=1\). It is found that the index of LESIQ\({}_{\text{V}}\) varies between 0.933 and 0.952 for the two turbulence models and the WALEM have littler \(\langle\
u_{\text{t}}\rangle/\
u\) near the wall compared to the DSLM. Figure 2: (**a**) Mean radial velocity and (**b**) turbulent kinetic energy at \(r/D=2\). The \(Nu\) distribution is predicted by LES with the WALEM and DSLM and compared with the experimental data of [PERSON] and [PERSON] [25], as shown in Figure 4. The global trend and variation are well captured for both the WALEM and DSLM. It is clearly seen in Figure 4 that an increase in \(Nu\) in region of \(r/D\) = 0-0.75 and a fast decay in the region after \(r/D\) = 0.75 are captured by the present LES simulations. The numerical results using WALEM agree better with the experimental results than those using the DSLM, especially the predicted peak at \(r/D\) = 0.75. However, the LES results using WALEM under-predict the experimental values at \(r/D\) < 0.25, about 18%, while the computed results using WALEM under-predict the experimental values at \(r/D\) > 0.5, about 7%. The big discrepancy exists between the predicted and experimental values in this region of \(r/D\) < 0.25, which may be explained by the fictitious zero value of subgrid-scale eddy viscosity in the near-wall boundary layer at this region, shown as Figure 3. Receiving an accurate simulation for large lumps of stagnant airflow near the stagnation point is still a big challenge in the computation of impinging flows at high Reynolds number values using the LES [23]. Furthermore, the experimental results reported by ref. [25] have measurement errors of less than 3%. Therefore, WALEM is more suitable for LES of swirling impinging jet flow in jet space and is selected in the following simulations.