## 2 Governing Equations and Subgrid-Scale Stress Modeling of LES
### The Selection of the DSLM and WALE Models  
DSLM and WALEM are used to simulate the test configuration with the swirl number \(S\) = 0.2 reported by [PERSON] and [PERSON] (2018). In ref. (2018), the swirl flow is generated by means of a helical insert and the swirl number is computed according to the formula  
\[S=\frac{2}{3}\frac{1-(D_{1}/D)^{3}}{1-(D_{1}/D)^{2}}|tan\theta \tag{14}\]  
where \(D\) and \(D_{1}\) are nozzle and vane pack hub diameters, respectively, and \(\theta\) is the swirling angle. The swirling jet has both tangential and axial velocity components compared with the conventional non-swirling jet. The combination of tangential and axial velocities enhances jet turbulent intensity, and hence it is deemed to have different swirling intensities and correspondingly different jet flow field characteristics. Thus far, lots of new nozzles with (intrusive) geometric inserts or directly guiding tangential flow have been proposed to excite the swirling jet ([PERSON] et al., 2018), and then there have been no unified understanding of the flow mechanism and flow characteristics of swirling impinging jets (SJI), and further studies are needed. Figure 1 shows the schematic of the computational domain. Here, the radial radius of the flat target surface is ten times the internal diameter of the cylindrical nozzle (\(D\) = 19.7 mm) with a helical insert as swirl generator. The jet spacing (\(H\)) from the nozzle exit to the flat target is \(2D\) and the height of jet space is \(2.5D\). The total height of the nozzle is \(6.5D\) with a smooth section of \(2D\) and an internal helical section of \(4.5D\). The swirl flow is generated by a helical insert and the swirl number is \(0.2\). It is noted that the internal helical section is extended to jet space and the length is \(0.5D\). The multi-block structured grid is drawn by applying the meshing software of ICEM CFD 17.0 as shown in Figure 1. In order to accurately capture the flow fields near the wall, the interface region of the jet space is chosen as a focus domain using a structured and refined grid above the target surface. The size of the first layer near the wall is \(0.075\) mm with the grid growth rate of \(1.07\), and the value of the y\({}^{+}\) function is about \(0.5\). The results of the grid-independent tests show that the suitable grid density is about \(20.55\) million cells. The length \(\times\) width \(\times\) height of the grid node numbers for the three-dimensional jet space is \(142\)\(\times\)\(145\)\(\times\)\(142\). At the nozzle inflow, a mean velocity is specified as the constant value of inlet mean velocity based on a given \(Re\) = 28,000 (\(U_{b}=Rev/D\) = 22.436 m/s, \(D\) = 19.7 mm; \(v\) is the kinematic viscosity of air at the jet exit). No fluctuations are added to this inflow; therefore, instantaneous velocities at this inflow are set equal to its mean velocities. For the solution of the energy equation, inflow temperature is set to \(25\)\({}^{\circ}\)C at the nozzle inlet to ensure a  
Figure 1: Computational domains and grid of a swirling impinging jet on a flat target: (**a**) computational domains, (**b**) computational grid. real ambient temperature condition. The top surface of the jet space is considered to be the inlet boundary of ambient airflow with a pressure of 1 atm and a temperature of 25 \({}^{\circ}\)C. The flat target is set as a non-slip surface with constant heat flux of 1000 W/m\({}^{2}\), and the nozzle walls are modeled as adiabatic walls with a non-slip boundary. Four edges of the jet space are modeled as the outlet boundary with the atmospheric pressure of 1 atm. Herein, the commercial software of ANSYS FLUENT 16.0 is used for calculation. The iterative algorithm of the fractional step method is used in the calculation process using the NITA scheme (i.e., the Non-Iterative Time Advancement option), and all the governing equations are time marched using a second-order with the accurate time-stepping scheme.