## 2 Calculation Model and Numerical Method
### Combination of IDDES Method and Transition Model  
The \(\gamma\)-\(\text{Re}_{\theta}\) model solves the intermission factor \(\gamma\) by adding two transport equations and combines the SST turbulence model by causing \(\gamma\) to act on the generation and destruction terms of the k equation. In terms of form, adding the IDDES method based on the SST model is simpler, and this can be performed by modifying the length scale function of the damage term of the turbulent kinetic energy transport equation in the SST turbulence model. The turbulent kinetic energy equation in the SST model is  
\[\frac{D(\rho k)}{Dt}=\tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}-\beta^{*} \rho\omega k+\frac{\partial}{\partial x_{j}}\left[(\mu+\sigma_{k}\mu_{t})\frac{ \partial k}{\partial x_{j}}\right] \tag{13}\]  
where \(D=\beta^{*}\rho\omega k\) is the damaged item, which should actually be \(D=\rho k^{3/2}/\widetilde{l}\). Because \(\widetilde{l}=k^{1/2}/(\beta^{*}\omega)\) in the RANS model, the equation is finally displayed as \(D=\beta^{*}\rho\omega k\), where \(\beta^{*}=0.09\). For the IDDES method, \(\widetilde{l}=l_{IDDES}\). To combine the IDDES method with \(\gamma\)-Re\({}_{\Theta}\), if the transition model is combined, it needs to be solved by using \(\gamma\)-Re\({}_{\Theta}\), and the four-equation transition model is obtained by modifying the length scale function of the damage term of the turbulent kinetic energy equation. The turbulent kinetic energy transport equation in the four-equation model is  
\[\frac{D(\rho k)}{Dt}=\gamma\tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}- \gamma\rho k^{3/2}/\widetilde{l}+\frac{\partial}{\partial x_{j}}\left[(\mu+ \sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{j}}\right] \tag{14}\]  
For the traditional \(\gamma\)-Re\({}_{\Theta}\) transition model, \(\widetilde{l}=k^{1/2}/(\beta^{*}\omega)\) in the IDDES framework based on the \(\gamma\)-Re\({}_{\Theta}\) transition model, and \(\widetilde{l}=l_{IDDES}\). So far, whether the pure IDDES method or framework based on \(\gamma\)-Re\({}_{\Theta}\) is used, the transition model only needs \(l_{IDDES}\) to be provided, and then the equation can be used.  
For details on developing the length-scaling function from DES to IDDES, please refer to the literature. We only provide the calculation of \(l_{IDDES}\) related to the grid scale:  
\[\Delta=\min\{\max[C_{w}d_{w},C_{w}h_{\max},h_{\min}],h_{\max}\} \tag{15}\]  
Here, \(h_{\max}=\max(\Delta_{x},\Delta_{y},\Delta_{z})\) and \(h_{\min}=\min(\Delta_{x},\Delta_{y},\Delta_{z})\). \(d_{w}\) is the distance from the wall and \(C_{w}\) is an empirical constant that is independent of the sublattice model and is generally taken as \(C_{w}=0.15\).  
\(l_{IDDES}\) is in the form of  
\[l_{IDDES}=\widetilde{f}_{d}l_{RANS}+(1-\widetilde{f}_{d})l_{LES} \tag{16}\]  
Here, \(l_{RANS}=k^{1/2}/(\beta_{k}\omega)\), \(l_{LES}=C_{DES}\Delta\), \(C_{DES}=(1-F_{1})C_{DES}^{k-\varepsilon}+F_{1}C_{DES}^{k-\omega}\), and \(C_{DES}^{k-\varepsilon}=0.61\), \(C_{DES}^{k-\omega}=0.78\).  
\[\widetilde{f}_{d}=\max\{(1-f_{dt}),f_{B}\} \tag{17}\]  
\[f_{dt}=1-\tanh[(C_{dt}r_{dt})^{3}] \tag{18}\]  
\[r_{dt}=\frac{\upsilon_{t}}{\max\{\sqrt{l_{ij}l_{ij}},10^{-10}\}\kappa^{2}{d_{w }}^{2}} \tag{19}\]  
where \(\upsilon_{t}\) is the turbulent viscosity coefficient and \(U_{ij}=\frac{\partial U_{i}}{\partial x_{j}}\). \(\kappa=0.41\) is the Karman constant. \(C_{dt}\) is the empirical constant, which is taken as 20 in IDDES based on the SST model. \(f_{B}\) is defined as follows:  
\[f_{B}=\min\{2\exp(-9\alpha^{2}),1.0\} \tag{20}\]  
Here, \(\alpha=0.25-d_{w}/h_{\max}\), and \(l_{IDDES}\) was determined.